text:
\begin{document}
\begin{center}{\large{\bf Physical Principles Based on Geometric Properties}}
\end{center}
\vspace*{1.5cm}
\begin{center}
A. C. V. V. de Siqueira
$^{*}$ \\
Departamento de Educa\c{c}\~ao\\
Universidade Federal Rural de Pernambuco \\
52.171-900, Recife, PE, Brazil.\\
\end{center}
\vspace*{1.5cm}
\begin{center}{\bf Abstract}

In this paper we present some results obtained in a previous paper
about the Cartan's approach to Riemannian normal coordinates and our
conformal transformations among  pseudo-Riemannian manifolds. We
also review the classical and the quantum angular momenta of a
particle obtained as a consequence of geometry, without postulates.
We present four classical principles, identified as new results
obtained from geometry. One of them has properties similar to the
Heisemberg's uncertainty principle and another has some properties
similar to the Bohr's principle. Our geometric result can be
considered as a possible starting point toward a quantum theory
without forces.
  \end{center}

 \vspace{3cm}

${}^*$ E-mail: acvvs@ded.ufrpe.br
\newline

\newpage

\section{Introduction}
$         $

In a previous paper dedicated only to mathematical results
\cite{1},we show how n-dimensional pseudo-Riemannian spaces are
related to each other by a conformal transformation. As a geometric
consequence, we obtained the classical angular momentum and the
quantum angular momentum operator of a particle, regardless of a
physical theory. In this paper, suggested by geometric results, we
build some new principles and consider the possibility of a new
starting point toward a quantum theory without forces.
 \newline
This paper is organized as follows. In Sec.$2$ we present normal
coordinates and elements of differential geometry. In Sec.$3$ we
show that all well-behaved n-dimensional pseudo-Riemannian metrics
in origin and in its neighborhood, in normal coordinates, are
conformal to an n-dimensional flat manifold and to an n-dimensional
manifold of constant curvature. In Sec.$4$, we make an embedding of
all n-dimensional pseudo-Riemannian manifold of constant curvature
in an n+1-dimensional flat manifold, obtaining, without postulates,
the quantum angular momentum operator of a particle as  a
consequence of geometry. In Sec.$5$, based on geometric properties,
we present some physical principles. Section $6$ is a continuation
of section $5$ with more concentration in quantum mechanics.
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\newpage
\section{\bf  Normal Coordinates}
$         $
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 $         $
 \setcounter{equation}{0}
 $         $
In this section we briefly present normal coordinates and  review
some elements of differential geometry for an n-dimensional
pseudo-Riemannian  manifold, \cite{2}, \cite{3}, \cite{4}.
\newline
Let us consider the line element
\begin{equation}
 ds^2= G_{\Lambda\Pi}du^{\Lambda}du^{\Pi},
\end{equation}
with
\begin{equation}
G_{\Lambda\Pi}=E_{\Lambda}^{(\mathbf{A})}E_{\Pi}^{(\mathbf{B})}\eta_{(\mathbf{A})(\mathbf{B})},
\end{equation}
where $ \eta_{(\mathbf{A})(\mathbf{B})}$ and $
E_{\Lambda}^{(\mathbf{A})}$ are flat metric and vielbein
components respectively.
\newline
We choose each
 $ \eta_{(\mathbf{A})(\mathbf{B})}$ as a plus or minus  Kronecker's  delta function.

Let us give the 1-form  $\omega^{(\mathbf{A})} $ by
\begin{equation}
\omega^{(\mathbf{A})}= du^{\Lambda} E_{\Lambda}^{(\mathbf{A})}.
\end{equation}
We now define Riemannian normal coordinates by
\begin{equation}
  u^{\Lambda}=v^{\Lambda}t,
\end{equation}
Substituting in (2.3)
\begin{equation}
  \omega^{(\mathbf{A})}= tdv^{\Lambda}
  E_{\Lambda}^{(\mathbf{A})}+dtv^{\Lambda}E_{\Lambda}^{(\mathbf{A})}.
\end{equation}
Let us define
\begin{equation}
 z^{(\mathbf{A})}=v^{\Lambda}E_{\Lambda}^{(\mathbf{A})},
\end{equation}
so that
\begin{equation}
\omega^{(\mathbf{A})}=dtz^{(\mathbf{A})}+tdz^{(\mathbf{A})}
+tE^{\Pi(\mathbf{A})}\frac{\partial{E_{\Pi(\mathbf{B})}}}{\partial{z^{(\mathbf{C})}}}z^{(\mathbf{B})}dz^{(\mathbf{C})}.\\
\end{equation}
We now make
\begin{equation}
 A^{(\mathbf{A})_{(\mathbf{B})(\mathbf{C})}}=tE^{\Pi(\mathbf{A})}\frac{\partial{E_{\Pi(\mathbf{B})}}}{\partial{z^{(\mathbf{C})}}},
\end{equation}
then
\begin{equation}
\varpi^{(\mathbf{A})}=
tdz^{(\mathbf{A})}+A^{({A})_{(\mathbf{B})(\mathbf{C})}}z^{(\mathbf{B})}dz^{(\mathbf{C})},
\end{equation}
with
\begin{equation}
\omega^{(\mathbf{A})}=dtz^{(\mathbf{A})}+\varpi^{(\mathbf{A})}.
\end{equation}
We have at $t=0$
\begin{equation}
 A^{({A})_{(\mathbf{B})(\mathbf{C})}}(t=0,z^{(\mathbf{D})})=0,
\end{equation}
\begin{equation}
\varpi^{(\mathbf{A})}(t=0,z^{(\mathbf{D})})=0 ,
\end{equation}
and
\begin{equation}
\omega^{(\mathbf{A})}(t=0,z^{(\mathbf{D})})=dtz^{(\mathbf{A})} .
\end{equation}
Consider, at an n+1-manifold, a coordinate system given by
$(t,z^{(\mathbf{A})})$. For each value of t we have a hyper-surface,
where $dt=0$ on each of them. We are interested in the hyper-surface
with $t=1$, where we verify the following equality
\begin{equation}
\omega^{(\mathbf{A})}(t=1,z)=\varpi^{(\mathbf{A})}(t=1,z).
\end{equation}
\newline
It is well known the following expression in a  vielbein basis
\begin{equation}
d\omega^{(\mathbf{A})}=-\omega^{(\mathbf{A})}_{(\mathbf{B})}\wedge\omega^{(\mathbf{B})}.
\end{equation}
Considering now the map  $\Phi$, between two manifolds M and N,
\newline
and two subsets, U of M and V of N, we have
\begin{equation}
\Phi:U\longrightarrow V.
\end{equation}
Defining now pull-back as follows, \cite{3}, \cite{4},
\begin{equation}
\Phi^\ast:F^p(V)\longrightarrow F^p(U),
\end{equation}
so that $ \Phi^\ast $ sends p-forms into p-forms.
\newline
It is well known that the exterior derivative commutes with
pull-back, so that
\begin{equation}
\Phi^\ast(d\omega^{(\mathbf{A})}_{(\mathbf{B})})=d\Phi^\ast(\omega^{(\mathbf{A})}_{(\mathbf{B})}),
\end{equation}
and
\begin{equation}
\Phi^\ast(d\omega^{(\mathbf{A})})=d\Phi^\ast(\omega^{(\mathbf{A})}).
\end{equation}
We also have
\begin{equation}
\Phi^\ast(\omega^{(\mathbf{A})}_{(\mathbf{B})}\wedge\omega^{(\mathbf{B})})=
\Phi^\ast(\omega^{(\mathbf{A})}_{(\mathbf{B})})\wedge\Phi^\ast(\omega^{(\mathbf{B})}).
\end{equation}
The equation (2.10) can be seen as pull-back,
\begin{equation}
\Phi^\ast(\omega^{(\mathbf{A})})=dtz^{(\mathbf{A})}+\varpi^{(\mathbf{A})}.
\end{equation}
It can be shown, by a simple calculation that
\begin{equation}
\Phi^\ast(\omega^{(\mathbf{A})}_{(\mathbf{B})})=
\varpi^{(\mathbf{A})}_{(\mathbf{B})}.
\end{equation}
\newpage
By the exterior derivative of (2.22), we obtain
\begin{eqnarray}
\nonumber
d(\Phi^\ast(\omega^{(\mathbf{A})}))=d(dtz^{(\mathbf{A})}+\varpi^{(\mathbf{A})})=dz^{(\mathbf{A})}\wedge(dt)\\
\nonumber+dt\wedge\frac{\partial(\varpi^{(\mathbf{A})})}{\partial(t)}\\
\end{eqnarray}
+ terms not involving  $dt$.
\vspace{1cm}
\newline
Making a pull-back of (2.15) and using (2.20) we have
\begin{equation}
\Phi^\ast(d\omega^{(\mathbf{A})})=\Phi^\ast(-\omega^{(\mathbf{A})}_{(\mathbf{B})}\wedge\omega^{(\mathbf{B})})=
-\Phi^\ast(\omega^{(\mathbf{A})}_{(\mathbf{B})})\wedge\Phi^\ast(\omega^{(\mathbf{B})}).
\end{equation}
Using (2.19), (2.22), (2.23) and (2.24) we have
\begin{equation}
\frac{\partial(\varpi^{(\mathbf{A})})}{\partial(t)}=dz^{(\mathbf{A})}+
\varpi^{(\mathbf{A})}_{(\mathbf{B})}z^{(\mathbf{D})}.
\end{equation}
We can, by a similar procedure to (2.19), and using the Cartan's
second structure equation, obtain the following result
\begin{equation}
\frac{\partial(\varpi_{(\mathbf{A})(\mathbf{B})})}{\partial(t)}=
R_{(\mathbf{A})(\mathbf{B})(\mathbf{C})(\mathbf{D})}z^{(\mathbf{C})}\varpi^{(\mathbf{A})}.
\end{equation}
Making a new partial derivative of (2.25), two partial derivatives
of (2.9), comparing the results and using (2.26) we have the
following equation
\begin{equation}
\frac{\partial^2(A_{(\mathbf{A}){(\mathbf{C})(\mathbf{D})}})}{\partial(t^2)}=
tz^{(\mathbf{B})}R_{(\mathbf{A})(\mathbf{B})(\mathbf{C})(\mathbf{D})}+
z^{(\mathbf{L})}z^{(\mathbf{M})}R_{(\mathbf{A})(\mathbf{L})(\mathbf{M})(\mathbf{N})}
A_{(\mathbf{P}){(\mathbf{C})(\mathbf{D})}}\eta^{(\mathbf{N})(\mathbf{P})}.
\end{equation}
It is easy to show that
\begin{equation}
A_{(\mathbf{A}){(\mathbf{C})(\mathbf{D})}}+A_{(\mathbf{A}){(\mathbf{D})(\mathbf{C})}}=0
\end{equation}
is the solution for all t.
\newline
Then,
\begin{equation}
A_{(\mathbf{A}){(\mathbf{C})(\mathbf{D})}}=-A_{(\mathbf{A}){(\mathbf{D})(\mathbf{C})}},
\end{equation}
so that,  we can rewrite (2.9) as
\begin{equation}
\varpi^{(\mathbf{A})}= tdz^{(\mathbf{A})}+
\frac{1}{2}A^{({A})_{(\mathbf{B})(\mathbf{C})}}(z^{(\mathbf{B})}dz^{(\mathbf{C})}-z^{(\mathbf{C})}dz^{(\mathbf{B})}).
\end{equation}
Let us  define
\begin{equation}
A_{(\mathbf{A}){(\mathbf{C})(\mathbf{D})}}=z^{(\mathbf{B})}B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}.
\end{equation}
The following result is obtained by substituting (2.31) in (2.27),
\begin{equation}
\frac{\partial^2(B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}})}{\partial(t^2)}=
tR_{(\mathbf{A})(\mathbf{B})(\mathbf{C})(\mathbf{D})}+
z^{(\mathbf{L})}z^{(\mathbf{M})}R_{(\mathbf{A})(\mathbf{B})(\mathbf{L})(\mathbf{N})}
B_{(\mathbf{P}){(\mathbf{M})(\mathbf{C})(\mathbf{D})}}\eta^{(\mathbf{N})(\mathbf{P})}.
\end{equation}
By a simple procedure we obtain the following solution
\begin{equation}
B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}+B_{(\mathbf{B}){(\mathbf{A})(\mathbf{C})(\mathbf{D})}}=0.
\end{equation}
Using (2.29), (2.31) and (2.33) we conclude that
$B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}} $ has the
same symmetries as the Riemann curvature tensor
\begin{equation}
B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}=-B_{(\mathbf{B}){(\mathbf{A})(\mathbf{C})(\mathbf{D})}}=
-B_{(\mathbf{A}){(\mathbf{B})(\mathbf{D})(\mathbf{C})}}.
\end{equation}
 Using (2.29) and  (2.31) we have
\begin{eqnarray}
\nonumber A_{(\mathbf{A}){(\mathbf{C})(\mathbf{D})}}dz^{(\mathbf{A})}z^{(\mathbf{C})}dz^{(\mathbf{D})}=\\
\nonumber +\frac{1}{4}B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}.\\
\nonumber .(z^{(\mathbf{B})}dz^{(\mathbf{A})}-z^{(\mathbf{A})}dz^{(\mathbf{B})}).\\
\nonumber .(z^{(\mathbf{C})}dz^{(\mathbf{D})}-z^{(\mathbf{D})}dz^{(\mathbf{C})}).\\
\end{eqnarray}
Now we can write the line element of the hyper-surface. We have
\begin{eqnarray}
\nonumber ds'^2=t^2\eta_{(\mathbf{A})(\mathbf{B})}dz^{(\mathbf{A})}dz^{(\mathbf{B})}+\\
 \nonumber+\frac{1}{2}\{\frac{1}{2}t\epsilon_{(\mathbf{B})}B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}+\\
 \nonumber+\eta^{(\mathbf{M})(\mathbf{N})}A_{(\mathbf{M}){(\mathbf{B})(\mathbf{A})}}A_{(\mathbf{N}){(\mathbf{C})(\mathbf{D})}}\}.\\
\nonumber.(z^{(\mathbf{B})}dz^{(\mathbf{A})}-z^{(\mathbf{A})}dz^{(\mathbf{B})})(z^{(\mathbf{C})}dz^{(\mathbf{D})}-z^{(\mathbf{D})}dz^{(\mathbf{C})}).\\
\end{eqnarray}
\newpage
The line elements of the manifold and the hyper-surface are equal
at $ t=1 $, where  $u^{\Lambda}=v^{\Lambda} $,
\begin{equation}
ds^2=ds'^2,
\end{equation}
and
\begin{eqnarray}
\nonumber ds^2=\eta_{(\mathbf{A})(\mathbf{B})}dz^{(\mathbf{A})}dz^{(\mathbf{B})}+\\
 \nonumber+\frac{1}{2}\{\frac{1}{2}\epsilon_{(\mathbf{B})}B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}+\\
 \nonumber+\eta^{(\mathbf{M})(\mathbf{N})}A_{(\mathbf{M}){(\mathbf{B})(\mathbf{A})}}A_{(\mathbf{N}){(\mathbf{C})(\mathbf{D})}}\}.\\
\nonumber.(z^{(\mathbf{B})}dz^{(\mathbf{A})}-z^{(\mathbf{A})}dz^{(\mathbf{B})})(z^{(\mathbf{C})}dz^{(\mathbf{D})}-z^{(\mathbf{D})}dz^{(\mathbf{C})}).\\
\end{eqnarray}
In the next section we build, by a simple procedure, the conformal
form of n-dimensional pseudo-Riemannian manifolds.
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\section{\bf Conformal Form of Riemannian Metrics }
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$         $
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 $         $
We now write (2.38) as
\begin{eqnarray}
 \nonumber ds^2=\eta_{(\mathbf{A})(\mathbf{B})}dz^{(\mathbf{A})}dz^{(\mathbf{B})}+\\
 \nonumber +\{\frac{1}{2}[\frac{1}{2}\epsilon_{(\mathbf{B})}B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}+\\
 \nonumber +\eta^{(\mathbf{M})(\mathbf{N})}A_{(\mathbf{M}){(\mathbf{B})(\mathbf{A})}}A_{(\mathbf{N}){(\mathbf{C})(\mathbf{D})}}]\}.\\
\nonumber .(z^{(\mathbf{B})}\frac{dz^{(\mathbf{A})}}{ds}-z^{(\mathbf{A})}\frac{dz^{(\mathbf{B})}}{ds})(z^{(\mathbf{C})}\frac{dz^{(\mathbf{D})}}{ds}-z^{(\mathbf{D})}\frac{dz^{(\mathbf{C})}}{ds}))ds^2.\\
\end{eqnarray}
It can also  be written in the form
\begin{eqnarray}
 \nonumber[1-\frac{1}{2}[\frac{1}{2}\epsilon_{(\mathbf{B})}B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}+\\
 \nonumber+\eta^{(\mathbf{M})(\mathbf{N})}A_{(\mathbf{M}){(\mathbf{B})(\mathbf{A})}}A_{(\mathbf{N}){(\mathbf{C})(\mathbf{D})}}].\\
\nonumber.(z^{(\mathbf{B})}\frac{dz^{(\mathbf{A})}}{ds}-z^{(\mathbf{A})}\frac{dz^{(\mathbf{B})}}{ds})(z^{(\mathbf{C})}\frac{dz^{(\mathbf{D})}}{ds}-z^{(\mathbf{D})}\frac{dz^{(\mathbf{C})}}{ds})]ds^2\\
 \nonumber =\eta_{(\mathbf{A})(\mathbf{B})}dz^{(\mathbf{A})}dz^{(\mathbf{B})}.\\
\end{eqnarray}
We now define the function
\begin{eqnarray}
L^{\mathbf{A})(\mathbf{B})}=(z^{(\mathbf{B})}\frac{dz^{(\mathbf{A})}}{ds}-z^{(\mathbf{A})}\frac{dz^{(\mathbf{B})}}{ds}),
\end{eqnarray}
which is the classical angular momentum  of a free particle.
\newline
The line element (3.2) can assume the following form
\begin{eqnarray}
 \nonumber\{1+\frac{1}{2}[\frac{1}{2}(\epsilon_{(\mathbf{B})}B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}+\\
 \nonumber+\eta^{(\mathbf{M})(\mathbf{N})}A_{(\mathbf{M}){(\mathbf{B})(\mathbf{A})}}A_{(\mathbf{N}){(\mathbf{C})(\mathbf{D})}}].\\
\nonumber.(L^{\mathbf{A})(\mathbf{B})}L^{\mathbf{C})(\mathbf{D})})\}ds^2\\
 \nonumber =(\eta_{(\mathbf{A})(\mathbf{B})}dz^{(\mathbf{A})}dz^{(\mathbf{B})}.\\
\end{eqnarray}
We now define the function
\begin{eqnarray}
\nonumber \exp(-2\sigma)=\{1+\frac{1}{2}[\frac{1}{2}(\epsilon_{(\mathbf{B})}B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}\\
 \nonumber +\eta^{(\mathbf{M})(\mathbf{N})}A_{(\mathbf{M}){(\mathbf{B})(\mathbf{A})}}A_{(\mathbf{N})){(\mathbf{C})(\mathbf{D})}})].\\
\nonumber .L^{(\mathbf{A})(\mathbf{B})}L^{(\mathbf{C})(\mathbf{D})}\},\\
\end{eqnarray}
so that, the line element assumes the form
\begin{equation}
ds^2=\exp(2\sigma)\eta_{(\mathbf{A})(\mathbf{B})}dz^{(\mathbf{A})}dz^{(\mathbf{B})}.
\end{equation}
The metric (3.6) is conformal to a flat manifold, and we conclude
that all n-dimensional pseudo-Riemannian metrics are conformal to
flat manifolds, when, in normal coordinates, the transformations are
well-behaved in the origin and in its neighborhood. It is important
to pay attention to the fact that a normal transformation and its
inverse are well-behaved in the region where geodesics  are not
mixed. Points where geodesics close or mix are known as conjugate
points of Jacobi's fields. Jacobi's fields can be used for this
purpose.
\newline
We can place (3.6) in the following form, \cite{1},
\begin{eqnarray}
\nonumber ds^2=\{1+\frac{1}{2}[\frac{1}{2}(\epsilon_{\beta}B_{\alpha{\beta\gamma\delta}})\\
 \nonumber +\eta^{\rho\sigma}A_{\rho{\alpha\beta}}A_{\sigma{\gamma\delta)}})].\\
\nonumber .L^{\alpha\beta}L^{\gamma\delta}\}^{-1}\eta_{\alpha\beta}d\Omega^{\alpha}d\Omega^{\beta}.\\
\end{eqnarray}
We now present the metric of a constant curvature manifold in the
well known form
\begin{eqnarray}
ds'^2=\{1+\frac{K\Omega^{\mathbf{\alpha}}\Omega^{\beta}\eta_{\alpha\beta}}{4}\}^{-2}d\Omega^{\rho}d\Omega^{\sigma}\eta_{\varrho\sigma}.
\end{eqnarray}
Because (3.7) and (3.8) are conformal to a flat manifold, there is a
conformal transformation between them,
\begin{eqnarray}
g'_{\alpha\beta}=(\exp2\psi)g_{\alpha\beta}.
\end{eqnarray}
More specifically,
\begin{eqnarray}
\nonumber\{1+\frac{1}{2}[\frac{1}{2}(\epsilon_{\beta}B_{\alpha{\beta\gamma\delta}})+\\
\nonumber
+\eta^{\rho\sigma}A_{\rho{\alpha\beta}}A_{\sigma{\gamma\delta)}})]L^{\alpha\beta}L^{\gamma\delta}\}=\\
 \nonumber
 =(\exp2\psi)\{1+\frac{K\Omega^{\mathbf{\alpha}}\Omega^{\beta}\eta_{\alpha\beta}}{4}\}^{2}.\\
 \end{eqnarray}
\newline
Note that (3.8) is an Einstein's space with a constant curvature,
where
\begin{eqnarray}
R'_{\alpha\beta}=\frac{R'}{n}g'_{\alpha\beta},
\end{eqnarray}
and $R'$ is the scalar curvature. Spaces, as the Schwarzschild's,
where
\begin{eqnarray}
R_{\alpha\beta}=0,
\end{eqnarray}
are  Einstein's spaces and are not maximally symmetric.
\newline
Einstein's spaces with a constant scalar curvature obey homogeneity
and isotropy conditions. They are maximally symmetric spaces.
\newline
We will be using the following definitions, \cite{5}
\begin{eqnarray}
\triangle_{1}{\psi}=g^{\mu\nu}{\psi}_{,\mu}{\psi}_{,\nu},
\end{eqnarray}
\begin{eqnarray}
{\psi}_{\mu\nu}={\psi}_{;\mu\nu}-{\psi}_{,\mu}{\psi}_{,\nu},
\end{eqnarray}
\begin{eqnarray}
\triangle_{2}{\psi}=g^{\mu\nu}{\psi}_{;\mu\nu}.
\end{eqnarray}
From (3.9), (3.13), (3.14), and (3.15) we obtain
\begin{eqnarray}
\nonumber {\psi}_{\mu\nu}=\frac{1}{(n-2)}(R_{\mu\nu})\\
\nonumber -\frac{1}{(2)(n-1)(n-2)}(g'_{\mu\nu}R'-g_{\mu\nu}R)\\
\nonumber-\frac{1}{2}\triangle_{1}{\psi}g_{\mu\nu}.\\
\end{eqnarray}
If $g'_{\mu\nu}$ is a metric of an Einstein's space, then (3.16) is
simplified to
\begin{eqnarray}
\nonumber {\psi}_{\mu\nu}=-\frac{1}{(n-2)}R_{\mu\nu}+\\
\nonumber +(\frac{1}{(2)(n-1)(n-2)}R+\frac{1}{(2n)(n-1)}R'(\exp2\psi)-\frac{1}{2}\triangle_{1}{\psi})g_{\mu\nu}.\\
\end{eqnarray}
In the following we consider the Einstein's equation
\begin{eqnarray}
\nonumber R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}={8\pi}T_{\mu\nu}.\\
\end{eqnarray}
Spaces as (3.11) with a non-constant scalar curvature do not obey
(3.18).
\newpage
\section{Embedding of Manifolds of  Constant Curvatures in Flat Manifolds}
 $                       $
 \setcounter{equation}{0}
 $         $
In this section we embed  the n-dimensional manifold (3.8) in an
n+1-dimensional flat manifold obtaining, as a geometric result,
without postulate, the quantum angular momentum of a particle. Other
results will be presented in another section.
\newline
We now consider a manifold (3.8) called S, embedded in an
n+1-dimensional flat manifold. The following constraint is obeyed
\cite{6},
\begin{eqnarray}
\eta_{\alpha\beta}x^{\alpha}x^{\beta}=K=\epsilon\frac{1}{R^2},
\end{eqnarray}
where K is the  scalar curvature of the n-dimensional manifold
(3.8), $\alpha, \beta = ( 1, 2,...,n+1 )$ and $ \epsilon=(+1, -1).$
For the special case of an n-sphere we use the following notation
$S^n$ for (3.8).
\newline
It is convenient that we use a local basis
$X_{\beta}=\frac{\partial}{\partial(x^{\beta})}.$
\newline
We consider a constant vector $\textbf{C}$ in the n+1-dimensional
manifold given by
\begin{eqnarray}
\eta_{\alpha\beta}C^{\alpha}X^{\beta}=\eta^{\alpha\beta}C_{\alpha}X_{\beta}=C,
\end{eqnarray}
where each  $C^{\alpha}$ is constant and  $\textbf{N}$ is a unitary
and normal vector to S. We use the symbol $<,>$ for the internal
product in the n+1-dimensional flat manifold  and $<,>'$ for S.
\newline
A constant vector $\textbf{C}$ can be  decomposed into two parts,
one in S and the other outside S as follows
\begin{eqnarray}
C=\bar{C}+<C,N>N.
\end{eqnarray}
From the definition of $\textbf{N}$ and (4.1) we obtain
\begin{eqnarray}
N^{\alpha}=\frac{x^{\alpha}}{R}
\end{eqnarray}
Let us construct  the covariant derivative of $\textbf{C}$. We have
a local basis and  a diagonal and unitary tensor metric, so that the
Christoffel symbols are null. Then the covariant derivative of
$\textbf{C}$ in the $\textbf{Y}$ direction is given by
\begin{eqnarray}
\nabla_{Y}C=0.
\end{eqnarray}
It is easy to show that
\begin{eqnarray}
\nabla_{Y}N=\frac{Y}{R}.
\end{eqnarray}
The Lie derivative of the metric tensor in S is given by \cite{6},
\begin{eqnarray}
L_{\bar{U}}g'=2\lambda_{U}g',
\end{eqnarray}
where $\textbf{U}$ is a constant vector in the flat manifold, and
$\lambda_{U}$ is the  characteristic function. For S the
characteristic function is given by
\begin{eqnarray}
\lambda_{U}=-\frac{1}{R}<U,N>.
\end{eqnarray}
Substituting (4.8) in (4.7) we have
\begin{eqnarray}
L_{\bar{U}}g'=-2\frac{1}{R}<U,N>g'.
\end{eqnarray}
In the region of S where $<U,N>$ is not null,  $ \bar{U} $ is a
conformal Killing vector and in the region where  $<U,N>$  is null,
 $\bar{U} $ is a Killing vector.
\newline
We now consider another constant vector $\textbf{V}$ in the flat
space. The Lie derivative of its projection in S is given by
\begin{eqnarray}
L_{\bar{U}}g'=-2\frac{1}{R}<U,N>g'.
\end{eqnarray}
As we consider a local basis and constant vectors $\textbf{U}$ and
$\textbf{V}$, the commutator is given by
\begin{eqnarray}
[U,V]=0.
\end{eqnarray}
Then,
\begin{eqnarray}
L_{[\bar{U},\bar{V}]}g'=-2\frac{1}{R}<[U,V],N>g'=0.
\end{eqnarray}
Regardless of $ \bar{U}$ and $\bar{V}$ being conformal Killing
vectors or Killing vectors, their commutator is a Killing vector. In
the following we will show that the commutator $[\bar{U},\bar{V}]$
is proportional to the quantum angular momentum of a particle.
\newline
Using (3.3) in the following commutator of elements of the basis, we
obtain
\begin{eqnarray}
\nonumber [\bar{U},\bar{V}]=\\
\nonumber =U^{\alpha}V^{\beta}[X_\alpha-<X_\alpha,N>N,X_\beta-<X_\beta,N>N]=\\
\nonumber =U^{\alpha}V^{\beta}[\bar{X}_\alpha,\bar{X}_\beta].\\
\end{eqnarray}
We now calculate the commutator of elements of the basis, by parts.
\newline
 We have by simple calculation
\begin{eqnarray}
<X_\alpha,N>N=\frac{1}{R}\eta_{\alpha\beta}x^{\beta}.
\end{eqnarray}
Substituting (4.14) in (4.13) we obtain
\begin{eqnarray}
\nonumber [\bar{X}_\alpha,\bar{X}_\beta]=\\
\nonumber =[X_\alpha,X_\beta]-[X_\alpha,\frac{1}{R}\eta_{\beta\sigma}x^{\sigma}N]+[X_\beta,\frac{1}{R}\eta_{\alpha\sigma}x^{\sigma}N]+\\
\nonumber +\frac{1}{R^2}[\eta_{\alpha\sigma}x^{\sigma}N, \eta_{\beta\sigma}x^{\sigma}N].\\
\end{eqnarray}
In a local basis we have
\begin{eqnarray}
[X_\alpha,X_\beta]=0,
\end{eqnarray}
\begin{eqnarray}
[\eta_{\alpha\sigma}x^{\sigma}N, \eta_{\beta\sigma}x^{\sigma}N]=0.
\end{eqnarray}
Substituting in (4.15) we obtain
\begin{eqnarray}
\nonumber [\bar{X}_\alpha,\bar{X}_\beta]=\\
\nonumber =
\frac{1}{R^2}(\eta_{\alpha\sigma}x^{\sigma}\frac{\partial}{\partial(x^{\beta})}-\eta_{\beta\sigma}x^{\sigma}\frac{\partial}{\partial(x^{\alpha})})\\
\nonumber =
\frac{1}{R^2}(x_\alpha\frac{\partial}{\partial(x^{\beta})}-x_\beta\frac{\partial}{\partial(x^{\alpha})})\\
\nonumber=-i\frac{1}{\hbar}\frac{1}{R^2}L_{\alpha\beta}.\\
\end{eqnarray}
Multiplying $ L_{\alpha\beta}$ by a vielbein basis  we obtain
\begin{eqnarray}
\nonumber
L_{(\mathbf{A})(\mathbf{B})}=\\
\nonumber =(i\hbar)(R^{2})R_{(\mathbf{A})(\mathbf{B})(\mathbf{C})(\mathbf{D})}x^{(\mathbf{D})}\eta^{(\mathbf{C})(\mathbf{M})}\frac{\partial}{\partial(x^{\mathbf{M})}}.\\
\end{eqnarray}
where
\begin{eqnarray}
 \hat{p}_{(\mathbf{M})}=(i\hbar)\frac{\partial}{\partial(x^{\mathbf{M})}},
\end{eqnarray}
is the quantum momentum operator of a particle, and
\begin{eqnarray}
\nonumber
R_{(\mathbf{A})(\mathbf{B})(\mathbf{C})(\mathbf{D})}=\\
\nonumber
=\frac{1}{R^2}[\eta_{(\mathbf{A})(\mathbf{D})}\eta_{(\mathbf{B})(\mathbf{C})}-\eta_{(\mathbf{A})(\mathbf{C})}\eta_{(\mathbf{B})(\mathbf{D})}],\\
\end{eqnarray}
is the curvature of S in the vielbein basis  and $
\eta_{(\mathbf{A})(\mathbf{C})} $ is diagonal. We consider as an
important observation that the association between the quantum
angular momentum operator and the constant curvature operator is
allowed in an orthogonal vielbein basis of a Cartesian coordinate,
regardless of having a curved or a flat manifold. We have used the
embedding of an n-dimensional manifold S in an n+1-dimensional flat
manifold, only to obtain the quantum angular momentum operator of a
particle, without postulates. We can rewrite (4.19) as follows
\begin{eqnarray}
\nonumber
L_{(\mathbf{A})(\mathbf{B})}=\\
\nonumber
=(i\hbar)[\eta_{(\mathbf{A})(\mathbf{D})}\eta_{(\mathbf{B})(\mathbf{C})}-\eta_{(\mathbf{A})(\mathbf{C})}\eta_{(\mathbf{B})(\mathbf{D})}].\\
\nonumber
.x^{(\mathbf{D})}\eta^{(\mathbf{C})(\mathbf{M})}\frac{\partial}{\partial(x^{\mathbf{M})}}.\\
\end{eqnarray}
Note that the coordinates in (3.7) are in the n+1-dimensional flat
manifold  and  $L_{\alpha\beta}\subset S$, so that
$L_{\alpha\beta}=0$  for $\alpha$ or $\beta$ is equal to $n+1.$
\newline
Racah has shown that \cite{7} the Casimir operators of  any
semisimple Lie group can be constructed from the quantum angular
momentum (4.22). Each multiplet of semisimple Lie group can be
uniquely characterized by the eigenvalues  of the Casimir operators.
\newline
We have built the quantum angular momentum operator from classical
geometric considerations. We can write the usual expression for the
eigenstates of Casimir operator without reference to quantum
mechanics, as follows
\begin{eqnarray}
\hat{C}\mid...>=C\mid...>.
\end{eqnarray}
This is simple and well known,\cite{7} for SO(3). There is an
interesting construction from the group theoretical point of view to
Dirac theory with or without Dirac's equation, \cite{8}.
\newline
In the following we calculate the Lie derivative of the so(p,n-p)
algebra. For the Lie group SO(p,n-p) we choose the signature
\newline
$(p,n-p)=(-,-,-,...-,+,+,..+)$, with the algebra
\begin{eqnarray}
\nonumber
[L_{(\mathbf{A})(\mathbf{B})},L_{(\mathbf{C})(\mathbf{D})}]=-i(\eta_{(\mathbf{A})(\mathbf{C})}L_{(\mathbf{B})(\mathbf{D})}+\eta_{(\mathbf{A})(\mathbf{D})}L_{(\mathbf{C})(\mathbf{B})}\\
\nonumber +\eta_{(\mathbf{B})(\mathbf{C})}L_{(\mathbf{D})(\mathbf{A})}+\eta_{(\mathbf{B})(\mathbf{D})}L_{(\mathbf{A})(\mathbf{C})}).\\
\end{eqnarray}
Considering the Lie derivative
\begin{eqnarray}
\nonumber \textbf{L}_{[L_{(\mathbf{A})(\mathbf{B})},L_{(\mathbf{C})(\mathbf{D})}]}g'=\\
\nonumber
=-R^{4}<[[X_{(\mathbf{A})},X_{(\mathbf{B})}],[X_{(\mathbf{C})},X_{(\mathbf{D})}]],N>g'=0,\\
\end{eqnarray}
where, for the orthogonal Cartesian coordinates, the vielbein is
given by
\begin{equation}
E_{\Lambda}^{(\mathbf{A})}=\delta_{\Lambda}^{(\mathbf{A})},
\end{equation}
we have
\begin{eqnarray}
\nonumber[X_{(\mathbf{A})},X_{(\mathbf{B})}]=[X_{\alpha},X_{\beta}]=0.\\
\end{eqnarray}
\newline
Note that $g'$ in S is form-invariant \cite{9} in relation to the
Killing's vector $\xi$, and in relation to the  algebra of
SO(p,n-p), as well. We conclude that the algebra of SO(p,n-p)  is a
Killing's object.
\newpage
The same is true for the algebra of the Lie group SO(n), where for
SO(n) we could  choose the following signature $(+,+,+...,+,+)$. The
constraint (4.1) is invariant for many of the classical groups. For
these groups it is possible to build operators, from the combination
of the quantum angular momentum operators, which are Killing's
objects in relation to $´g'$. Therefore, the metric is
form-invariant in relation to this algebra. It is interesting to see
some of these groups in the Cartan's list of irreducible Riemannian
globally symmetric spaces, \cite{4}, and in \cite{10}.
\newline
Note that we start from a normal coordinate transformation. In other
words, in the region where the transformation (2.4) is well-behaved,
we can build (3.6) and by a conformal transformation we have (3.8),
which was essential to obtain the quantum angular momentum operator
from geometry.
\newpage
\begin{center}
\section{Physical Principles Based on Geometric Properties}
 $                       $
\end{center}
 \setcounter{equation}{0}
 $         $
From a different point of view, Dirac \cite{12} embedded the De
Sitter space in a five-dimensional flat manifold. He has considered
functions and fields living in a five-dimensional flat manifold and
has built a procedure to project them in the De Sitter space. The
Dirac procedure implies  the need for the quantum momentum and the
quantum angular momentum postulates. Other authors have used the
Dirac's idea or variants of it, as in \cite{13}. To obtain the
quantum angular momentum from geometric considerations, we have
considered only constant vectors in an n+1-dimensional flat
manifold. There are many procedures to define or introduce
functions, fields and geometric objects in (3.8). In the following,
we reconsider the qualitative analyzes of (4.9) made in section $4$.
In the region where $<U,N>$ does not vanish  $\bar{U}$ is a
conformal Killing vector and in the region where it vanishes,
$\bar{U}$ is a Killing vector. In other words, we have Killing and
conformal Killing vectors living in (3.8). For our objective we need
only Killing objects as the quantum angular momentum.
\newline
If we make the Lie derivative of (3.9) in relation to $\xi$, we
obtain
\begin{eqnarray}
L_{\xi}g'=(2\xi(\psi)g+L_{\xi}g)(\exp2\psi).
\end{eqnarray}
More specifically, (5.1) can be written as
\begin{eqnarray}
\nonumber
 L_{\xi}[\{1+\frac{K\Omega^{\mathbf{\alpha}}\Omega^{\beta}\eta_{\alpha\beta}}{4}\}^{(-2)}\eta]=\\
\nonumber L_{\xi}[(\exp2\psi)\{1+\frac{1}{2}[\frac{1}{2}(\epsilon_{\beta}B_{\alpha{\beta\gamma\delta}})+\\
\nonumber
+\eta^{\rho\sigma}A_{\rho{\alpha\beta}}A_{\sigma{\gamma\delta)}}]L^{\alpha\beta}L^{\gamma\delta}\}^{(-1)}\eta].\\
\end{eqnarray}
We now consider the following condition
\begin{eqnarray}
2\xi(\psi)g+L_{\xi}g=0,
\end{eqnarray}
which implies the following
\begin{eqnarray}
L_{\xi}g'=0,
\end{eqnarray}
which is a definition of a Killing vector. If  $\xi$ obeys (5.3) we
conclude that $\xi$ is a  Killing vector in (3.8). Note that $\xi$
is a conformal Killing vector in (3.7). The equation (5.3) shows how
a Killing vector in (3.8) will be in (3.7). We conclude that the
best way toward our objective will be from a postulate as follows,
associated to conditions of minimal energy : \textbf{In (3.8) nature
always choose Killing objects.}. Based on this postulate we will
build four classical principles, where one of them is identified as
a classical version of the Heisemberg's uncertainty principle and
another as a classical version of the Bohr's non-radiation
postulate. The third principle is not new and is associated to the
electric neutrality of some stable systems. The fourth  can be
interpreted as an equivalence between two descriptions of a
particle's motion. The first one as the motion due to the presence
of forces and the second as a consequence of geometry, as in
Einstein's gravitation. For this we assume only constant vectors in
an n+1-dimensional flat manifold, where (3.8) will be embedded.
\newline
The equations (2.27) and (2.32) tell us that if the curvature is
null, $ A_{(\mathbf{A}){(\mathbf{C})(\mathbf{D})}}$ and
$B_{(\mathbf{A}){(\mathbf{B})(\mathbf{C})(\mathbf{D})}}$ are null.
In this case, the equation (3.4) implies a null angular momentum. We
conclude that any free particle in a curved manifold will be always
in movement, with angular momentum not null regardless of wether or
not we consider a physical theory.
\newline
The equation (3.6) tells us that $ ds^2$ is conformal to a flat
manifold and to (3.8). An observer in (3.8) will see the space as
being homogeneous and isotropic in the small region where the
transformation (2.4) is well-behaved. With this condition, Ricci
principal directions of space will be indeterminate so that in that
region the position of the particle is uncertain. In the conjugate
points of Jacobi's fields, the transformation (2.4) fails because
geodesics cross, mix or touch each other. Therefore, close to a
conjugate point we will not have indetermination in the Ricci
principal directions and the uncertainty in the position of the
particle disappears. If (3.8) is valid in all points of the space,
in each point there will be an indetermination of Ricci principal
directions and consequently a total uncertainty in the position of
the particle. This resembles a property of the Heisenberg's
uncertainty principle and could be seen as a classical version.
\newline
The metric (3.8) will be form-invariant for  a displacement $ \xi $
which is a Killing's vector. In this metric \cite{9}, a scalar
function will be constant or null, there will be Killing's vectors
only, and tensors will be a combination of the metric tensor. In
these conditions, the electromagnetic fields will be trivial and
there will not be radiation. In the neighborhoods of the conjugate
points the transformations in normal coordinates fail and we will
not have an indetermination of Ricci principal directions and the
electromagnetic fields will not be trivial, being a radiative field.
This is similar to the Bohr's postulate for radiation and could be
seen as a classical version.
\newline
In the metric (3.8) there are no forces generated by fields in the
region where the transformation (2.4) is well-behaved. Particles
move free of forces. In the local system, there are ordinary  forces
generated by fields. If we consider a Kalusa-Klein theory, where
gauge fields are present in the metric, particles are free in the
local coordinates and in (3.6). In the local system, there are
fields while in (3.8) there are not.
\newline
We can consider this as a principle, creating  an equivalence
between two descriptions of motion, which are possible by normal
transformations. The first description, in local coordinates, is
related to the conception of force generating fields. The second is
related to the conception of  motion without forces.
\newline
We believe that this principle is going toward the  Einstein's dream
because it points to  the possibility of thinking in  physics
without forces as in Einstein's gravity.
\newline
We notice that the conjugate points of the Jacobi's fields can be a
consequence of  geometric singularities, as it is in the origin of
the Schwarzschild's geometry, \cite{11}, where the curvature
diverges, but it can also be due to the construction of the
coordinates, as it is in the case of a maximally symmetric space,
where the curvature is finite in all points. In the second case we
have an indetermination of Ricci principal directions, and in the
first we do not.
\newline
Considering the momentum-energy tensor of matter and electromagnetic
fields,
\begin{eqnarray}
\nonumber T_{\alpha\beta}=\frac{1}{4\pi}(F_{\alpha\sigma}F_{\beta}{^{\sigma}}-\frac{1}{4}g'_{\alpha\beta}F_{\varrho\sigma}F^{\rho\sigma})+t_{\alpha\beta},\\
\end{eqnarray}
where $ t_{\alpha\beta} $ is associated to  electric charges and
$T(F_{\alpha\sigma})=T_{\alpha\beta\mathbf{em}}$ to the
electromagnetic fields, with
\begin{eqnarray}
\nonumber T_{\alpha\beta\mathbf{em}}=\frac{1}{4\pi}(F_{\alpha\sigma}F_{\beta}{^{\sigma}}-\frac{1}{4}g'_{\alpha\beta}F_{\varrho\sigma}F^{\rho\sigma}).\\
\end{eqnarray}
Then,
\begin{eqnarray}
\nonumber T_{\alpha\beta}=T_{\alpha\beta\mathbf{em}}+t_{\alpha\beta}.\\
\end{eqnarray}
As  $g'$ is form-invariant in (3.8), the electromagnetic vector $
A_{\mu}$ will be null in (3.8),
\begin{eqnarray}
A_{\mu}=0.
\end{eqnarray}
\newline
From Maxwell's equations we have
\begin{eqnarray}
F^{\rho\sigma}{_{;\rho}}=-J^{\sigma}.
\end{eqnarray}
Using(5.9) in (5.10) we obtain
\begin{eqnarray}
 J^{\sigma}=0.
\end{eqnarray}
We conclude that in (3.8) the sum of all charges is zero, as well as
it is the sum of all currents.
\begin{center}
\newpage
\section{Geometric Properties Based on Quantum Principles and Quantum Principles Based on Geometric Properties }
 $                       $
\end{center}
 \setcounter{equation}{0}
 $         $
In this section we analyze some results obtained in section 5 which
resemble some postulates of quantum mechanics. Part of the
development of our results is qualitative because only some
applications, like SO(3) and the Casimir  eigenvalues, for instance,
can be easily calculated. Considerations of more complex systems are
qualitative at the moment. Therefore, from a theoretical point of
view, there is a gap.
\newline
We recall that in the region where there are no  conjugate points of
Jacobi's fields, it is possible to build a transformation (2.4)
between the ordinary metric and (3.6), and a conformal
transformation between (3.6) and (3.8). Because $g'$ is
form-invariant in the region where (3.8) is defined, there are no
fields nor radiation. The quantum angular momentum, which  is a
Killing's object, appears as a geometric consequence of embedding
(3.8) in an n+1-dimensional flat manifold. Particles will be in a
free motion, but confined in (3.8). In this context, where forces do
not exist, the particle confinement is due to the manifold geometry.
This resembles the Heisenberg's principle of quantum mechanics.
\newline
We consider as an important observation, made in section $5$, that
the association between the quantum angular momentum operator and
the constant curvature operator is allowed in a orthogonal vielbein
basis of a Cartesian coordinate, even for a flat manifold. This
suggests that, even in a flat spacetime, we can consider the
intrinsic angular momentum, or spin $\frac{1}{2}$ of a free massive
particle, as a quantum object living in an manifold of constant
curvature, embedded in this flat spacetime.
\newline
From the geometric point of view, some operations with the quantum
angular momentum, as sums and products, suggest the same operations
with curvature. We can define some procedures in differential
geometry by operations with quantum angular momenta. As an example,
consider the algebra (4.24) of the group SO(p,n-p)
\begin{eqnarray}
\nonumber
[L_{(\mathbf{A})(\mathbf{B})},L_{(\mathbf{C})(\mathbf{D})}]=-i(\eta_{(\mathbf{A})(\mathbf{C})}L_{(\mathbf{B})(\mathbf{D})}+\eta_{(\mathbf{A})(\mathbf{D})}L_{(\mathbf{C})(\mathbf{B})}\\
\nonumber +\eta_{(\mathbf{B})(\mathbf{C})}L_{(\mathbf{D})(\mathbf{A})}+\eta_{(\mathbf{B})(\mathbf{D})}L_{(\mathbf{A})(\mathbf{C})}).\\
\end{eqnarray}
We can substitute  (4.19) in (6.1) obtaining a representation of the
algebra in terms of  curvature operators. Substituting (4.19) in
(4.25) we will have the form-invariance of the metric tensor $g'$,
in relation to the algebra  so(p,n-p), in terms of the curvature
operators. Any other possible operation among quantum angular
momenta, here defined, can be placed in terms of curvatures. This
offers some curious procedures in differential geometry by simple
operations with quantum angular momentum, which may not be possible
by geometric methods. We do not know if this is known in the
specialized literature. The association between the quantum angular
momentum and differential geometry can be useful in geometry, as
well as in physics.
\newline
In the following, we make more qualitative considerations. We know
that mass and energy curve the spacetime. From the postulate of
section $5$, the existence of electric charges is allowed in (3.8)
provided that the total sum is zero. For each charge there is an
associated particle . The particles confinement in the metric (3.8)
is not due to forces, it is a consequence of the form-invariance of
$g'$. Obviously, in the region where there is a transition between
well-behaved and not well-behaved transformations (2.4), we also
have a transition from the condition where there are no forces to an
ordinary description by forces. This can be seen as a small
deformation in (3.8). The intensity of the deformation could be
responsible for some polarization or emission.
\newline
For a stable and isolated  system, like SO(3), we can write the
eigenvalues of the Casimir operators by (4.23). Particles will be
confined if the metric is (3.8). As we have seen, the postulate
implies the electric neutrality of (3.8) and  the confinement of the
particles.  We imagine that an incident particle curves and deforms
(3.8), then, there will be a transition from the well-behaved to the
not well-behaved transformation (2.4). This can be seen as a
transition, emission or scattering. Mass and energy of each particle
curve the spacetime, then particles in (3.8) will be responsible for
the confinement. We conclude that each particle contributes to the
confinement, which is proportional to its mass if the resultant
metric is (3.8). In this case a proton curves the spacetime with
more intensity than an electron because its mass is bigger. As a
consequence, the associated curvatures obey
\begin{eqnarray}
K_{\mathbf{p}}>K_{\mathbf{e}}.
\end{eqnarray}
But
\begin{eqnarray}
K=\frac{1}{R^2},
\end{eqnarray}
then
\begin{eqnarray}
R_{\mathbf{p}}<R_{\mathbf{e}}.
\end{eqnarray}
Note that the region of the electron  confinement is bigger than
that of the proton. This is compatible with the Heisemberg's
uncertainty principle and with the experimental evidence of nuclear
and atomic dimensions. Protons live in  nucleus and electrons live
outside. Starting from the above point of view, if the metric
deformation, caused by mass, energy and motion of some interacting
particles, generates a resultant metric given by (3.8), there will
not be any transition, emission, or scattering. If the metric
deformation generates a different metric from (3.8), there will be
transition, emission or scattering. This resembles the Bohr's
principle of quantum mechanics. In this paper we are using the
classical point of view and Riemannian manifolds. Considering an
ionized positive system, we conclude that the metric will not be
(3.8) because the sum of all charges is zero only for (3.8).
Although, the Einstein's equation is well-behaved for an ionized
system, we know that it is not valid for small regions. Obviously,
in the limit of a classical distribution of charges, the Einstein's
equation  is a good theory, with a negative curvature in the
interior of a classical distribution of charges, and null curvature
outside, obeying the weak energy condition, \cite{14}. For coherence
with this classical limit, we consider that a microscopic ionized
positive system has a negative curvature, so that the Jacobi
equation in a Fermi-Walker transported vielbein basis  will have a
form similar to an inverted oscillator, \cite{15}. A positive
charged incident particle will have in this space a trajectory which
will be interpreted as a repulsive electric force. If we have a
negative charged incident particle interacting with an ionized
positive system, the sum of all charges will be zero and by the
postulate above presented, another system with positive curvature
will be in process. The conditions of minimal energy of (3.8) will
be obeyed, resulting in energy loss (fotons) and a new electrically
neutral system (3.8) will be formed. This process will be seen as an
attractive electric force. If the new system is not electrically
neutral, therefore having a net positive charge, internal parts of
the system will be electrically neutral, shielding parts of the
system. The resultant geometry can be very complex. It will be
impossible to understand geometric details only by geometric
methods. However, the appropriate and usual quantum mechanics
operations with angular momenta can be converted in geometric
operations as we have seen in the beginning of this section. This
quantum mechanics  approach to geometry, associated to the
Landau-Raychaudhuri equation \cite{14} can be very useful.
\newline
The Heisemberg's and the Bohr's postulates are part of a theory
involving  force and interaction, known as quantum mechanics. We
have not exchanged the Heisemberg's postulate and the Bohr's
postulate by geometric properties of classical nature. Actually, we
have made classical geometric descriptions with  similar properties,
as much as possible, to the quantum postulates. We offered the
possibility of obtaining a quantum mechanics without forces, from
only one postulate , simple systems, and geometry. From our point of
view this theory will preserve the essence of quantum mechanics,
differing from the usual one in some aspects above presented . This
competitive classical geometric theory, if possible, will avoid the
incompatibilities between quantum mechanics and relativity,
verifying the success of both. This would represent a program of
very unexpected results and would not be attractive enough to
researchers because forces are considered as spacetime deformations.
Although we have presented a geometric alternative to electric
forces, unfortunately this is an unsolvable problem at the moment,
so that force will be present in many situations. The conceptions of
force and non-force may be seen, provisionally, as complementary. We
believe that our results can be considered as a possible starting
point to a quantum theory without forces.
\newpage


text:
\begin{document}
\begin{CJK*}{GB}{}

\title{The Favored Classical Variables to \\Promote to Quantum Operators}
\author{John R. Klauder\footnote{klauder@phys.ufl.edu} \\
Department of Physics and Department of Mathematics \\
University of Florida,
Gainesville, FL 32611-8440}
\date{ }

\bibliographystyle{unsrt}

\maketitle
\end{CJK*}

\begin{abstract}
Classical phase-space variables are normally chosen to promote to quantum operators in order to
quantize a given classical system. While classical variables can exploit coordinate transformations
to address the same problem, only one set of quantum operators to address the same problem can give the correct analysis. Such a choice leads to the need to find the favored classical variables in order to achieve a valid quantization. This article addresses the task of how such favored variables are found that can be used to properly solve a given quantum system. Examples, such as non-renormalizable
scalar fields and gravity, have profited by initially changing which classical variables to promote to quantum operators.
\end{abstract}
{\bf Key words:} canonical, spin, and affine quantization, coherent states

\section{A Brief Story of Three Valid Quantization Procedures}
\subsection{Canonical quantization}
Conventional phase-space variables, such as $p$ and $q$, where $-\infty <p,q<\infty$, with Poisson
brackets $\{q,p\}=1$, are natural candidates to promote to basic quantum operators in the procedures that canonical quantization employs. However, traditional coordinate transformations, such as $p\ra
\o{p}$ and $q\ra\o{q}$, with $-\infty<\o{p},\o{q}<\infty$ and Poisson brackets $\{\o{q},\o{p}\}=1$
are also qualified, in principle, as natural candidates to promote to basic quantum operators. This article resolves the problem of deciding which pair of classical variables to promote to basic quantum operators in order to achieve a valid quantization.

Dirac gave us the clue about which classical variables to choose for canonical quantization \cite{dirac}, although he did not prove his proposal. Dirac proposed that the proper choice of variables were those that were `Cartesian coordinates'.
This is not obvious because phase space does not have a metric to exhibit Cartesian coordinates.
To fulfill Dirac's rule a metric space that admits Cartesian coordinates must be added to these
particular coordinates. This request requires that the metric of that two-dimensional space must
be given by an expression such as $d\si(p,q)^2= \omega^{-1} \,dp^2+\omega\,dq^2$ for some constant $\omega$.
This space admits a simple shift of coordinates, e.g., $p\ra p+a$ and $q\ra q+b$, where $a$ and $b$ are constants. Moreover, and this is a property that we wish to feature, the shift of these two variables has brought one to a different point on the flat surface that is identical in its surroundings as it was before the shift occurred. Clearly the identity in surroundings for a two-dimensional flat surface
is a property that distinguishes it from almost all other surfaces that are not completely flat.
Also the Poisson brackets $\{q+b, p+a\}=\{q, p\}=1$ are automatically fulfilled after the shift in location.
If we denote the promotion to quantum operators, e.g., $p\ra P$ and $q\ra Q$, it also follows that
$p+a\ra P+a\one$ and $q+b\ra Q+b\one$, which leads to $[Q+b\one,P+a\one]=[Q,P]=i\hbar\one$. In brief,
a flat surface and the choice of Cartesian coordinates, with or without $a$ and $b$ shifts, leads to acceptable classical variables to promote to become the basic quantum operators.

The only problem deals with how that flat surface, which is linked to both the classical and quantum realms, comes about. As a potential link
to other forms of quantization (the spin and affine versions), we call our flat,
two-dimensional surface a `constant zero curvature'.

\subsubsection{The role of canonical coherent states}
We choose to name the favored classical variables that are promoted to quantum operators as $p$
and $q$, and the valid quantization of these variables named as $P$ and $Q$. We confirm this choice by
first introducing canonical coherent states given by
   \bn |p,q\>\equiv e^{-iqP/\hbar}\,e^{ipQ/\hbar}\,|\omega\>\;,\en
   where $(Q+iP/\omega)\,|\omega\>=0$. It follows that $\<p,q|P|p,q\>=\<\omega|(P+p\one)|\omega\>=p$ and $\<p,q|Q|p,q\>=\<\omega|(Q+q\one)|\omega\>=q$, which is a clear connection between the quantum and classical {}
   basic variables. We finalize this connection with  a Fubini-Study metric \cite{77} that
   involves a tiny ray distance (minimized over non-dynamical phases) between two infinitely close canonical coherent states given by
   \bn &&d\sigma(p,q)^2\equiv2\hbar[\,|\!|\,d|p,q\>|\!|^2-|\<p,q|\,d|p,q\>|^2]\no \\
   &&\hskip4.1em=\omega^{-1}\, dp^2+\omega\,dq^2\;.\en

Observe that this process has given us Cartesian coordinates. These variables clearly are invariant even if we choose $p+a$ and $q+b$. These favored coordinates are Cartesian coordinates and they are promoted to valid basic quantum operators, as Dirac had predicted.

\subsection{Spin quantization}
The surface of an ideal three-dimensional ball is two-dimensional and spherical with a constant radius; we can say that it has a `constant positive curvature'. Again, like a flat space,
the properties at any point on the spherical surface are exactly like those at any other point on the surface. This is the space on which the spin variables appear. There are three spin operators, $S_1$, $S_2$, and $S_3$, which belong to the groups $SO(3)$ or $SU(2)$. These operators obey certain rules, such as
$[S_1,S_2]=i\hbar\,S_3$, and natural permutations, as well as $S_1^2+S_2^2+S_3^2=\hbar^2 s(s+1)\one_{2s+1}$,
where $2s+1$ is the dimension of the vectors, and the spin $s$ values are $(1,2,3,\ldots)/2$. Some
basic vectors are $|s,m\>$, where $S_3|s,m\>=m|s,m\>$ and $-s\leq m\leq s$,   $(S_1+iS_2)|s,m\>=|s,m+1\>$, as well as $(S_1+iS_2)|s,s\>=0$.

\subsubsection{The role of spin coherent states}
The spin coherent states are given by
   \bn |\th,\vp\>\equiv e^{-i\vp S_3/\hbar}\,e^{-i\th S_2/\hbar}\,|s,s\>\;, \en
where $-\pi<\vp\leq\pi$, and $0\leq\th\leq\pi$. We also introduce $q=(s\hbar)^{1/2}\,\vp$
and $p=(s\hbar)^{1/2}\,\cos(\th)$. It follows that
  \bn &&d\si(\th,\vp)^2\equiv 2\hbar\,[\,|
  \!|\,d|\th,\vp\>|\!|^2-|\<\th,\vp|\,d|\th,\vp\>|^2\,] \no \\
  &&\hskip4.3em =(s\hbar)[d\th^2+\sin(\th)^2\,d\vp^2\,
  ]\;,\label{ff} \en
  or we can say that
  \bn d\si(p,q)^2\equiv 2\hbar\,[\,|\!|\,d|p,q\>|\!|^2-|\<p,q|\,d|p,q\>|^2\,] \no \\
  &&\hskip-16em =(1-p^2/s\hbar)^{-1}
  dp^2+(1-p^2/s\hbar)\,dq^2\;. \label{ss} \en
  Equation (\ref{ff})makes it clear that we are dealing with a spherical surface with a radius of $(s\hbar)^{1/2}$. Equation (\ref{ss}) makes it clear that if $s\ra\infty$, in which case both  $p$ and $q$ span the real line, we will have recovered properties of canonical quantization.

  So far we have obtained surfaces with a constant zero curvature and a constant positive curvature. Could there be more? Could there be surfaces with a `constant negative curvature'?

  \subsection{Affine quantization}
  One of the simplest problems to quantize is an harmonic oscillator for which
   $-\infty<p,q<\infty$,
  but it is not so simple if $0<q<\infty$. To solve the second version requires a new method of quantization called affine quantization, which, as we will discover, involves a constant negative curvature. To introduce this procedure let us focus on the classical term $p\, dq$ which is part of a classical action functional. Instead of these variable's range being $-\infty<p,q<\infty$, let us assume
  $q$ is limited to $0<q<\infty$, and we want to change variables. As a first step, let us consider
  $ p\, dq=pq\, dq/q=pq\,d\ln(q)$. While $q$ must be positive, $\ln(q)$ covers the whole real line.
  Although that $p\ra pq$ and $q\ra\ln(q)$ both cover the whole real line, we instead just choose $pq$ and $q$ as our new variables. A potential quantization of this pair of variables could involve $q\ra Q$, with $0<Q<\infty$, and $pq\ra(PQ+QP)/2\equiv D$. Note, if $0<Q<\infty$, then $P$ cannot be self adjoint; however, thanks to $Q$, $D$ can be self adjoint, which is a very important advantage. The
  two basic operators for affine quantization then are $D$ and $Q$, for
  which $[Q,D]=i\hbar\,Q$.\footnote{Besides $0<q,Q<\infty$, one may also consider $-\infty<q,Q<0$, as
  well as $-\infty<q\neq0,Q\neq0<\infty$.}

 \subsubsection{The role of affine coherent states}
 The affine coherent states, where both $q$ and $Q$ have been chosen dimensionless for simplicity, are given by
      \bn |p;q\>\equiv e^{ipQ/\hbar}\,e^{-i\ln(q)\,D/\hbar}\,|b\>\;,  \en
   where $[(Q-1)+iD/b\hbar]\,|b\>=0$. For these variables we find that $\<p;q|Q|p;q\>=
   \<b| qQ|b\>=q$ and $\<p;q|D|p;q\>=\<b| D+pqQ|b\>=pq$. It follows that
      \bn &&d\si(p,q)^2\equiv 2\hbar\,[\,|\!|\,d|p;q\>|\!|^2-|\<p;q|\,d|p;q\>|^2\,] \no   \\
      &&\hskip4.1em =(b\hbar)^{-1}\,q^2\,dp^2 +(b\hbar)\,q^{-2}\,dq^2 \;. \en
      This expression for the Fubini-Study metric is that of a constant negative curvature, an amount of $-2/(b\hbar)$, which is also geodetically complete \cite{p}. Surfaces that are constant negative curvature are only visible in our three-dimensional space at the `center point' only, namely at
      $q=1$, where it appears that in one direction the surface goes down and 90 degrees away the surface goes up. In reality that is what happens at all points for a constant negative curvature,
      but we can not see that effect. Let us give a $q\ra q+b'$ test, with $b'>0$; we use $b'$ so as
      to make sure this is different from the $b$ that labels the fiducial vector. In that case, we now have
      $(b\hbar)^{-1}(q+b')^2 dp^2+(b\hbar)(q+b')^{-2}dq^2$. This version now implies that  $-b'<q<\infty$.
      Indeed, we can let $b'\ra\infty$ and at the same time, let the factor $b$ in the fiducial vector
      become large ($b\propto b'^2$) in which case we can arrange that the negative curvature $-2/b\hbar\ra0$,
      in such a way that the Fubini-Study metric effectively passes to $B^{-1}dp^2+B\,dq^2$, namely, the constant zero curvature case, with a non-dynamical positive $B$. Basically, we have effectively changed the affine operators $Q$ and $D$ to canonical operators, by
      $Q\ra Q+b'\one$ and $D\ra D+b'P$ so that, as $b'\ra\infty$,
       $[Q+b'\one, D+b'P]/b'=i\hbar(Q+b'\one)/b'\ra
      [Q,P]=i\hbar\one$.

      \section{Summary}
      Our quantization of classical variables has now been completed. It was shown that classical variables that represent the coordinates of constant positive, zero, and negative curvatures,
      complete the natural forms of surface and these three divisions include a different variety of
      classical systems that can be quantized. Affine quantization, as a special procedure to quantize systems, has not yet become universally well known and exploited; it deserves more attention.
      Besides the harmonic oscillator with $0<q<\infty$ (see \cite{t}), there are other
      problems for which affine quantization can work well. Quite
      recently, a surprisingly transparent version of affine quantization has been used for non-renormalizable covariant scalar fields and for Einstein's gravity \cite{k}.

      Others are encouraged to see what affine quantization can do for their own
      quantization problems.

config: arxiv


TITLE: Projective dimension over noetherian rings
QUESTION [4 upvotes]: Let $R$ a commutative Noetherian ring and $M$ a finitely generated $R$-module such that $\operatorname{pd}_R(M)=d < \infty$, where $\operatorname{pd}_R(M)$ denotes the projective dimension of $M$.
Can I prove that there exists a projective resolution of $M$
$0 \to P_d \to P_{d-1} \to \cdots \to P_1 \to P_0 \to M \to 0$ where each $P_i$ is finitely generated?

REPLY [4 votes]: Yes; this follows from two key facts:
Fact 1 If $R$ is Noetherian and $M$ is finitely generated then $M$ has a projective resolution consisting of finitely generated modules.
Proof Idea. Construct a free resolution in the standard way, and use the fact that $R$ is Noetherian to verify that the kernels are finitely generated at each step.
Fact 2 If $M$ has projective dimension $d < \infty$, then for any projective resolution
$$\dots \to F_d \xrightarrow{f_d} \dots \to F_0 \xrightarrow{f_0} M \to 0,$$
$\ker f_d$ is projective.
Proof Idea. Use the generalized version of Schanuel's lemma with the assumed projective resolution of length $d$ and the given projective resolution.

A complete proof of these facts can be found in https://www.math.purdue.edu/~iswanso/homologicalalgebra.pdf (Theorem 6.5 and Proposition 6.6).


TITLE: 3d integral under the limitation of norm of two vectors
QUESTION [1 upvotes]: I would like to integrate
$$ \int d^3 \vec{k} \int d^3 \vec{q} \frac{1}{q^2} $$
under the condition that
$$ |\vec{k}| \leq A\ \ \ \text{and}\ \ \ |\vec{k} + \vec{q}| \leq A$$
where $A$ is some constant.
Is this executable? I have no idea how to handle (angle term of) the second condition.

REPLY [1 votes]: The integral wrt $\mathbf k$ is the volume of the intersection of two balls centered at the origin and at $-\mathbf q$. The formula for the volume of a solid of revolution gives
$$V = 2 \pi \int_{q/2}^A (A^2 - x^2) dx \, [q < 2 A] =
\frac \pi {12} (q + 4 A) (q - 2 A)^2 \, [q < 2 A].$$
Then the original integral is
$$2 \pi \int_0^{2 A} \int_0^\pi V \sin \theta \, d\theta dq =
4 \pi^2 A^4.$$


TITLE: There is some known deficiency through the Lebesgue Integral?
QUESTION [1 upvotes]: The Integral in the Riemann sense has a lot of deficiencies, and the Lebesgue Integral can solve almost all of them.
I know that over limited intervals, Lebesgue Integral is a generalization of the Riemann Integral. But
$$\int_a^\infty\frac{\sin(x)}{x}dx$$
converges in Riemann sense, but it is not true in the Lebesgue sense. So, we could consider this like a deficiency.
But... There is some known deficiency through the Lebesgue Integral over limited intervals?
I read (in wikipedia) that in some cases, the Henstock–Kurzweil Integral is more general than the Lebesgue one. Is it a deficiency? What kind of cases are they talking about?

REPLY [0 votes]: One thing you might consider a deficiency of the Lebesgue integral is that there are integrable functions whose derivatives are not integrable. For example (taken from Priestley's Introduction to Integration) $f(x)=x^2 \sin(x^{-2})$ is integrable on $(0,1]$ (since it extends continuously to $[0,1]$) but its derivative
$$ f'(x) = 2x\sin(x^{-2})-2x^{-1}\cos(x^{-2})$$
is not integrable on $(0,1]$. The first term is integrable, the second isn't because the integral of its absolute value diverges.


TITLE: Full isomorphism-closed reflective subcategory of Set
QUESTION [2 upvotes]: What are the full isomorphism-closed reflective subcategories of $\textbf{Set}$?
In the book "The Joy of Cats" it is mentioned (p. 58) that there are precisely three such subcategories.  I can see two of them are:

the whole category $\textbf{Set}$
the subcategory consisting of all singletons

What would be the third one?

REPLY [4 votes]: The subcategory containing the singletons and the empty set seems to work.


TITLE: Are cyclotomic polynomials irreducible modulo a prime?
QUESTION [1 upvotes]: I am actually interested on cyclotomic polynomials of the form $x^n+1$ (i.e. $n$ is a power of 2, for large $n$).
Are these polynomials irreducible modulo a prime $q$? I already saw that $X^2+1$ is not irreducible modulo 5 because $Z_5$ contains roots of -1. What about $x^{512}-1$? Is there a general criterion?
Thanks

REPLY [1 votes]: For any $n \in \mathbb{N}$ and $p$ a prime that does not divide $n$,
$$ \Phi_n(x) \text{ is irreducible in } \mathbb{Z}_p \iff p \text{ is a primitive root modulo } n$$
Wikipedia article: Primitive root modulo n.
For powers of $2$, only $2$ and $4$ have primitive roots. Primitive roots modulo $n$ correspond to generators of $\mathbb{Z}_n^*$. For $n=2^k$ and $k\geq 3$, $\mathbb{Z}_n^*$ is not cyclic and thus there are no primitive roots modulo $n$.
Thus for higher powers of $2$, the cyclotomic polynomial will be reducible modulo any odd prime.


text:
\begin{document}

\begin{frontmatter}
\pretitle{Research Article}

\title{Distance from fractional Brownian motion with associated Hurst
index $0<H<1/2$ to the subspaces of Gaussian martingales involving power
integrands with an arbitrary positive exponent}

\author[a]{\inits{O.}\fnms{Oksana}~\snm{Banna}\thanksref{cor1}\ead[label=e1]{okskot@ukr.net}\orcid{0000-0002-9730-4654}}
\thankstext[type=corresp,id=cor1]{Corresponding author.}
\author[b]{\inits{F.}\fnms{Filipp}~\snm{Buryak}\ead[label=e1]{filippburyak2000@gmail.com}}
\author[b]{\inits{Yu.}\fnms{Yuliya}~\snm{Mishura}\ead[label=e2]{myus@univ.kiev.ua}\orcid{0000-0002-6877-1800}}
\address[a]{\institution{Kyiv National Taras Shevchenko University}, Faculty of Economics,
Volodymyrska~64, 01601 Kyiv, \cny{Ukraine}}
\address[b]{\institution{Kyiv National Taras Shevchenko University}, Faculty of Mechanics and Mathematics,
Volodymyrska~64, 01601 Kyiv, \cny{Ukraine}}

\runtitle{Distance from fBm to the subspaces of Gaussian martingales}

\begin{abstract}
We find the best approximation of the fractional Brownian motion with the Hurst
index $H\in (0,1/2)$ by Gaussian martingales of the form
$\int _{0}^{t} s^{\gamma }dW_{s}$, where $W$ is a Wiener process,
$\gamma >0$.
\end{abstract}

\begin{keywords}
\kwd{Fractional Brownian motion}
\kwd{martingale}
\kwd{approximation}
\end{keywords}

\begin{keywords}[MSC2010]
\kwd{60G22}
\kwd{60G44}
\end{keywords}

\received{\sday{22} \smonth{4} \syear{2020}}
\revised{\sday{6} \smonth{6} \syear{2020}}
\accepted{\sday{6} \smonth{6} \syear{2020}}
\publishedonline{\sday{23} \smonth{6} \syear{2020}}

\end{frontmatter}

\section{Introduction}
\label{sec1}

The subject of the present paper is a fractional Brownian motion (fBm) \index{fractional Brownian motion}
$B^{H}=\{B_{t}^{H}, t\ge 0\}$ with the Hurst index
$H\in  \left(0, \, \frac{1}{2}\right)$.\index{Hurst index} Generally speaking, a fBm with the Hurst
index $H\in (0,1)$\index{Hurst index} is a Gaussian process with zero mean and the covariance
function of the form

\begin{equation*}
{\mathrm{E}}B_{t}^{H} B_{s}^{H} =\frac{1}{2}(t^{2H}+s^{2H}-\vert t-s\vert ^{2H}).
\end{equation*}

Its properties are rather different for
$H\in \left (0, \, \frac{1}{2}\right )$ and
$H\in \left (\frac{1}{2}, \, 1\right )$. In particular,
$H\in \left (0, \, \frac{1}{2}\right )$ implies short-term dependence.
In contrast, $H\in \left (\frac{1}{2}, \, 1\right )$ implies long-term
dependence. Moreover, technically it is easier to deal with fBms having
$H\in \left (\frac{1}{2}, \, 1\right )$. Due to this and many other reasons,
fBm with $H\in \left (\frac{1}{2}, \, 1\right )$ has been much more intensively
studied in the recent years. However, the financial markets in which trading
takes place quite often, demonstrate the presence of a short memory, and
therefore the volatility in such markets (so called rough volatility) is
well modeled by fBm with $H\in \left (0, \, \frac{1}{2}\right )$, see e.g.
\cite{Volatility}. Thus interest to fBm with small Hurst indices\index{Hurst index} has substantially
increased recently. Furthermore, it is well known that a fractional Brownian
motion\index{fractional Brownian motion} is neither a Markov process nor semimartingale, and especially it
is neither martingale nor a process with independent increments\index{independent increments} unless
$H=\frac{1}{2}$. That is why it is naturally to search the possibility
of the approximation of fBm in a certain metric by simpler processes, such
as Markov processes, martingales, semimartingales or a processes of bounded
variation. As for the processes of bounded variation and semimartingales,
corresponding results are presented in \cite{Andr05,Andr06} and
\cite{Thao03}. In the papers
\cite{BM08,BMishura10,MishuraB08,5stars} approximation of a fractional
Brownian motion\index{fractional Brownian motion} with Gaussian martingales\index{Gaussian martingales} was studied and summarized in
the monograph \cite{MonoBMRS19}, but most of problems were considered only
for $H\in \left (\frac{1}{2}, \, 1\right )$, for the reasons stated above.

In the present paper we continue to consider the approximation of a fractional
Brownian motion\index{fractional Brownian motion} by Gaussian martingales\index{Gaussian martingales} but concentrate on the case
$H\in \left (0, \frac{1}{2}\right )$.

Let $(\Omega , \mathcal{F}, \mathrm{P})$ be a complete probability space
with a filtration $\mathbb{F} = \{ \mathcal{F}_{t}\}_{t\geq 0}$ satisfying
the standard assumptions. We start with the Molchan representation\index{Molchan representation} of fBm
via the Wiener process\index{Wiener process} on a finite interval. Namely, it was established
in \cite{Norros} that the fBm
$\{B_{t}^{H} , \mathcal{F}_{t}, \,t\geq 0\}$ can be represented as

\begin{equation}
\label{BtHK}
B_{t}^{H} =\int \limits _{0}^{t} {z(t,s)} dW_{s},
\end{equation}

where $\{W_{t} ,\,\,t\in [0,\,T]\}$ is a Wiener process,\index{Wiener process}

\begin{equation*}

\begin{gathered}
z(t,s)=c_{H} \bigg (t^{H-1/2} s^{1/2-H}(t-s)^{H-1/2}
\\
-(H-1/2) s^{1/2-H} \int \limits _{s}^{t} u^{H-3/2}(u-s)^{H-1/2}du
\bigg ),
\end{gathered}

\end{equation*}

is the Molchan kernel,\index{Molchan kernel}

\begin{equation}
\label{cH}
c_{H} =\left (
\frac{2H\cdot \Gamma (\frac{3}{2}-H)}{\Gamma (H+\frac{1}{2})\cdot \Gamma (2-2H)}
\right )^{1/2},
\end{equation}

and $\Gamma (x)$, $x>0$, is the Gamma function.\index{Gamma function}

Let us consider a problem of the distance between a fractional Brownian motion\index{fractional Brownian motion}
and the space of square integrable martingales (initially not obligatory
Gaussian), adapted to the same filtration. So, we are looking for a square
integrable $\mathbb{F}$-martingale $M$ with the bracket that is absolutely
continuous w.r.t. (with respect to) the Lebesgue measure such that it minimizes
the value

\begin{equation*}
\rho _{H}(M)^{2}:=\sup _{t\in [0,T]} {\mathrm{E}}(B_{t}^{H}-M_{t})^{2}.
\end{equation*}

We observe first that $B^{H}$ and $W$ generate the same filtration, so
any square integrable $\mathbb{F}$-martingale $M$ with the bracket that
is absolutely continuous w.r.t. the Lebesgue measure, admits a representation

\begin{equation}
\label{itorep}
M_{t} = \int _{0}^{t} a(s) dW_{s},
\end{equation}

where $a$ is an $\mathbb{F}$-adapted square integrable process such that
$\langle M\rangle _{t}=\int _{0}^{t}a^{2}(s)ds$. Hence we can write, see
\cite{5stars},

\begin{align*}
{\mathrm{E}}(B_{t}^{H} - M_{t})^{2} &= {\mathrm{E}}\left (\int _{0}^{t} (z(t,s)-a(s))
dW_{s}\right )^{2} = \int _{0}^{t} {\mathrm{E}}(z(t,s)-a(s))^{2} ds
\\
&= \int _{0}^{t} (z(t,s)- {\mathrm{E}} a(s))^{2} ds + \int _{0}^{t}
\mathrm{Var}\; a(s) ds.
\end{align*}

Consequently, it is enough to minimize $\rho _{H}(M)$ over continuous
\emph{Gaussian} martingales.\index{continuous Gaussian martingales} Such martingales have orthogonal and therefore
independent increments.\index{independent increments} Then the fact that they have representation \eqref{itorep} with a non-random $a$ follows, e.g., from
\cite{Skorohod}.

Now let $a:[0,T]\to \mathbb{R}$ be a nonrandom measurable function of the
class $L_{2} [0,T]$; that is, $a$ is such that the stochastic integral
$\int \limits _{0}^{t} {a(s)dW_{s} } $, $t\in [0,\;T]$, is well defined
w.r.t. the Wiener process $\{W_{t} ,\;\;t\in [0,\;T]\}$\index{Wiener process} (this integral
is usually called the Wiener integral if the integrand is a nonrandom function).
So, the problem is to find

\begin{equation*}
\mathop{\inf }\limits _{a\in L_{2} [0,T]} \mathop{\sup }\limits _{0
\le t\le T} {\mathrm{E}}\left (B_{t}^{H} -\int \limits _{0}^{t} {a(s)dW_{s}
} \right )^{2}= \mathop{\inf }\limits _{a\in L_{2} [0,T]} \mathop{
\sup }\limits _{0\le t\le T} \int \limits _{0}^{t} (z(t,s)- a(s) )^{2}
ds .
\end{equation*}

Note that the expression

to be minimized does not involve neither the fractional
Brownian motion\index{fractional Brownian motion} nor the Wiener process,\index{Wiener process} so the problem becomes purely analytic.
Moreover, since the problem posed in a general form is not observable and
accessible for solution, we restrict ourselves to one particular subclass
of functions. We study the class

\begin{equation*}
\{a(s)=s^{\gamma }, \gamma >0\}.
\end{equation*}

Our main result is Theorem \ref{thdistpowf}, which shows where
$\mathop{\max }\limits _{t\in [0, 1]} {\mathrm{E}}\left (B^{H}_{t} - \int _{0}^{t}
a(s) dW_{s} \right )^{2}$ could be reached, depending on $\gamma >0$. We
also provide remarks after the theorem.

\section{Distance from fBm with $H\in (0, 1/2)$ to the subspaces of Gaussian martingales\index{Gaussian martingales} involving power integrands}
\label{sec2}

Consider a class of power functions with an arbitrary positive exponent.
Thus, we now introduce the class

\begin{equation*}
\{a(s)=s^{\gamma }, \gamma >0\}.
\end{equation*}

For the sake of simplicity, let $T = 1$.

\begin{thm}
\label{thdistpowf}
Let $a=a(s)$ be a function of the form $a(s)=s^{\gamma }$, $\gamma >0$,
$H \in (0, 1/2)$. Then:

\begin{itemize}

\item[\upshape(\textit{i})] For all $\gamma >0$ the maximum
$\mathop{\max }\limits _{t\in [0, 1]} {\mathrm{E}}\left (B^{H}_{t} - \int _{0}^{t}
s^{\gamma }dW_{s} \right )^{2}$ is reached at one of the following points:
$t=1$ or $t=t_{1}$, where

\begin{align*}
t_{1} &= \Bigg (\;c_{H} B\left (\gamma -H+\frac{3}{2}, H+\frac{1}{2}
\right )(\gamma +1)
\\
&-\sqrt{c_{H}^{2}\left (B\left (\gamma -H+\frac{3}{2}, H+\frac{1}{2}
\right )(\gamma +1)\right )^{2}-2H}\;\; \Bigg )^{
\frac{1}{\gamma -H+\frac{1}{2}}}.
\end{align*}

\item[\upshape(\textit{ii})] For any $H\in (0,1/2)$ there exists
$\gamma _{0}=\gamma _{0}(H)> 0$ such that for $\gamma >\gamma _{0}$ the maximum

\begin{align*}
&\mathop{\max }\limits _{t\in [0, 1]} {\mathrm{E}}\left (B^{H}_{t} - \int _{0}^{t}
s^{\gamma }dW_{s} \right )^{2}
\\
&= t_{1}^{2H}-2 t_{1}^{\gamma +\frac{1}{2}+H} c_{H} B\left (\gamma -H+
\frac{3}{2},H+\frac{1}{2}\right ) \frac{\gamma +1}{\gamma +\frac{1}{2}+H}+
\frac{1}{2\gamma +1} t_{1}^{2\gamma +1}
\end{align*}

and is reached at the point $t_{1}$. Here $B(x,y)$, $x,y>0$, is a beta function.

\end{itemize}

\end{thm}

\begin{proof}
According to Lemma 2.20 \cite{MonoBMRS19}, the distance between the fractional
Brownian motion\index{fractional Brownian motion} and the integral $\int _{0}^{t}s^{\gamma }dW_{s}$ w.r.t.
Wiener process\index{Wiener process} $t\in [0, 1]$ equals

\begin{align}
E &\left (B^{H}_{t} - \int _{0}^{t} s^{\gamma }dW_{s}\right )^{2} = E
\left (B^{H}_{t} \right )^{2}-2E\left (\int _{0}^{t} z(t,s) dW_{s}
\int _{0}^{t} s^{\gamma }dW_{s} \right )
\nonumber
\\
&+E\left (\int _{0}^{t} s^{\gamma }dW_{s}\right )^{2} = t^{2H}-2\int _{0}^{t}
z(t,s)s^{\gamma } ds + \int _{0}^{t} s^{2\gamma } ds
\nonumber
\\
&=t^{2H} - 2t^{\gamma +H+\frac{1}{2}}c_{H}B\left (\gamma -H+
\frac{3}{2},H+\frac{1}{2}\right )
\frac{\gamma +1}{\gamma +H+\frac{1}{2}}
\nonumber
\\
&+\frac{t^{2\gamma +1}}{2\gamma +1} := h(t,\gamma ),
\label{htgam}
\end{align}

where $c_{H}$ is taken from \eqref{cH}.\vadjust{\goodbreak}

Let us calculate the partial derivative of $h(t,\gamma )$ in $t$:

\begin{align*}
\frac{\partial h(t,\gamma )}{\partial t} = t^{2H-1}&\Big (2H-2t^{
\gamma -H+\frac{1}{2}}c_{H}
\\
&\cdot B\left (\gamma -H+\frac{3}{2},H+\frac{1}{2}\right )(\gamma +1)+t^{2(
\gamma -H+\frac{1}{2})}\Big ).
\end{align*}

Let us verify whether there is $t\in [0, 1]$ such that
$\frac{\partial h(t,\gamma )}{\partial t}=0$, i.e.

\begin{equation*}
t^{2(\gamma -H+\frac{1}{2})} -2t^{\gamma -H+\frac{1}{2}}c_{H} B\left (
\gamma -H+\frac{3}{2},H+\frac{1}{2}\right )(\gamma +1)+ 2H=0.
\end{equation*}

Changing the variable $t^{\gamma -H+\frac{1}{2}}=: x$, we obtain the following
quadratic equation:

\begin{equation}
\label{eqquadrat}
x^{2}-2xc_{H} B\left (\gamma -H+\frac{3}{2},H+\frac{1}{2}\right )(
\gamma +1)+2H=0.
\end{equation}

The discriminant $D=D(\gamma )$ of the quadratic equation \eqref{eqquadrat} equals

\begin{align}
D(\gamma ) &=4c_{H}^{2}\left (B\left (\gamma -H+\frac{3}{2},H+
\frac{1}{2}\right )(\gamma +1)\right )^{2}-8H
\nonumber
\\
&= 8H\left (\left (B\left (\gamma -H+\frac{3}{2},H+\frac{1}{2}\right )(
\gamma +1)\right )^{2}
\frac{\Gamma (\frac{3}{2}-H)}{\Gamma (H+\frac{1}{2})\Gamma (2-2H)}-1
\right )
\nonumber
\\
&= 8H\left (
\frac{\Gamma (H+\frac{1}{2})\Gamma (\frac{3}{2}-H)}{\Gamma (2-2H)}
\left (\frac{\Gamma (\gamma -H+\frac{3}{2})}{\Gamma (\gamma +1)}
\right )^{2}-1\right ).
\label{eqdiscr}
\end{align}

Now we are going to show that $D(0)>0$ and $D(\gamma )$ is increasing in $\gamma >0$. For this we study separately the function
$ f(H):=
\frac{\Gamma (H+\frac{1}{2})(\Gamma (\frac{3}{2}-H))^{3}}{\Gamma (2-2H)}$
for $H \in (0,\frac{1}{2})$.

\begin{figure}[h]
\includegraphics{156f01}
\caption{The behavior of $f(H)$ for $H \in (0,\frac{1}{2})$}
\label{fig:fH}
\end{figure}

\begin{figure}[h]
\includegraphics{156f02}
\caption{The behavior of $D(0)$ as a function of $H \in (0,\frac{1}{2})$}
\label{fig:D0}
\end{figure}

Let us calculate the values of this function at the following points:
$f(0)=\frac{\pi ^{2}}{8}>1$, $\smallskip f(\frac{1}{2})=1$. To establish that
$f(H)$ is decreasing in $H$, consider Lemma \ref{lem:gamder} and the following calculation:

\begin{align}
(\ln f(H))_{H}' &=\left (\ln
\frac{\Gamma (H+\frac{1}{2})(\Gamma (\frac{3}{2}-H))^{3}}{\Gamma (2-2H)}
\right )_{H}'
\nonumber
\\
&= \left (\ln \Gamma \left (H+\frac{1}{2}\right )+3\ln \Gamma \left (
\frac{3}{2}-H\right )-\ln \Gamma \left (2-2H\right )\right )_{H}'
\nonumber
\\
&= \frac{\Gamma '(H+\frac{1}{2})}{\Gamma (H+\frac{1}{2})}-3
\frac{\Gamma '(\frac{3}{2}-H)}{\Gamma (\frac{3}{2}-H)}+2
\frac{\Gamma '(2-2H)}{\Gamma (2-2H)}
\nonumber
\\
&= \int _{0}^{1} \frac{1-t^{H-\frac{1}{2}}}{1-t} dt-C-3\left (\int _{0}^{1}
\frac{1-t^{\frac{1}{2}-H}}{1-t} dt-C\right )
\nonumber
\\
&\qquad +2\left (\int _{0}^{1} \frac{1-t^{1-2H}}{1-t} dt-C\right )
\nonumber
\\
&= \int _{0}^{1}
\frac{1-t^{H-\frac{1}{2}}-3+3t^{\frac{1}{2}-H}+2-2t^{1-2H}}{1-t} dt
\nonumber
\\
&= \int _{0}^{1} \frac{3t^{\frac{1}{2}-H}-2t^{1-2H}-t^{H-\frac{1}{2}}}{1-t} dt
\nonumber
\\
&=\int _{0}^{1}
\frac{t^{H-\frac{1}{2}}(3t^{1-2H}-2t^{\frac{3}{2}-3H}-1)}{1-t} dt.
\label{eqlnfh}
\end{align}

Let $t \in (0,1)$. Obviously, in this case $t^{H- \frac{1}{2}}>0$ and
$ 1-t >0$. Changing the variables in \eqref{eqlnfh} as
$z:=t^{\frac{1}{2}-H}$, we get

\begin{equation*}
3z^{2}-2z^{3}-1= -(1-z)^{2}(2z+1),
\end{equation*}

and this function is negative for all $z \in (0,1)$. Hence,
$(\ln f(H))'<0 $ and it means that $f(H)$ is decreasing. Furthermore,
$ f(H)>1$ for every $H \in (0,\frac{1}{2})$. The behavior of $ f(H)$ is presented
in Figure \ref{fig:fH}.

So, we proved that $D(0)>0$ (the behavior of $D(0)$ as a function of
$H$ is presented in Figure \ref{fig:D0}), and it follows from Lemma~\ref{lem:z-gam}
that $D(\gamma )$ is increasing in $\gamma >0$ for any
$H\in (0,1/2)$. Therefore, for every $H \in (0,\frac{1}{2})$ and
$\gamma >0$ we have that the quadratic equation \eqref{eqquadrat} has two
roots.

More precisely, if you use standard notations for the coefficients of the
quadratic equation, then coefficient $a$ at $x^{2}$ in \eqref{eqquadrat} is strongly positive, $a=1$, coefficient at $x$ equals
$b=-2c_{H}B(\gamma -H+\frac{3}{2},H+\frac{1}{2})(\gamma +1)$ and is negative,
and $c=2H>0$. We conclude that our quadratic equation has two positive
roots $x_{1}=\frac{-b-\sqrt{b^{2}-4ac}}{2a}$ and
$x_{2}=\frac{-b+\sqrt{b^{2}-4ac}}{2a}$, $x_{1}\le x_{2}$.

According to our notations, we let
$t_{i} :=x_{i}^{\frac{1}{\gamma -H+\frac{1}{2}}}$, $i=1,2$. Since
$x=t^{\gamma -H+\frac{1}{2}}\in [0,1]$ for $t \in [0,1]$ and the left-hand side
of \eqref{eqquadrat} is negative for $x \in (x_{1},x_{2})$, we get the
following cases:

\begin{itemize}

\item[(\textit{i})] Let $x_{1}<1$ and $x_{2}<1$. Then
$\max \limits _{t \in [0,1]}h(t,\gamma )$ can be achieved at one of two
points: $t=t_{1}$ or $t= 1$.

\item[(\textit{ii})] Let $x_{1}<1$ and $x_{2}\ge 1$. Then
$\max \limits _{t \in [0,1]}h(t,\gamma )$ is achieved at point
$t=t_{1}$.

\item[(\textit{iii})] Let $x_{1}\ge 1$ (and consequently $x_{2}>1$). Then
$\max \limits _{t \in [0,1]}h(t,\gamma )$ is achieved at point
$t= 1$.

\end{itemize}

Now, we rewrite the discriminant \eqref{eqdiscr} in the following form:

\begin{align}
D &=4\left (c_{H}^{2}\left (B\left (\gamma -H+\frac{3}{2},H+
\frac{1}{2}\right )(\gamma +1)\right )^{2}-2H\right )
\nonumber
\\
&=: 4(d_{H}^{2}(\gamma )-2H)>0,
\label{eq:Dtr}
\end{align}

where
$d_{H}(\gamma )=c_{H}B(\gamma -H+\frac{3}{2},H+\frac{1}{2})(\gamma +1)$.
From Lemma \ref{lem:d_H}, $x_{1}<\sqrt{2H}<1$, so case (\textit{iii}) never occurs.

According to Lemma \ref{lem:d_H}, the biggest of two roots,
$x_{2}=d_{H}(\gamma )+\sqrt{d_{H}^{2}(\gamma )-2H}$, is increasing in
$\gamma >0$ and $x_{2}>1 \Leftrightarrow d_{H}>\frac{1}{2}+H$. Moreover, it
follows from Lemma \ref{lem:d_H}, (\textit{iv}) that
$d_{H}(\gamma ) \to +\infty $ as $\gamma \to +\infty $. Therefore, for
all $H \in (0,\frac{1}{2})$ there exists $\gamma _{0}(H)>0$ such that for all
$\gamma >\gamma _{0}$ we have $x_{2}>1$.

It means that for $\gamma >\gamma _{0}$ our maximum is reached at the point
$t_{1} = (x_{1})^{\frac{1}{\gamma +\frac{1}{2}-H}}$. Finally,

\begin{align*}
&\mathop{\max }\limits _{t\in [0, 1]} {\mathrm{E}}\left (B^{H}_{t} - \int _{0}^{t}
s^{\gamma }dW_{s} \right )^{2}
\\
&= t_{1}^{2H}-2 t_{1}^{\gamma +\frac{1}{2}+H} c_{H} B\left (\gamma -H+
\frac{3}{2},H+\frac{1}{2}\right ) \frac{\gamma +1}{\gamma +\frac{1}{2}+H}+
\frac{1}{2\gamma +1} t_{1}^{2\gamma +1},
\end{align*}

where $t_{1} = (x_{1})^{\frac{1}{\gamma +\frac{1}{2}-H}}$.
\end{proof}

\begin{remark}
\label{rem:gam0curve}
The implicit equation $d_{H}(\gamma )=\frac{1}{2}+H$ considered as the equation
for $\gamma _{0}$ as a function of $H$ gives us the relation between
$H \in \left (0,\frac{1}{2}\right )$ and respective $\gamma _{0}>0$ which,
by virtue of the foregoing, is determined unambiguously. The form of the
algebraic curve $\gamma _{0}=\gamma _{0}(H)$\index{algebraic curve} is presented in Figure \ref{fig:DH}.

\begin{figure}[h]
\includegraphics{156f03}
\caption{The algebraic curve $\gamma _{0}=\gamma _{0}(H)$}
\label{fig:DH}
\end{figure}

\end{remark}

Consider one of the coefficients that are present in \eqref{htgam},

namely,

\begin{align*}
c^{1}_{H,\gamma } &=2c_{H}B\left (\gamma -H+\frac{3}{2},H+\frac{1}{2}\right )
\frac{\gamma +1}{\gamma +H+\frac{1}{2}}
\\
&= 2\left (
\frac{2H\Gamma \left (\frac{3}{2}-H\right )}{\Gamma \left (H+\frac{1}{2}\right )\Gamma \left (2-2H\right )}
\right )^{1/2}
\frac{\Gamma \left (\gamma -H+\frac{3}{2}\right )\Gamma \left (H+\frac{1}{2}\right )}{\Gamma \left (\gamma +1\right )\left (\gamma +H+\frac{1}{2}\right )}.
\end{align*}

Let $\gamma =0$. Then

\begin{align*}
c^{1}_{H,0} &=2\left (
\frac{2H\Gamma \left (\frac{3}{2}-H\right )}{\Gamma \left (H+\frac{1}{2}\right )\Gamma \left (2-2H\right )}
\right )^{1/2}
\frac{\Gamma \left (\frac{3}{2}-H\right )\Gamma \left (H+\frac{1}{2}\right )}{H+\frac{1}{2}}
\\
&= \frac{2}{H+\frac{1}{2}}\left (
\frac{2H\Gamma \left (\frac{3}{2}-H\right )}{\Gamma \left (H+\frac{1}{2}\right )\Gamma \left (2-2H\right )}
\right )^{1/2}
\frac{\pi \left (\frac{1}{2}-H\right )}{\sin \left (\pi \left (\frac{1}{2}-H\right )\right )}.
\end{align*}

For $H=\frac{1}{2}$, one has $c_{1/2,0}^{1}=2$. Obviously,
$c_{H,0}^{1} \rightarrow 0$ as $H \downarrow 0$. It means that
$c_{H,0}^{1}$ is small in some neighborhood of zero. Having this in mind,
we establish some sufficient condition for $h(t,0)$ to get its maximum
at point 1.

\begin{lemma}
\label{lem:sufmaxat1}
Let $c_{H,0}^{1}<1$. Then $h(t_{1},0)<h(1,0)$.
\end{lemma}

\begin{proof}
Indeed, $h(t_{1},0)=t_{1}^{2H}-c_{H,0}^{1}t_{1}^{1/2+H}+t_{1}$, while
$h(1,0)=2-c_{H,0}^{1}$. Then the inequality $h(t_{1},0)<h(1,0)$ is equivalent
to the following one:

\begin{align*}
c_{H,0}^{1}\left (1-t_{1}^{1/2+H}\right ) &< 2-t_{1}-t_{1}^{2H}.
\end{align*}

If $c_{H,0}^{1}<1$, then
$c_{H,0}^{1} \bigl(1-t_{1}^{1/2+H} \bigr)<1-t_{1}^{1/2+H} < 1-t_{1} <
2-t_{1}-t_{1}^{2H}$.
\end{proof}

\begin{remark}
\label{rem:tosufmax}
As it was mentioned before, $c_{1/2,0}^{1}=2$, and so for
$H = \frac{1}{2}$ the condition of Lemma \ref{lem:sufmaxat1} is not satisfied.
In this case $d_{1/2}(0)=1$, and $t_{1}=t_{2}=1$, so that we have the equality
$h(t_{1},0)=h(1,0)$.

However, the question what will be for $\gamma =0$ and
$H_{0}<H<\frac{1}{2}$, where $H_{0}$ is such a value that for
$0<H<H_{0}$, $c_{H,0}^{1}<1$, is open. In order to fill this gap, we provide
the numerical results with some comments.

Consider two fuctions $h(t_{1},\gamma )$ and $h(1,\gamma )$ as functions
of $\gamma $ and $H$. We already know that
$\mathop{\max }\limits _{t\in [0, 1]} h(t,\gamma ) = \max \{h(t_{1},
\gamma ),h(1,\gamma )\}$. The projection of the surface of
$\max \{h(t_{1},\gamma ),h(1,\gamma )\}$ on the $(H,\gamma )$-plane is presented
in Figure \ref{fig:surfaceprojection}. Points, where $h(t,\gamma )$ reaches
its maximum at $t=1$ are represented in green color, and points where
$h(t,\gamma )$ reaches its maximum at $t=t_{1}$ are represented in brown
color. The black curve is  the algebraic curve
$\gamma _{0}=\gamma _{0}(H)$, which is presented in Figure \ref{fig:DH}.

\begin{figure}[h]
\includegraphics{156f04}
\caption{The projection of the surface of
$\mathop{\max }\limits _{t\in [0, 1]} h(t,\gamma )$}
\label{fig:surfaceprojection}
\end{figure}

\end{remark}

\begin{appendix}

\section*{Appendix section}
\label{secA}

In the proof of Theorem \ref{thdistpowf}, we use these auxiliary results.

\begin{lemma}
\label{lem:srtinc}
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a strictly convex function
of one variable. Take the function
$g(x_{1}, x_{2}) = \frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}}$, $x_{1} \neq x_{2}$, $x_{1},x_{2} \in \mathbb{R}$. Then $g(x_{1}+\alpha ,x_{2}+\alpha )$ is strictly
increasing in $\alpha > 0$.
\end{lemma}

\begin{lemma}
\label{lem:gamder}
Let $\Gamma (x)$ be the Gamma function.\index{Gamma function} Then

\begin{equation*}
(\ln \Gamma (x))' = \frac{\Gamma '(x)}{\Gamma (x)}=\int _{0}^{1}
\frac{1-t^{x-1}}{1-t} dt-C,
\end{equation*}

where $C$ is a fixed constant.
\end{lemma}

Proofs of Lemma 2 and Lemma 3 could be found in \cite{Fichtenholz}.

\begin{lemma}
\label{lem:z-gam}
If $H \in \left (0,\frac{1}{2}\right )$, then the function
$z(\gamma ):=
\frac{\Gamma  (\gamma -H+\frac{3}{2} )}{\Gamma (\gamma +1)}$ is
increasing in $\gamma >0$.
\end{lemma}

\begin{proof}
For every $\gamma >0$ we have $z(\gamma )>0$. Let us calculate

\begin{align*}
\ln z(\gamma )&=\ln
\frac{\Gamma \left (\gamma -H+\frac{3}{2}\right )}{\Gamma (\gamma +1)}=
\\
&= \left (\frac{1}{2}-H\right )
\frac{\ln \Gamma \left (\gamma -H+\frac{3}{2}\right ) -
\ln \Gamma (\gamma +1)}{\left (\gamma -H+\frac{3}{2}\right )-(\gamma +1)}=:
\left (\frac{1}{2}-H\right )\omega (\gamma ).
\end{align*}

According to Lemma \ref{lem:srtinc} and the fact that
$\ln (\Gamma (x))$ is strictly convex we have that
$\omega (\gamma )$ is increasing. Since
$\left (\frac{1}{2} - H\right )>0$, it is clear that $z(\gamma )$ is increasing
in $\gamma >0 $.
\end{proof}

\begin{lemma}
\label{lem:d_H}
Let
$d_{H}(\gamma )=c_{H}B(\gamma -H+\frac{3}{2},H+\frac{1}{2})(\gamma +1)$,
and
$x_{1}=d_{H}(\gamma )-\sqrt{d_{H}^{2}(\gamma )-2H}$, $x_{2}=d_{H}(
\gamma )+\sqrt{d_{H}^{2}(\gamma )-2H}$ be roots of the quadratic equation \eqref{eqquadrat}.
Then for all $\gamma >0$, $H\in \bigl(0,\frac{1}{2}\bigr)$ the following
statements hold.

\begin{itemize}

\item[\upshape\textit{i})] $d_{H}(\gamma )$ is increasing in $\gamma >0$.

\item[\upshape\textit{ii})] $x_{1}<\sqrt{2H}$ and $x_{2}>\sqrt{2H}$.

\item[\upshape\textit{iii})] $d_{H}(\gamma )>\frac{1}{2}+H \Leftrightarrow x_{2}>1$ and
$x_{1}<2H$.

\item[\upshape\textit{iv})] $d_{H}(\gamma )\rightarrow \infty $ as
$\gamma \rightarrow \infty $.

\end{itemize}

\end{lemma}

\begin{proof}
(\textit{i}) Note that $d_{H}(\gamma )$ is increasing in $\gamma >0$ since

\begin{equation*}
d_{H}(\gamma )=c_{H} \Gamma \left (H+\frac{1}{2}\right )
\frac{\Gamma \left (\gamma -H+\frac{3}{2}\right )}{\Gamma \left (\gamma +1\right )},
\end{equation*}

where $c_{H} \Gamma \left (H+\frac{1}{2}\right )>0$ for all
$H \in \left (0,\frac{1}{2}\right )$, and according to Lemma \ref{lem:z-gam},
$
\frac{\Gamma  (\gamma -H+\frac{3}{2} )}{\Gamma  (\gamma +1 )}$
is increasing in $\gamma >0$.

(\textit{ii}) Discriminant \eqref{eqdiscr} satisfies the following relation:

\begin{align*}
0<D\left (0\right ) &= 4\left (d_{H}^{2}\left (0\right )-2H\right ).
\end{align*}

Therefore, $d_{H}(\gamma )>\sqrt{2H}$ for all $\gamma >0$. Also, we can
rewrite $x_{1}=d_{H}(\gamma )-\sqrt{d_{H}^{2}(\gamma )-2H}$ and
$x_{2}=d_{H}(\gamma )+\sqrt{d_{H}^{2}(\gamma )-2H}$. Hence
$x_{2}>\sqrt{2H}$. Transform the value $x_{1}$ to the following form:

\begin{equation*}
x_{1}=
\frac{d_{H}^{2}(\gamma )-(d_{H}^{2}(\gamma )-2H)}{d_{H}(\gamma )+\sqrt{d_{H}^{2}(\gamma )-2H}}=
\frac{2H}{x_{2}}<\sqrt{2H}.
\end{equation*}

(\textit{iii}) Let us assume that $x_{2}>1$ (or, what is equivalent,
$x_{1}=\frac{2H}{x_{2}}<\break 2H$). In turn, this is equivalent to the relation
$d_{H}(\gamma )+\sqrt{d_{H}^{2}(\gamma )-2H}>1$, or
$\sqrt{d_{H}^{2}(\gamma )-2H}>1-d_{H}(\gamma )$. The latter inequality
can be realized in one of two cases:

\begin{equation}
\label{eqmox1}
d_{H}^{2}(\gamma )-2H>1-2d_{H}(\gamma )+d_{H}^{2}(\gamma ),\quad 1-d_{H}(
\gamma )\ge 0;
\end{equation}

or

\begin{equation}
\label{eqmox2}
1-d_{H}(\gamma )<0.
\end{equation}

The couple of inequalities \eqref{eqmox1} is equivalent to
$ 1\ge d_{H}(\gamma )>\frac{1}{2}+H$. Therefore, inequalities \eqref{eqmox1} and \eqref{eqmox2}, taken together, indicate that

$d_{H}(\gamma )>\frac{1}{2}+H$ if and only if $x_{2}>1$.

(\textit{iv}) The value $d_{H}(\gamma )$ can be presented as

\begin{equation*}
d_{H}(\gamma )=c_{H}
\frac{\Gamma \left (\gamma -H+\frac{3}{2}\right )\Gamma \left (H+\frac{1}{2}\right )}{\Gamma \left (\gamma +2\right )}
\left (\gamma +1\right ),
\end{equation*}

where $c_{H}\Gamma \left (H+\frac{1}{2}\right )>0$ is a fixed constant, and for
all $H \in \left (0,\frac{1}{2}\right )$,

\begin{align*}
\frac{\Gamma \left (\gamma -H+\frac{3}{2}\right )}{\Gamma \left (\gamma +2\right )}
\left (\gamma +1\right )&=
\frac{\Gamma \left (\gamma -H+\frac{3}{2}\right )}{\Gamma \left (\gamma +1\right )}
\\
&=
\frac{\sqrt{\frac{2\pi }{\gamma -H+\frac{3}{2}}}\left (\frac{\gamma -H+\frac{3}{2}}{e}\right )^{\gamma -H+\frac{3}{2}}\left (1+O\left (\frac{1}{\gamma -H+\frac{3}{2}}\right )\right )}{\sqrt{\frac{2\pi }{\gamma +1}}\left (\frac{\gamma +1}{e}\right )^{\gamma +1}\left (1+O\left (\frac{1}{\gamma +1}\right )\right )}
\\
&\sim \frac{1}{e^{\frac{1}{2}-H}}
\frac{\left (\gamma -H+\frac{3}{2}\right )^{\gamma -H+1}}{\left (\gamma +1\right )^{\gamma +\frac{1}{2}}}
\rightarrow \infty , \quad \gamma \rightarrow \infty ,
\end{align*}

which follows from the Stirling's approximation for Gamma function.\index{Gamma function}
\end{proof}
\end{appendix}