text:
\begin{document}

\date{}
\title{\textbf{On the Conformal change of a Douglas space of second kind with special $(\alpha, \beta )$-metric}}
\maketitle
\begin{center}
\author{\textbf{Gauree Shanker$^1$, Sruthy Asha Baby$^{2}$ }}
\end{center}

\begin{center}

$^1$Department of Mathematics and Statistics\\[0pt]
School of Basic and Applied sciences,\\[0pt]
Central University of Punjab, Bathinda-151001, India\\[0pt]
Email: gshankar@cup.ac.in \\[0pt]
$^{2}$Department of Mathematics and Statistics\\[0pt]
Banasthali University, Banasthali\\[0pt]
Rajasthan-304022, India\\[0pt]
Email: sruthymuthu123@gmail.com.

\textbf{Abstract}
\end{center}

{\small The notion of a Douglas space of second kind of a Finsler space
with $(\alpha, \beta)$-metric was introduced by I. Y. Lee [\ref{CD-5}].
Since then, so many geometers have studied this topic e. g., [\ref{CD-14}]. In
this paper, we prove that a Douglas space of second kind with special $
(\alpha, \beta)$-metric $\alpha +\epsilon \beta + k \frac{\beta^2}{\alpha }$
is conformally transformed to a Douglas space of second kind. Further, we
obtain some results which prove that a Douglas space of second kind with
certain $(\alpha, \beta)$-metrics such as Randers metric, Kropina metric,
first approximate Matsumoto metric and Finsler space with square metric is
conformally transformed to a Douglas space of second kind. }\newline

\textbf{Mathematics Subject Classification:} 53B40, 53C60.

\textbf{Keywords and Phrases:} Conformal change, Douglas space of second
kind, Finsler space with $(\alpha ,\beta )$-metric, special $(\alpha ,\beta )
$-metric, Berwal sapce.

\section{Introduction}

The notion of Douglas space was introduced by S. Bacso and M. Matsumoto [\ref
{CD-2}] as a generalization of Berwald space from view point of geodesic
equations. Also, they consider the notion of Landsberg space as a
generalization of Berwald space. Recently, the notion of weakly-Berwald
space as another generalization of Berwald space was introduced by S. Bacso
and B. Szilagyi [\ref{CD-11}]. It is remarkable that a Finsler space is a
Douglas space if and only if the Douglas tensor $D^i_{hjk}$ vanishes
identically [\ref{CD-3}].\newline

The theories of Finsler spaces with an $(\alpha, \beta)$-metric and Berwald
spaces with an $(\alpha, \beta)$-metrics ([\ref{CD-16}], [\ref{CD-7}], [\ref
{CD-15}]) have contributed alot to the development of Finsler geometry [\ref{CD-8}
]. The conformal theory of Finsler spaces was introduced by M. S. Kneblman [
\ref{CD-12}] in 1929 and this theory has been investigated in detail by M.
Hashiguchi [\ref{CD-4}]. Later on Y.D. Lee [\ref{CD-6}] and B. N. Prasad [
\ref{CD-10}] found conformally invariant tensors in the Finsler space with $
(\alpha, \beta)$-metric under conformal $\beta$-change.\newline

The purpose of the present paper is to prove that a Douglas space of second
kind with a special $(\alpha, \beta)$-metric given by $L = \alpha + \epsilon
\beta + k \frac{\beta^2}{\alpha}$, where $\epsilon$ and k are constants is
conformally transformed to a Douglas space of second kind. Further, we obtain
some results which prove that the Douglas space of second kind with certain $
(\alpha, \beta)$-metrics such as Randers metric, Kropina metric, first
approximate Matsumoto metric and Finsler space with square metric is
conformally transformed to a Douglas space of second kind.

\section{Preliminaries}

A Finsler space $F^n$ = $(M^n, L(\alpha, \beta))$ is said to be with an $
(\alpha, \beta)$-metric, if $L(\alpha, \beta)$ is a positively homogenous
function of $\alpha$ and $\beta$ of degree one, where $\alpha^2 =
a_{ij}(x)y^iy^j$ and $\beta = b_i(x)y^i$. The space $R^n = (M^n, \alpha)$ is
called the Riemannian space associated with $F^n$. We shall use the
following symbols [\ref{CD-8}]:
\begin{eqnarray*}
&&b^i=a^{ir}b_r,\ \ b^2 = a^{rs}b_rb_s \\
&& 2r_{ij} = b_{i:j} + b_{j:i},\ \ 2s_{ij} = b_{i:j} - b_{j:i}, \\
&& s^i_j = a^{ir}s_{rj},\ \ s^i_j = a^{ir}s_{rj},\ \ s_j = b_rs^r_j.
\end{eqnarray*}
The Berwald connection $B\Gamma = \{G^i_{jk}, G^i_j\}$ of $F^n$ plays an
important role in the present paper. Denote by $B^i_{jk}$, the difference
tensor of $G^i_{jk}$ from $\gamma^i_{jk}$:\newline
\begin{equation}
G^i_{jk} (x,y)= \gamma^i_{jk} (x) + B^i_{jk} (x,y).
\end{equation}
With the subscript 0, transvecting by $y^i$, we have
\begin{equation}
G^i_j = \gamma^i_{oj} + B^i_j \ \ \ and\ \ 2 G^i = \gamma^i_{00} + 2B^i,
\end{equation}
and then $B^i_j = \dot{\partial}_jB^i$ and $B^i_{jk} = \dot{\partial}_k B^i_j
$.

The geodesics of an n-dimensional Finsler space $F^n$ = $(M^n, L)$ are given
by the system of differential equations [\ref{CD-4}]
\begin{equation}
\frac{d^2x^i}{dt^2}y^j - \frac{d^2x^j}{dt^2}y^i + 2 (G^iy^j - G^jy^i) = 0;
y^i=\frac{dx^i}{dt},
\end{equation}
in a parameter t. The spray function $G^i (x,y)$ is given by
\begin{equation}
2G^i(x,y)= g^{ij}(y^r\dot{\partial}_j \partial_r F - \partial_j F)
=\gamma^i_{jk}y^jy^k,
\end{equation}
where $\dot{\partial_i} = \frac{\partial }{\partial x^i}$, F = $\frac{L^2}{2}
$, $\gamma^i_{jk}$ are the Christoffel symbols constructed from $g_{ij}(x,y)$
with respect to $x^i$ and $g^{ij}(x,y)$ is the inverse of the metric tensor
$g_{ij}(x,y)$.

It is well known [\ref{CD-2}] that a Finsler space $F^n$ becomes a Douglas space if and only if
the Douglas tensor
\begin{equation}
D^h_{ijk} = G^h_{ijk} - \frac{1}{n + 1}(G_{ijk} y^h + G_{ij}\delta^h_k +
G_{jk}\delta^h_i + G_{ki}\delta^h_j),
\end{equation}
vanishes identically, where $G^h_{ijk} = \dot{\partial_k G^h_{ij}}$ is the h$
\nu$-curvature tensor of the Berwald connection $B\Gamma$.\newline

Further, a Finsler space $F^n$ is said to be a Douglas space [\ref{CD-9}], if \newline
\begin{equation}
D^{ij} = G^i(x,y)y^j - G^j(x,y)y^i,  \label{C.2.1}
\end{equation}
are homogenous polynomials in $(y^i)$ of degree three.

Differentiating (\ref{C.2.1}) by $y^m$ and contracting m and j in
the obtained equation, we have
\begin{equation}
D^{im}_m = (n + 1)G^i -G^{m}_m y^i.  \label{C.2.2}
\end{equation}
Thus a Finsler space $F^n$ becpmes a Douglas space of the second kind if and only if (
\ref{C.2.2}) are homogenous polynomials in $(y^i)$ of degree two.

\begin{definition}
A Finsler space $F^n$ is said to be a Douglas space of second kind if $
D^{im}_m = (n+1)G^i -G^{im}_m y^i$ is a homogenous polynomial in ($y^i$) of
degree two.
\end{definition}

On the other hand, a Finsler space with an $(\alpha,\beta)$-metric is said
to be a $Douglas\ space\ of\ second\ kind$ if and only if
\begin{equation}
B^{im}_m = (n + 1)B^i - B^m_m y^i,
\end{equation}
are homogenous polynomials in $(y^i)$ of degree two, where $B^m_m$ is given
by [\ref{CD-13}].

Furthermore, differentiating the above with respect to $y^h$, $y^j$ and $y^k$
, we get
\begin{equation}
B^{im}_{hjkm} = B^i_{hjk} = 0.
\end{equation}

\begin{definition}
A Finsler space $F^n$ with $(\alpha, \beta)$-metric is said to be a Douglas
space of second kind, if $B^{im}_m = (n + 1)B^i - B^m_m y^i$ is a homogenous
polynomial in $(y^i)$ of degree two.
\end{definition}

\section{Douglas Space of second kind with $(\protect\alpha, \protect\beta)$
-metric}

In this section, we deal with the condition for a Finsler space with an $
(\alpha, \beta)$-metric to be a Douglas space of second kind.

Let $G^i(x,y)$ be the spray function of a Finsler space $F^n$ with an $(\alpha, \beta)$
-metric. According to [\ref{CD-7}], $G^i(x,y)$ is written in the form\newline
\begin{equation*}
2G^i = \gamma^i_{00} + 2 B^i,
\end{equation*}
\begin{equation}
B^i = \frac{\alpha L_\beta}{L_\alpha}s^i_0 + C^* \Bigl[\frac{\beta L_\beta}{
\alpha L}y^i - \frac{\alpha L_{\alpha\alpha}}{L_{\alpha}} \Bigl(\frac{y^i}{
\alpha} - \frac{\alpha b^i}{\beta} \Bigr)\Bigr],  \label{C.3.1}
\end{equation}
where
\begin{eqnarray}
&&C^* = \frac{\alpha \beta (r_{00}L_\alpha - 2 \alpha s_0L_{\beta})}{2
(\beta^2 L_{\alpha} + \alpha \gamma^2 L_{\alpha \alpha})},  \notag \\
&& \gamma^2 = b^2 \alpha^2 - \beta^2.  \label{C.3.2}
\end{eqnarray}
Since $\gamma^i_{00} = \gamma^i_{jk}(x)y^jy^k$ is hp(2), equation (\ref
{C.3.1}) yields
\begin{equation}
B^{ij} = \frac{\alpha L_{\beta}}{L_{\alpha}}(s^i_0 y^j - s^j_0 y^i) + \frac{
\alpha^2 L_{\alpha \alpha}}{\beta L_{\alpha}} C^*(b^iy^j - b^jy^i).
\label{C.3.3}
\end{equation}
By means of (\ref{C.2.1}) and (\ref{C.3.3}), we have the following lemma [
\ref{CD-9}] :

\begin{lemma}
A Finsler space $F^n$ with an $(\alpha, \beta)$-metric becomes a Douglas space if
and only if $B^{ij} = B^i y^j - B^j y^i$ are hp(3).
\end{lemma}

Differentiating (\ref{C.3.3}) with respect to $y^h$, $y^k$, $y^p$ and $y^q$,
we have $D^{ij}_{hkpq} = 0$, which are equivalent of $D^{im}_{hkpm} = (n+1)
D^i_{hkp} = 0$. Thus, if a Finsler space $F^n$ satisfies the condition $
D^{ij}_{hkpq} = 0$, we call it Douglas space. Further differentiating (\ref
{C.3.3}) by $y^m$ and contracting m and j in the obtained equation, we
obtain
\begin{eqnarray}
B^{im}_m &=& \frac{(n+1)\alpha L_{\beta} s^i_0}{L_{\alpha}} + \frac{\alpha
\{(n+1)\alpha^2 \Omega L_{\alpha \alpha}b^i + \beta \gamma^2 A y^i\}r_{00}}{
2\Omega^2}  \notag \\
&&- \frac{\alpha^2 {\{(n+1)\alpha^2 \Omega L_{\beta}L_{\alpha \alpha}b^i +
By^i\}s_0}}{L_{\alpha} \Omega^2} - \frac{\alpha^3 L_{\alpha \alpha}y^i r_0}{
\Omega},  \label{C.3.4}
\end{eqnarray}
where\newline
$\Omega$ = $(\beta^2 L_{\alpha} + \alpha \gamma^2 L_{\alpha \alpha}),\
provided\ that\ \Omega \neq 0$,\newline
$A$ = $\alpha L_{\alpha} L_{\alpha \alpha \alpha} + 3 L_{\alpha }L_{\alpha
\alpha} - 3 \alpha (L_{\alpha \alpha})^2$,
\begin{equation}
B = \alpha \beta \gamma^2 L_{\alpha }L_{\beta}L_{\alpha \alpha \alpha} +
\beta \{(3\gamma^2 - \beta^2)L_{\alpha} -4\alpha \gamma^2 L_{\alpha
\alpha}\}L_{\beta}L_{\alpha \alpha } +\Omega L L_{\alpha \alpha}.
\label{C.3.5}
\end{equation}
We use the following result [\ref{CD-5}]:

\begin{theorem}
The necessary and sufficient condition for a Finsler space $F^n$ with an $
(\alpha, \beta)$-metric to be a Douglas space of second kind is that, $
B^{im}_m$ are homogenous polynomials in $(y^m)$ of degree two, where $
B^{im}_m$ is given by (\ref{C.3.4}) and (\ref{C.3.5}), provided that $\Omega
\neq 0$.
\end{theorem}

\section{Conformal change of Douglas space of second kind with $(\protect
\alpha, \protect\beta)$-metric.}

In the present section, we find the condition on conformal change, so that a
Douglas space of second kind with $(\alpha, \beta)$-metric is conformally transformed to a Douglas space
of second kind.

Let $F^n$= $(M^n, L)$ and \={F}$^n$ = ($M^n$, \={L}) be two Finsler spaces on
the same underlying manifold $M^n$. If we have a function $\sigma(x)$ in
each coordinate neighbourhoods of $M^n$ such that \={L}(x,y) = $e^\sigma
L(x,y)$, then $F^n$ is called conformal to \={F}$^n$ and the change L $
\rightarrow$\={L} of metric is called a conformal change.

A conformal change of $(\alpha, \beta)$-metric is given as $(\alpha, \beta)$
$\rightarrow$ ($\bar{\alpha},\bar{\beta}$) , where $\bar{\alpha} =e^{\sigma}
\alpha$ and $\bar{\beta}= e^{\sigma}\beta$. Therefore, we have
\begin{eqnarray}
\bar{a}_{ij} &=& e^{2\sigma} a_{ij}, \ \ \bar{b_i} = e^\sigma b_i \\
\bar{a}^{ij} &=& e^{-2\sigma} a^{ij}, \ \ \bar{b^i} = e^{-\sigma}b^i
\label{C.4.1}
\end{eqnarray}
and $b^2$ = $a^{ij}b_i b_j$ = $\bar{a}^{ij} \bar{b}_i\bar{b}_j$. Thus, we
state the following theorem for further use:\newline

\begin{theorem}
A Finsler space with $(\alpha, \beta)$-metric with the length b of $b_i$
with respect to the Riemannian metric $\alpha$ is invariant under any
conformal change of $(\alpha, \beta)$-metric.
\end{theorem}

From (\ref{C.4.1}), it follows that the conformal change of Christoffel
symbols is given by [\ref{CD-4}]:
\begin{equation}
\bar{\gamma}_{jk}^i = \gamma_{jk}^i + \delta^i_j \sigma_k + \delta^i_k
\sigma_j - \sigma^i a_{jk},  \label{C.4.2}
\end{equation}
where $\sigma_j = \partial_j \sigma$ and $\sigma^i = a^{ij}\sigma_j$.

From (\ref{C.4.1}) and (\ref{C.4.2}), we have the following identities:
\newline

\begin{equation*}
\bar{\triangledown}_j \bar{b_i} = e^{\sigma} ( \triangledown_j b_i + \rho
a_{ij} -\sigma_i b_j ),
\end{equation*}
\begin{equation*}
\bar{r}_{ij} = e^{\sigma} [r_{ij} + \rho a_{ij} - \frac{1}{2} (b_i \sigma_j
+ b_j \sigma_i)],
\end{equation*}
\begin{equation*}
\bar{s}_{ij} = e^{\sigma} [s_{ij} + \frac{1}{2} (b_i \sigma_j -b_j
\sigma_i)],
\end{equation*}
\begin{equation*}
\bar{s}^i_j = e^{-\sigma}[s^i_j + \frac{1}{2} (b^i \sigma_j -b_j \sigma^i)],
\end{equation*}
\begin{equation}
\bar{s}_j = s_j + \frac{1}{2}(b^2 \sigma_j - \rho b_j),  \label{C.4.3}
\end{equation}
where $\rho = \sigma_r b^r$.\newline
From (\ref{C.4.2}) and (\ref{C.4.3}), we can easily obtain the following:
\begin{eqnarray}
\bar{\gamma}_{00}^i &=& \gamma_{00}^i + 2 \sigma_0 y^i - \alpha^2 \sigma_j,
\\
\bar{r}_{00} &=& e^{\sigma} (r_{00} + \rho \alpha^2 - \sigma_0 \beta), \\
\bar{s}^i_0 &=& e^{-\sigma}[s^i_0 + \frac{1}{2} (\sigma s_0 b^i - \beta
\sigma^i)], \\
\bar{s_0} &=& s_0 + \frac{1}{2} (\sigma_0 b^i - \rho \beta).  \label{C.4.4}
\end{eqnarray}
Next, we find the conformal change of $B^{ij}$ given in (\ref{C.3.3}). We
have $\bar{L}(\alpha, \beta)$ = $e^{\sigma} L(\alpha, \beta)$, and
\begin{equation}
\bar{L}_{\bar{\alpha}} = L_{\alpha},\ \ \bar{L}_{\bar{\alpha}\bar{\alpha}}=
e^{-\sigma}L_{\alpha \alpha},\ \ \bar{L}_{\bar{\beta}} = L_{\beta},\ \ \bar{
\gamma}^2= e^{2\sigma}\gamma^2.  \label{C.4.5}
\end{equation}
By using (\ref{C.3.2}), (\ref{C.4.4}), (\ref{C.4.5}) and theorem (3.1), we
obtain
\begin{equation}
\bar{C}^* = e^{\sigma} (C^* + D^*),
\end{equation}
where
\begin{equation}
D^* = \frac{\alpha\beta [(\beta \alpha^2 - \sigma_0 \beta)L_{\alpha} -
\alpha (b^2 \sigma_0 - \rho \beta)L_{\beta}]}{2(\beta^2 L_\alpha + \alpha
\gamma^2 L_{\alpha \alpha})}.
\end{equation}
Hence, under the conformal change $B^{ij}$ can be written as:
\begin{eqnarray*}
\bar{B}^{ij} &=& \frac{\alpha L_{\beta}}{L_{\alpha}}(s^i_0y^j - s^j_0y^i) +
\frac{\alpha^2 L_{\alpha \alpha}}{\beta L_{\alpha} }C^* (b^iy^j-b^jy^i) \\
&&+ \Bigl(\frac{\alpha \sigma_0 L_{\beta}}{L_{\alpha} }+ \frac{\alpha^2
L_{\alpha \alpha}}{\beta L_{\alpha}} D^*\Bigr) (b^iy^j-b^jy^i) - \frac{
\alpha \beta L_{\beta}}{2 L_{\alpha}} (\sigma^i y^j - \sigma^j y^i), \\
&=& B^{ij} + C^{ij},
\end{eqnarray*}
where
\begin{equation*}
C^{ij} = \Bigl(\frac{\alpha \sigma_0 L_{\beta}}{L_{\alpha} }+ \frac{\alpha^2
L_{\alpha \alpha}}{\beta L_{\alpha}} D^*\Bigr) (b^iy^j-b^jy^i) - \frac{
\alpha \beta L_{\beta}}{2 L_{\alpha}} (\sigma^i y^j - \sigma^j y^i).
\end{equation*}
From (\ref{C.3.5}), we have
\begin{equation}
\bar{\Omega} = e^{2\sigma}\Omega, \ \ \bar{A} = e^{-\sigma}A,\ \ \bar{B}=
e^{2\sigma}B.
\end{equation}
Now, we apply conformation transformation to $B^{im}_m$, and obtain
\begin{equation}
\bar{B}^{im}_m = B^{im}_m + K^{im}_m,
\end{equation}
where
\begin{eqnarray}
2 K^{im}_m &=& \frac{(n + 1) \alpha L_{\beta}}{L_\alpha}(\sigma_0 b^i -
\beta \sigma^i) + \alpha \Bigl\{ \frac{(n+1) \alpha^2 \Omega L_{\alpha
\alpha}b^i + \beta \gamma^2 A y^i}{\Omega^2}\Bigr\}(\rho \alpha^2 - \sigma_0
\beta)  \notag \\
&& - \Bigl[\frac{\alpha^2\{(n+1) \alpha^2 \Omega L_{\beta} L_{\alpha \alpha}
b^i + By^i\}}{L_{\alpha} \Omega^2} - \frac{\alpha^3 L_{\alpha \alpha} y^i}{
\Omega} \Bigr] (b^2\sigma_0 - \rho \beta).  \label{C.4.7}
\end{eqnarray}
Thus, we have the following result:

\begin{theorem}
The necessary and sufficient condition for a conformal change of Douglas
space of the second kind to be a Douglas space of second kind, is that $
K^{im}_m (x)$ are homogenous polynomial in ($y^i$) of degree two.
\end{theorem}

\section{Conformal change of Douglas space of second kind with special $(
\protect\alpha, \protect\beta)$-metric $L = \protect\alpha +\protect\epsilon
\protect\beta + k \frac{\protect\beta^2}{\protect\alpha}$}

Let us consider a Finsler space with special $(\alpha, \beta)$-metric
\begin{eqnarray}
L = \alpha +\epsilon \beta + k \frac{\beta^2}{\alpha},
\end{eqnarray}
where $\epsilon$ and $k$ are constants.\newline
Then, from (29), we can easily find
\begin{eqnarray}
L_{\alpha} &=& 1- \frac{k\beta^2}{\alpha^2},  \notag \\
L_\beta &=& \epsilon +\frac{ 2k\beta}{\alpha}, \\
L_{\alpha \alpha} &=& 2 k \frac{\beta^2}{\alpha^3},  \notag \\
L_{\alpha \alpha \alpha} &=& \frac{-6k\beta^2}{\alpha^4}.  \notag
\end{eqnarray}
Hence, from (\ref{C.3.5}), we have
\begin{eqnarray}
\Omega &=& \frac{-3k\beta^4 + (1 + 2kb^2)\alpha^2 \beta^2}{\alpha^2},  \notag
\\
A &=& \frac{-12k^2 \beta^4}{\alpha^5}, \\
B &=& \frac{2k}{\alpha^6} \Bigl\{ (1 +2kb^2)\alpha^4 \beta^4 -6\epsilon k
b^2 \alpha^3 \beta^5 -2k (2 + 7kb^2)\alpha^2 \beta^6  \notag \\
&& + 6 \epsilon k \alpha \beta^7 + 15k^2 \beta^8\Bigr\}.  \notag
\end{eqnarray}
Thus, $K^{im}_m$ in (\ref{C.4.7}), reduces to
\begin{equation}
2K^{im}_m = \frac{(n+1)(\epsilon \alpha^3 + 2k \alpha \beta)(\sigma_0 b^i -
\beta \sigma^i)}{(\alpha^2 -k\beta^2)} + p_1 + p_2 + p_3 + p_4,
\end{equation}
where
\begin{eqnarray*}
p_1 &=& \frac{(n+1) b^i}{\{k\beta^4 - (1-2kb^2)\alpha^2 \beta^2\}^2}\Bigl\{
2k\rho(1-2kb^2)\alpha^6 \beta^4 - 2k^2 \rho \alpha^4 \beta^6- 12k^2 \rho b^2
\alpha^2 \beta^7 y^i \\
&&- (1-2kb^2)2k \sigma_0 \alpha^4 \beta^5 + (\sigma_0 - 6 \rho y^i)2k^2
\alpha^2 \beta^7 + 12k^2 \sigma_0 b^2 \alpha^2 \beta^6 y^i -12k^2
\sigma_0\beta^8 y^i\Bigr\}, \\
p_2 &=& \frac{-12 k^2 \beta^5 \gamma^2(\rho \alpha^2 - \sigma_0 \beta)y^i}{
\{(1+2kb^2)\alpha^2 \beta^2 - 3 k\beta^4\}^2}, \\
p_3 &=& \frac{2k(b^2 \sigma_0 \alpha^2 y^i - \rho \alpha^2 \beta y^i)}{
(\alpha^2 -k\beta^2)\{(1+2kb^2)\alpha^2 \beta^2- 3k\beta^4\}^2}\Bigl\{ (1
+2kb^2)\alpha^4 \beta^4 -6\epsilon k b^2 \alpha^3 \beta^5 \\
&&-2k (2 + 7kb^2)\alpha^2 \beta^6 + 6 \epsilon k \alpha \beta^7 + 15k^2
\beta^8\Bigr\}, \\
p_4 &=& \frac{2kb^2 \alpha^2 y^i \sigma_0 - 2k\rho \alpha^2 \beta y^i}{
(1+2kb^2)\alpha^2 - 3k \beta^2}.
\end{eqnarray*}
which shows that $K^{im}_m$ is a homogenous polynomial in $(y^i)$ of degree
two. Hence, we have the following theorem:\newline

\begin{theorem}
\label{Theorem 1-CD} A Douglas space of second kind with special $ \left( \alpha, \beta\right)  $-metric
$L = \alpha +\epsilon \beta + k \frac{\beta^2}{\alpha}$, where $
\epsilon$ and k are constants, is conformally transformed to a Douglas space
of second kind.
\end{theorem}

From theorem \ref{Theorem 1-CD}, we can easily prove that a Douglas space of
second kind for certain Finsler spaces with $(\alpha, \beta )$-metric is
conformally transformed to a Douglas space of second kind. We have the
following cases.

\begin{enumerate}
\item[\textbf{Case (i).}] If $\epsilon$ = 1 and $k$ = 0, the special $ \left( \alpha, \beta\right)  $-metric reduces to $L = \alpha
+ \beta$, which is the well known Randers metric. In this case, $2K^{im}_m$ reduces to
\begin{equation}
2K^{im}_m = (n+1)\alpha(\sigma_0 b^i - \beta \sigma^i),
\end{equation}
which shows that $K^{im}_m$ is a homogenous polynomial in $(y^i)$ of degree
two.\newline
Hence, we have the following:

\begin{corollary}
A Douglas space of second kind with Randers metric $L = \alpha +
\beta$, is conformally transformed to a Douglas space of second kind.
\end{corollary}

\item[\textbf{Case (ii).}] If $\epsilon$ = 0 and $k$ = 1, the special $ \left( \alpha, \beta\right) $-metric reduces to $L =
\alpha + \frac{\beta^2}{\alpha}$. In this case, $2k^{im}_m$ reduces to
\begin{equation}
2K^{im}_m = \frac{(n+1)( 2 \alpha \beta)(\sigma_0 b^i - \beta \sigma^i)}{
(\alpha^2 -\beta^2)} + q_1 + q_2 + q_3 + q_4,
\end{equation}
where
\begin{eqnarray*}
q_1 &=& \frac{(n+1) b^i}{\{\beta^4 - (1-2b^2)\alpha^2 \beta^2\}^2}\Bigl\{
2\rho(1-2b^2)\alpha^6 \beta^4 - 2 \rho \alpha^4 \beta^6- 12 \rho b^2
\alpha^2 \beta^7 y^i \\
&&- (1-2b^2)2 \sigma_0 \alpha^4 \beta^5 + (\sigma_0 - 6 \rho y^i)2 \alpha^2
\beta^7 + 12 \sigma_0 b^2 \alpha^2 \beta^6 y^i -12 \sigma_0\beta^8 y^i\Bigr\}
, \\
q_2 &=& \frac{-12 \beta^5 \gamma^2(\rho \alpha^2 - \sigma_0 \beta)y^i}{
\{(1+2b^2)\alpha^2 \beta^2 - 3 \beta^4\}^2}, \\
q_3 &=& \frac{2(b^2 \sigma_0 \alpha^2 y^i - \rho \alpha^2 \beta y^i)}{
(\alpha^2 -\beta^2)\{(1+2b^2)\alpha^2 \beta^2- 3\beta^4\}^2}\Bigl\{ (1
+2b^2)\alpha^4 \beta^4 -2 (2 + 7b^2)\alpha^2 \beta^6 \\
&& + 15 \beta^8\Bigr\}, \\
q_4 &=& \frac{2b^2 \alpha^2 y^i \sigma_0 - 2\rho \alpha^2 \beta y^i}{
(1+2b^2)\alpha^2 - 3\beta^2}.
\end{eqnarray*}
This shows that $K^{im}_m$ is a homogenous polynomial in $(y^i)$ of degree
two.\newline
Hence, we have the following:

\begin{corollary}
A Douglas space of second kind with special $(\alpha, \beta)$-metric $L = \alpha + \frac{\beta^2}{\alpha}$ is conformally transformed to a
Douglas space of second kind.
\end{corollary}

\item[\textbf{Case (iii).}] If $\epsilon$ = 1 and $k$ = 1, the special $ \left( \alpha, \beta\right)  $-metric reduces to the form $L =
\alpha+ \beta + \frac{\beta^2}{\alpha}$, which is the first approximate Matsumoto metric. In this case, $2K^{im}_m$ reduces to
\begin{equation}
2K^{im}_m = \frac{(n+1)( \alpha^3 + 2 \alpha \beta)(\sigma_0 b^i - \beta
\sigma^i)}{(\alpha^2 -\beta^2)} + r_1 + r_2 + r_3 + r_4,
\end{equation}
where
\begin{eqnarray*}
r_1 &=& \frac{(n+1) b^i}{\{\beta^4 - (1-2b^2)\alpha^2 \beta^2\}^2}\Bigl\{
2\rho(1-2b^2)\alpha^6 \beta^4 - 2 \rho \alpha^4 \beta^6- 12 \rho b^2
\alpha^2 \beta^7 y^i \\
&&- (1-2b^2)2 \sigma_0 \alpha^4 \beta^5 + (\sigma_0 - 6 \rho y^i)2 \alpha^2
\beta^7 + 12 \sigma_0 b^2 \alpha^2 \beta^6 y^i -12 \sigma_0\beta^8 y^i\Bigr\}
, \\
r_2 &=& \frac{-12 \beta^5 \gamma^2(\rho \alpha^2 - \sigma_0 \beta)y^i}{
\{(1+2b^2)\alpha^2 \beta^2 - 3 \beta^4\}^2}, \\
r_3 &=& \frac{2(b^2 \sigma_0 \alpha^2 y^i - \rho \alpha^2 \beta y^i)}{
(\alpha^2 -\beta^2)\{(1+2b^2)\alpha^2 \beta^2- 3\beta^4\}^2}\Bigl\{ (1
+2b^2)\alpha^4 \beta^4 -6 b^2 \alpha^3 \beta^5 \\
&&-2 (2 + 7b^2)\alpha^2 \beta^6 + 6 \alpha \beta^7 + 15\beta^8\Bigr\}, \\
r_4 &=& \frac{2b^2 \alpha^2 y^i \sigma_0 - 2\rho \alpha^2 \beta y^i}{
(1+2b^2)\alpha^2 - 3\beta^2}.
\end{eqnarray*}
This shows that $K^{im}_m$ is a homogenous polynomial in $(y^i)$ of degree
two.\newline
Hence, we have the following:

\begin{corollary}
A Douglas space of second kind with first approximate Matsumoto metric $L
= \alpha+ \beta + \frac{\beta^2}{\alpha}$ is conformally transformed to a
Douglas space of second kind.
\end{corollary}

\item[\textbf{Case (iv).}] If $\epsilon$ = 2 and $k$ = 1, the special $ \left( \alpha, \beta\right)  $-metric reduces to the form $L = \frac{
(\alpha+ \beta)^2}{\alpha}$, which is known as square metric. In this case, $
2K^{im}_m$ reduces to
\begin{equation}
2K^{im}_m = \frac{(n+1)(2 \alpha^3 + 2 \alpha \beta)(\sigma_0 b^i - \beta
\sigma^i)}{(\alpha^2 -\beta^2)} + s_1 + s_2 + s_3 + s_4,
\end{equation}
where
\begin{eqnarray*}
s_1 &=& \frac{(n+1) b^i}{\{\beta^4 - (1-2b^2)\alpha^2 \beta^2\}^2}\Bigl\{
2\rho(1-2b^2)\alpha^6 \beta^4 - 2 \rho \alpha^4 \beta^6- 12 \rho b^2
\alpha^2 \beta^7 y^i \\
&&- (1-2b^2)2 \sigma_0 \alpha^4 \beta^5 + (\sigma_0 - 6 \rho y^i)2 \alpha^2
\beta^7 + 12 \sigma_0 b^2 \alpha^2 \beta^6 y^i -12 \sigma_0\beta^8 y^i\Bigr\}
, \\
s_2 &=& \frac{-12 \beta^5 \gamma^2(\rho \alpha^2 - \sigma_0 \beta)y^i}{
\{(1+2b^2)\alpha^2 \beta^2 - 3 \beta^4\}^2}, \\
s_3 &=& \frac{2(b^2 \sigma_0 \alpha^2 y^i - \rho \alpha^2 \beta y^i)}{
(\alpha^2 -\beta^2)\{(1+2b^2)\alpha^2 \beta^2- 3\beta^4\}^2}\Bigl\{ (1
+2b^2)\alpha^4 \beta^4 -12 b^2 \alpha^3 \beta^5 \\
&& -2 (2 + 7b^2)\alpha^2 \beta^6 +12 \alpha \beta^7 + 15 \beta^8\Bigr\}, \\
s_4 &=& \frac{2b^2 \alpha^2 y^i \sigma_0 - 2\rho \alpha^2 \beta y^i}{
(1+2b^2)\alpha^2 - 3\beta^2},
\end{eqnarray*}
which shows that $K^{im}_m$ is a homogenous polynomial in $(y^i)$ of degree
two.\newline
Hence, we have the following:

\begin{corollary}
A Douglas space of second kind with square metric $L = \frac{(\alpha+
\beta)^2}{\alpha}$ is conformally transformed to a Douglas space of second
kind.
\end{corollary}
\end{enumerate}

config: arxiv


TITLE: Asymptotics of the number of elements in the intersection of two growing sets
QUESTION [3 upvotes]: Let $[n]:=\{1,\dots,n\}$ and $0\leq p_n\leq n$. Fix any subset $A_n$ of $[n]$ with $p_n$ elements. The number of subsets $B$ of $[n]$ with $p_n$ elements that are disjoint from $A$ is $\binom{n-p_n}{p_n}$ and so the number of subsets $B$ with $p_n$ elements that have a non-empty intersection with $A$ is given by $\binom{n}{p_n}-\binom{n-p_n}{p_n}$ which asymptotically behaves like $\binom{n}{p_n}$ if $\sqrt{n}\ll p_n$. Precisely: $$\lim_{n\to\infty}\frac{\binom{n}{p_n}-\binom{n-p_n}{p_n}}{\binom{n}{p_n}}=\begin{cases}1 &\text{if }\sqrt{n}\ll p_n\\
0&\text{if }p_n\ll\sqrt{n}\\
e^{-a^2}&\text{if }\lim_{n\to\infty}\frac{p_n}{\sqrt{n}}=a\end{cases}$$
To put it differently, any two subsets of $[n]$ with $p_n$ elements typically have non-empty intersection if $p_n$ grows faster than $\sqrt{n}$.
Now, I am interested in the typical number of elements in their intersection if $p_n$ grows faster than $\sqrt{n}$. My feeling would be that this number also grows with $n$. Precisely, I believe (numerics support this) that if $q_n$ grows slower than $p_n/\sqrt{n}$ then eventually all sets $A_n$, $B_n$ with $p_n$ elements each share at least $q_n$ elements. To put it in formulas, if $A_n$ is any set with $p_n$ elements, then $$\lim_{n\to\infty }\frac{\lvert\{B_n\subset[n]~~\lvert~~\lvert B_n\lvert=p_n\text{ and }\lvert A_n\cap B_n\lvert\geq q_n\}\rvert}{\binom{n}{p_n}}=1.$$
Since there is no exact combinatorial count (only in terms of generalised hypergeometric functions) on the number of $B_n$ of a given size that share at least $q_n$ elements with some fixed sets, I would need some good lower bound on it. Has anybody seen those asymptotics worked out? Any references would be welcome.

REPLY [2 votes]: The problem statement exactly corresponds to the definition of the hypergeometric distribution.  With this key-phrase in hand, it is easy to locate an extensive literature. Start with wikipedia for basic stuff like the mean, variance, normal approximation, etc.. The book "Univariate Discrete Distributions" by Johnson, Kotz and Kemp contains a lot more.  A recent lovely paper of Matthew Skala has a survey of tail inequalities.


TITLE: Check whether W is a T-invariant subspace of V
QUESTION [1 upvotes]: $V$ = $C([0,1])$
$T(f(t))$ =[ $\int\limits_0^1f(x)dx$]$t$
$W$ = {$f \in V $ : $f(t)$ = $at+b$ $for$ $some$ $a$ $and$ $b$}
I got $T(f(t))$ for $f(t)$ = $at+b$ is,
[ $\int\limits_0^1(ax+b)dx$]$t$ = $(a/2 + b)$$t$ ,which does not belong to $W$.
Hence $W$ is not a $T$-invariant subspace of $V$.
But my book says otherwise i.e. $W$ is a $T$-invariant subspace of $V$.
I can't figure out where i went wrong ?

REPLY [1 votes]: Why doesn't $\left(\frac a2+b\right)t$ belong to $W$? It is of the form $ct+d,$ where $c,d\in\Bbb R$, yes?


TITLE: Book about the foundation of math?
QUESTION [4 upvotes]: I am an undergraduate engineering physics student since 1 1/2 years back. I've taken classes in real analysis, complex analysis and linear algebra.
When one tries to read about math on Wikipedia the articles talk about mathematical structures, mathematical objects and so on. I'd like to know more about this, the "foundations of mathematics".
I want to know if there are any appropriate books, given my background, that would be possible to use for self-study?

REPLY [2 votes]: A classic text that used to be (and still may be) a standard undergraduate text for courses in the foundations of mathematics is Introduction to the Foundations of Mathematics by Raymond Louis Wilder. You can find this book in virtually every U.S. college/university library, and apparently (from what I've just read online) Wilder's book is being reprinted by Dover Publications. I think this book would be a very good fit for what you're looking for and your background. (I've had a copy of the 1965 edition for nearly 30 years, so I'm somewhat familiar with it.)


TITLE: Can a smooth vector field in $\mathbb{R}^{3}$ be interpreted as a map that smoothly distorts space?
QUESTION [0 upvotes]: So basically I've been studying some basic differential geometry and I've learnt that a smooth map of the form $\gamma : \mathbb{R} \longrightarrow \mathbb{R}^{3}$
can be interpreted as a map that takes the real line and smoothly embeds it in $\mathbb{R}^{3}$, with the result being some smooth curve in $\mathbb{R}^{3}$.
Similarly a smooth function $\phi : \mathbb{R}^{2} \longrightarrow \mathbb{R}^{3}$ can be interpreted as taking the 2 dimensional Euclidean plane and smoothly embedding it in $\mathbb{R}^{3}$ with the result being a smooth surface.
Now I'm assuming that a function $F: \mathbb{R}^{3} \longrightarrow \mathbb{R}^{3}$ can be interpreted as a smooth distortion of space but a smooth map of this sort is what we usually interpret as as a vector field on $\mathbb{R}^{3}$ (assuming $\mathbb{R}^{3}$ is equipped with a vector space structure).
To pick a slightly physics oriented example, can a vector field like the Electric Field be thought of as a smooth distortion of space? If so then how would we interpret vector field operations like the divergence or curl?
All of this seems hugely unintuitive to me and I'm not even sure that what I said makes any sense so a little guidance would be hugely appreciated.

REPLY [1 votes]: The reason you are confused is that Euclidean space carries a natural "connection," which makes it possible for us to view points as vectors and vice versa. In a general manifold this ambiguity is not possible.
As Moishe Cohen points out, the two pictures are very different. One way to view a smooth map from 3-space into itself is as a smooth volume distortion. (If the map is a diffeomorphism, you in fact just get a smooth change of coordinates.) Another way is to view such a map as assigning a tangent vector to each point of space. This point of view is only possible becaus the standard connection gives a canonical isomorphism of each tangent space $T_p\mathbb{R}^3$ with $\mathbb{R}^3$.
The distinction is clearer in the setting of general manifolds. Let $M$ be a smooth manifold. A smooth vector field on $M$ is a smooth assignment of a tangent vector to each point of $M$. That is, a smooth vector field smoothly assigns to each point $p$ a tangent vector in the tangent space $T_pM$. More formally, a vector field is a smooth section of the tangent bundle, i.e. a smooth map $v:M\to TM$ such that $\pi\circ v=\text{Id}_M$. Note, in particular, that a vector field is NOT a map from $M$ into itself. When you think of a map from 3-space into 3-space as a vector field, you are really using the canonical affine connection on $M=\mathbb{R}^3$ to identify each tangent space $T_pM$ with $M$ itself.
The upshot is that a vector field is a map $M\to TM$, whereas a smooth distortion of points is a map $M\to M$. (The two types of maps aren't totally unrelated. As Alfred Yerger hints at, a vector field on a manifold induces a flow, which is a one-parameter family of diffeomorphisms $M\to M$ given by flowing along the integral curves of the vector field.)