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\section*{0) Introduction and
Survey of Results}\label{pg-0}
\
\s

The discovery of new algebraic structures in the form of quantum groups and
braided tensor categories (BTC's) has led to new insights in several areas of
mathematical physics, and, in particular, has stimulated renewed interest in
low dimensional topology. This development was set off with the observation
that the braid group representations obtained from quantum groups or BTC's
can be used to construct knot and link invariants. The general strategy
followed here
is to associate to the ``regular'' singularities of a generic flat projection
of a link, namely crossings, maxima, and minima, the defining  braid- and
rigidity-morphisms of a BTC. To the higher order singularities of
projections of codimension one, (triple crossings, crossings at equal
heights, saddles, tangential strands, etc.)
which connected  all generic projections to each other, one associates the respective relations
that occur in the axioms of a
BTC, i.e., the Artin relations (or hexagonal- and pentagonal equations), the
rigidity constraints, inverse braid isomorphisms, etc.
\medskip

Soon thereafter it was realized that the same structures can be used to define invariants of closed, compact three manifolds. The works of Turaev-Viro and
of Reshetikhin-Turaev brought forward two types of invariants constructed from
tensor categories, which are related to each other. In this article we shall
focus on the second type described in [RT], since it is closer related to link invariants.
\medskip

The starting point of the construction there are surgery presentations of a
manifold by a framed link $\li\hookrightarrow S^3\,$. In the description
of [Wc] the components of the link are the curves along which two handles
$\hh^2$ are attached to a four ball. The resulting three fold is then given as
 the boundary  $M=\partial\bigl(D^4\cup\hh^2\cup\ldots\cup\hh^2\bigr)\,$.
From $\li$ the three fold  $M=M_{\li}\,$ can also be
described  directly by a surgery
along the components of $\li\,$, as in [Li].
\medskip

The manifold invariant of [RT] is then defined as a weighted sum over the
link invariants, described previously, evaluated on $\li\,$. The fact that
still needs to be checked is  that links presenting the same manifold
yield the same invariant. In particular we need to know how two links
$\li_1$ and $\li_2$ with  $M_{\li_1}=M_{\li_2}$
can be related.
\medskip

An answer to this question is given by Kirby's theorem, [Ki].
It states that $\li_2$
can be obtained from  $\li_1$ by a series of two types of moves in the
class of links in
$S^3\,$. They are the two-handle slides ( ${\cal O}_2$-moves ) and the signature move (or ${\cal O}_1\,$-move ), which corresponds to the replacement
$W\to W\#{\bf CP}^2\,$ for the bounding four fold.
 This result is improved by the result in [FR], where it is shown
that it suffices to consider only a special type of  ${\cal O}_2$-moves,
namely the $\kappa$-moves.
\medskip

The main technical concern in both works is
that the moves can be chosen such that one never leaves the class of
presentations that use only two handles. For this purpose one also has
to restrict the
class of three folds on which we surger to connected and simply connected ones
(like $S^3\,$). If simply-connectedness is given up, Proposition
\ref{pp-red1} tells us that the Kirby formulation
(but not that of [FR] )  can
be salvaged by including an additional move, which is introduced as  the $\eta$-move in
Section 2.2.2). Clearly, for disconnected manifolds not even the results of
[Wc] and [Li] hold, since surgeries along links do not change connectedness.
\medskip

Consequently, the point of view chosen in the mentioned
references involves a number
difficulties when dealing with the constructions of [RT]:
First the  algebraic verification of  the $\kappa$-move is somewhat involved,
since several pieces of the  $SL(2,{\bf Z})\,$ representation, that is associated  to any modular BTC, have to be constructed by hand.
Moreover, the verification is not like in the case of links a more or less
obvious  identification of elementary topological and algebraic constituents.
Instead, it relies on the non trivial result in [FR], that presents all two handle slides as compositions of $\kappa$-moves.
In the study of non semisimple invariants interpretations of two handle slides
as the defining equation of an integral of a Hopf algebra have been found in [HKRL]. This, however, does not simplify the verification that the formulae
given in [RT] are actually integrals.
\medskip

In [RT] not only an invariant but more generally  the construction of a TQFT, i.e., fiber functors on cobordism categories, is proposed.
A rigorous construction of presentations of cobordisms, which generalizes
the theory for closed manifolds, has so far been missing.
For the special case of invertible cobordisms, which are given by elements
of the mapping class group, tangle presentations  have been constructed in
[MP]using
explicit presentations in terms of generators and relations, as in [Wj].
 In order to generalize the surgery presentation as in [Wc], [Li], [Ki], and [FR], and for nice descriptions of the composition of cobordisms it is most
convenient to start surgery on unions of handlebodies. These  are neither
connected nor simply connected and therefore do not fit in the
conventional Kirby calculus.\footnote{After completion of the
first version of this papaer the author received the preprint [Sw].
The approach presented there is similar to the description of [RT], where
the spines of embedded handlebodies are treated on the same footing as surgery
ribbons. Presentations are thus reduced to the ordinary Kirby calculus. Unlike
the description we derive here, the embeddings of the boundaries are variable, which makes the formulation of composition rules (let alone presentations of cobordism categories)  more difficult.}
\medskip

In order to shed more light on both of the outlined problems, it is most
instructive, if we refrain from insisting on
presentations, where only two handles are attached, but also include
one handles into the surgery description.
The manifolds we shall consider are thus of the form   $M\,=\,\partial\bigl(N^{(4)}\cup\hh^1\cup\ldots\cup\hh^1\cup\hh^2\cup\ldots\cup
\hh^2\,\bigr)\,$, and we make no connectedness assumptions on the boundary of
$N^{(4)}$. The attaching data is now encoded into, what we call
 a ``bridged link'',
which is embedded in the  manifold $\li\hookrightarrow M_0=\partial N^{(4)}\,$,
to be surgered on. The additional data that enters a bridged link are
pairs of small spheres in $M_0$, that indicate, where the one handles are to
 be attached. In analogy with presentations of groups this is like including
more  generators into the description and thereby providing more freedom in
choosing a practical set of relations.
\medskip

The one surgered manifold,
$\breve M\,=\,\partial\bigl(N^{(4)}\cup\hh^1\cup\ldots\cup\hh^1\bigr)\,$,
is obtained by gluing the spheres of each pair together along an
orientation reversing
homeomorphism, after the interior three-balls have been removed.
 In particular, this type of surgery allows for a change of connectedness.
To get from $\breve M\,$ to $M\,$ we attach the remaining two handles, i.e.,
we surger along an ordinary, framed link $\breve \li$ in $\breve M\,$. The preimage of the components of $\breve \li$ in $M_0\,$ are ribbons that may
end in one surgery sphere and emerge at the respective spot on the partner
sphere. A typical example of a bridged link is given in figure~\ref{fig-typp}.
\medskip

Manipulations with similar types of attaching data have appeared in the papers
of, e.g., [FR]. A frequently used notation for one handle attachement due
to Kirby is that of a ``dotted circle''. However, the isoptopy classes of
such attaching data are generally smaller, because associated surgery
spheres cannot be moved independently.
\medskip

With our definition, the analogue of the presentation
theorems in [Wc] and [Li] is the following:
\begin{theorem}\label{ONE}
Suppose $M$ and $M_0$ are compact, oriented three folds, and $M$ is
connected. Also assume there is a homeomorphism of boundaries:
$$
\psi\,:\,\partial M_0 \isto \partial M\;.
$$
Then there is a bridged link $\li$ in the interior of $M_0$ such that $\psi$
can be extended to a homeomorphism
$$
\tilde \psi\,:\,\bigl(M_0)_{\li}\isto M\;.
$$
\end{theorem}
\medskip

The ``relations'' between presentations, which we propose to choose,  are
given by the following five moves. See also, Proposition \ref{pp-quint-mv}
in Section 2.3).

\begin{theorem}\label{TWO}
If for two bridged links $\li_1$ and $\li_2$ in the same compact manifold
$M_0\,$ we have that $\bigr(M_0\bigl)_{\li_1}\,=\,\bigr(M_0\bigl)_{\li_2}\,$,
then they are related by a sequence of the following moves:

\begin{enumerate}

\item \ub{Isotopies}: Regular isotopies of the bridged link $\li\hookrightarrow M_0\,$, where the ends of the ribbons stay attached to the spheres.
\item \ub{Signature or ${\cal O}_1$ -move}:  (as in [Ki])

\item \ub{ Isolated Cancellation} : If a component of the bridged link consists
of  a pair of spheres and a single ribbon which penetrates the spheres in
only
one pair of points, then this component can be discarded from the diagram.
(This corresponds to figure~\ref{fig-canc},
where the outer ribbons, $a$,$b$, and $c$, are omitted.)

\item \ub{Modification (or Handle Trading)} : This corresponds to replacing
a one handle of the four fold by a two handle.
The operation on the bridged link is shown in
figure~\ref{fig-modi} and explained in Section 2.2.2.5).

\item \ub {One Slides and Isotopies over Components} : If a pair of surgery spheres (say $Lo$ and $Lo'$) lies in two different
  components of $M_0$, then we can push another surgery sphere ( $Hi$) through as shown in figure~\ref{fig-1slid}. An  isotopy of the link in a fixed $\breve M\,$ gives rise isotopies of ribbons through spheres as indicated in figure
\ref{fig-isot}.

(In the description of planar ribbon diagrams those are broken into opposite braid group actions on the
the ribbons attached to the two surgery spheres, and pushing through loops.)

\end{enumerate}

\end{theorem}

Since the original motivation of this work was to find cobordism
presentations, the language and organization used in the derivation of
these results sometimes suggests a
specialization to the case where $M_0\,$ is a union of handle bodies
$\H\,$. As no references to the special properties of this choice are
ever made, the proofs may be read literally in full generality.
\medskip

All of the proposed moves in Theorem~\ref{TWO} are local. In particular, we have no two handle
slides or $\eta$-moves
in the list. Note also, that the fifth move can be omitted if $M_0\,$ is
connected.
\medskip

 Diagrammatically,
the modification-move is similar to the $\kappa$-move, but its topological
derivation and its interpretation are far more elementary.
Also, the algebraic computations involved
in the invariance proof of [RT] become much easier, see Section 4.2.
The proof of invariance we give here is thus closer in spirit to the
proof of invariance for links as described in the beginning of the
introduction. The relations between surgery and algebraic data are
summarized in the table at the end of Section 4.2.1). In [KL] this
correpondence is sytematically put to use, in order to construct
extended TQFT's and the elementary cobordisms that belong to the
algebraic generators are more explicitly identified.
\medskip

Interestingly, we find - analogous to the correspondence in [HKRL] between
2-handles and integrals of Hopf algebras - an interpretation of 1-handles
as cointegrals. Details and implications for non-semisimple TQFT's will
be discussed in separate papers.
\medskip

As a byproduct of purely topological considerations, we will show that the modularity condition of
[Tu] on the abelian
BTC we start with can be equivalently replaced by the weaker condition:
$$
1\in im(S) \qquad\qquad \bigl(\;{\rm or,\quad  equivalently},\; S^2(1)=1\;, etc.\bigr)\;,
$$
where the $S$-matrix is as usual defined by  the traces over the monodromies.
If we think of $S$ as a generalized Fourier transform, this condition can
be seen as an analogue of the point separation condition  of the
Stone-Weierstrass theorem. For the non-semisimple version of this
condition see again [KL].
\medskip

The second application of the bridged link calculus
is the derivation of tangle presentations of
connected cobordisms - or a central extension, $1\to\Omega_4\to
\wct\to\ct\to 1\,$, thereof described in Chapter 1.
We introduce so called ``standard presentations''  on unions of standard
handle bodies with  a fixed one-handle structure of the bounding
four fold. The moves under these
restrictions are derived in Chapter 3. An additional move that has to
be added to those for closed manifolds is  the so called the $\sigma$-move,
which also appeared in the
combinatorial description of the mapping class group in [MP].
In fact, our presentation will entail a 3+1 dimensional, Cerf-theoretical
 proof of the results in [MP],
which does not use the explicit presentations from [Wj]. The category
\mbox{${\cal T}\!\!g\,$} is described in the following theorem:
\medskip

\begin{theorem}\label{THREE}
The category $\,\wct\,$ is naturally isomorphic to the category \mbox{$\,\widetilde{{\cal
T}\!\!g\,}\,$} of ordered sets of grouped, admissible, ribbon tangles in ${\bf
R}\times [0,1]\,$, modulo relations.

\noindent
The composition rules in \mbox{$\widetilde{{\cal T}\!\!g\,}$} are as descibed in
Sections 1.3) and 3.2.3).

\noindent
The five equivalence relations are
\begin{enumerate}
\item[] \quad Isotopies
\item[] \quad The $\tau$-move \quad\qquad (see figure \ref{fig-tau-move})
\item[] \quad The $\sigma$-move \quad\qquad (see figure \ref{fig-sigma})
\item[] \quad The $\kappa$-move \quad\qquad (as a special ${\cal O}_2\,$, see [FR])
\item[] \quad The $\,^0\bigcirc\mkern-15mu\bigcirc^0\,$-move \quad(a special $\eta$-move,
see Section 2.2.2.8))
\end{enumerate}
The functor $\wct\to\ct$ is presented by a corresponding quotient
\mbox{$\widetilde{{\cal T}\!\!g\,}\to {\cal T}\!\!g\,$} of tangle
categories.
In \mbox{${\cal T}\!\!g\,$} we have in addition the ${\cal O}_1\,$-move, which
allows us to omit the $\,^0\bigcirc\mkern-15mu\bigcirc^0\,$-move.

\end{theorem}

In Section 4.1) tangle presentations are used to derive presentations of
general mapping tori.
We also discuss a diagram that is of interest in connection with the
composition rules of the tangle categories. This tangle element gives rise to
a canonical idempotent in a given quasitriangular Hopf algebra $H$,
whose image on a general representation of $H$ is the maximal, self conjugate subrepresentation.
\medskip

The proofs of Theorem \ref{TWO} and related statements are based on the
theory of stratifications of function spaces and their topology as
developed by Cerf in [Ce]. The relevant facts are reviewed in Section 2.1).
Another indication that the inclusion of one handles yields a more coherent
picture is given by the observation that the space of codimension one Morse
functions on a four fold with singularities of index one and two is
connected, where as
the corresponding space with only index two singularities is disconnected.
(see Lemma~\ref{lm-12-cerf}).
\medskip

In Theorem \ref{TWO} the listed moves have specific meanings as to
which part of the presentation they affect: The moves 1), 3) and 5)
correspond to the elementary deformations of functions on a fixed
bounding four fold. The ${\cal O}_1$-move is as usual the elementary move
that changes the cobordism class of the four fold in $\Omega_4$, i.e., the
class in the central extension $\wct\,$, by connected summing with  a ${\bf
C}P^2\,$.
Only the modification changes the four fold. The elementary operation it
involves is the attachement of a five dimensional 2-handle to
$W\times[0,1]\,$ along a curve ${\cal C}\subset W\times\{1\}\,$. If
$\cal C$ intersects the attaching data of a ( four dimensional ) 1-handle
$\hh^1\subset W$ in exactly one point, then the surgered manifold $(W)_{\cal
C}\,$ is given by trading $\hh^1\,$ for a two handle which is attached
along an annulus surrounding $\cal C\,$.

\medskip

In Section 4.3) we also describe the presentations for categories
$\wct(N)\,$ of compact
surfaces with $N$ boundary components, which we work out completely
in [KL]. The associated
tangles contain $N$ additional strands.
A new non-trivial operation on the set $\{\wct(N)\}_N\,$  is the glue tensor product
$\otimes_{glue}\,$, where we do not just take the
disjoint unions of surfaces
but also sew them  along some of their boundary components. The main
difficulty we encounter here is that the glue tensor of two standard
surfaces is not canonically identified as a standard surfaces
anymore.
We briefly discuss the role of double categories, introduced in [KL]
to describe the two gluing operations.
\s

\paragraph{Acknowledgments:}\

I thank Daniel Altschuler, Dror Bar-Natan,
Raoul Bott, Dale Husemoller, Kiyoshi
Igusa, Vaughan Jones, Louis Kauffman, David Kazh\-dan, Robion Kirby,
Volodimir Lyu\-ba\-shen\-ko, David Radford, Nicolai Re\-she\-tik\-hin,
Justin Roberts,
Stephen Sawin, Volker Schomerus, and Vladimir Turaev  for interest, comments, and discussion.

I wish to thank Jim Harmon for helping with the conversion of eps-files.
Pictures were created with {\em FreeHand} and  {\em Xfig}.

The author is partially supported by NSF grant DMS-9305715.


TITLE: Partition function for 1D XY model
QUESTION [0 upvotes]: The Hamiltonian for 1D XY Model for $N$ Ising spins is written as, $$H=\sum_{i=1}^{N} \vec{S_i} . \vec{S_{i+1}} =\sum_{i=1}^{N} cos(\theta_i-\theta_{i+1})$$
Here we will implement periodic boundary condition as $\vec{S_{N+1}}=\vec{S_{1}}$.
Using the Transfer Matrix method we can arrive at the partition function as, $$
\begin{align}
Z&=\int_{0}^{2\pi} \Big(\prod_{i=1}^{N} d\theta_{i}\Big) \langle \theta_1 |T|\theta_2 \rangle \langle \theta_2 |T|\theta_3 \rangle\dots \langle \theta_N |T|\theta_1 \rangle\\
&=\int_{0}^{2\pi}d\theta_{1} \langle \theta_1 |T^N|\theta_1 \rangle =Tr(T^N)
\end{align}
$$
And we identify $\langle \theta_i |T|\theta_{i+1} \rangle =e^{\beta J cos(\theta_i-\theta_{i+1})}$.
Now my question is how to find out the partition function $Z(\beta J)$, given that $f(\theta_i) \sim e^{in\theta_i}$ diagonalizes the matrix? And what is the integral form of the matrix $T$?

REPLY [2 votes]: The eigenvalue corresponding to $e^{in\theta}$ is  a multiple of $I_n(\beta J)$ where $I_n$ is the modified Bessel function. Just use the definition of this function as an integral to see that this is so.


TITLE: A question about vector fields and divergence
QUESTION [0 upvotes]: I am reading the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf in order to gain a basic understanding about tensors. I had some difficulties about understanding some definitions.
1) At the end of the page 499 and at the start of 500, the definition of a field is made as a function of space coordinates $\textbf{x}=(x_1,x_2,x_3)$. If the function is a vector it is named a vector field, if it is a scalar it is named as a scalar field.
So is this definition equivalent to this: For example a vector field on $\textbf{x}=(x_1,x_2,x_3)$,of 3 dimensions is the set of vectors$$\begin{pmatrix}A_1(x_1,x_2,x_3)\\A_2(x_1,x_2,x_3)\\A_3(x_1,x_2,x_3)\end{pmatrix}$$
And a scalar vector field is just the real function $A_1(x_1,x_2,x_3)$?
2)They define the differential operator as ${\partial_a}=\frac{\partial}{\partial x_a}$ then say the divergence of the vector field $\textbf{A}=\textbf{A}(\textbf{x})=\sum_{a=1}^3 A_a(\textbf{x})\hat{x_a}$ is equal to
$$\nabla .\textbf{A} = \sum_{a=1}^3 \partial_aA_a $$
Since they defined the partial derivative in a weird way, I got confused in this statement. This looks like the dot product of differential vector $(\frac{\partial A_1(x_1,x_2,x_3)}{\partial x_1},\frac{\partial A_2(x_1,x_2,x_3)}{\partial x_2},\frac{\partial A_3(x_1,x_2,x_3)}{\partial x_3})$ and vector field evaluated at $(x_1,x_2,x_3)$, namely: $(A_1(x_1,x_2,x_3),A_2(x_1,x_2,x_3),A_3(x_1,x_2,x_3))$. Is this really what it is meant in that definiton?
Thanks in advance

REPLY [1 votes]: Right, they make it look like a vector, called the Nabla operator, $\nabla$. You can do a "dot product" with a vector field, yielding a scalar (the divergence), but also a "cross product" to yield a vector (the rotational), or a "scalar product" (with a scalar field) giving a vector (the gradient).
Nabla can be applied to the gradient of a scalar field, giving the Laplacian: symbolically, $\Delta=\nabla\nabla=\nabla^2$.
This is a notational convenience, but it makes sense.


TITLE: Does there exist a Banach space with no complemented closed subspaces?
QUESTION [9 upvotes]: I know that every Hilbert space can be decomposed as the direct sum of two non-trivial closed subspaces, eg. taking the kernel and range of any non-trivial bounded projection. But I don't know what happens in Banach spaces.
Does there exist a Banach space $B$ with no complemented closed subspaces? In other words, such that there do not exist closed subspaces $U,V\subset B$ with $B=U\oplus V$?

REPLY [9 votes]: Finite-dimensional subspaces are always complemented; this follows from the Hahn–Banach theorem.
Proof. Let $F$ be a finite-dimensional subspace of a Banach space $X$ and let $\{x_1, \ldots, x_n\}$ be a basis for $F$. Choose $f_1, \ldots, f_n\in F^*$ to be the coordinate functionals for the basis $\{x_1, \ldots, x_n\}$. Use the Hahn–Banach theorem to extend $f_j$ to bounded linear functionals on $X$; call any such extension $f_j$ too ($j\leqslant n$). Set
$$Px = \sum_{j=1}^n \langle f_j, x\rangle x_j\quad (x\in X).$$
Then $P$ is a projection with ${\rm im}\, P=F$. ${ \rm \square}$
The non-trivial variant of the question is really about the possibility of non-trivial decompositions into subspaces which are both infinite-dimensional (the spaces which lack this property exist and are called indecomposable Banach spaces). There is a whole industry concerning such spaces and there exist even spaces with a stronger property, i.e., there are spaces whose all infinite-dimensional suspaces are indecomposale (the so-called hereditarily indecomposable spaces).
There are examples of compact, Hausdorff spaces $K$ for which the Banach space $C(K)$ is indecomposale. (Of course such spaces cannot be hereditarily indecomposale as $C(K)$ always contains a copy of $c_0$ unless $K$ is finite.)
This is a nice introduction to hereditarily indecomposale spaces.


TITLE: Does probability measure on cartesian product of finite set allow "nothing" to happen?
QUESTION [0 upvotes]: That is, suppose we have a finite set $X$, and a probability measure $P:X\times X\to [0,1]$
Is it possible for "nothing" to occur?
I.e., suppose $X=\{A,B\}$,
Then we know the following must be defined: $P(A,A); P(A,B); P(B,A), P(B,B)$.
Must these sum up to 1, or is it possible that "nothing" occurs (and takes up there remaining probability).
If "nothing" is not allowed to occur, what could we change to make it possible? Add the empty set to $X$?

REPLY [0 votes]: Measure of an empty set is 0 by definition. Measure of the set X on which you define a probability is 1, also by definition of a probability measure. If you add the empty set to X, then it may be the case that the measure (probability) of the set containing an empty set ({∅}) is non-zero, but the measure of the empty set itself (∅) is still zero.


TITLE: Will the product of a real valued matrix and its transpose always have a real eigenvector?
QUESTION [2 upvotes]: I'm trying to solve this really interesting problem, taken from "Berkeley Problmes in Mathematics" by Souza and Silva
https://www.amazon.com/Berkeley-Problems-Mathematics-Problem-Books/dp/0387204296
Trying to make headway, I've noticed that if the last two statements are correct, then $M^T(Mu)=\sigma^2 u$, so in other words, there exists an eigenvector $u$ with the eigenvalue $\sigma^2$.
Therefore, I have two questions, and I would dearly love a proof or counter-example:

Is it true that $M^TM$ must have at least a (real) eigenvector?
I strongly suspect this $\sigma$ they talk of is the operator norm of $M$. Is this true?
Is this a dead-end way to try to solve this problem. Should I try something else?

Of course, everything is real-valued here.

REPLY [0 votes]: yes, obvious once you realize $M^TM$ is symmetric.
yes, can you prove it?
I think this is a good way to do it. Be somewhat careful in finding condition (2) since $v$ is a unit vector. I have not ideas besides doing this by construction from condition (3). have fun.


text:
\begin{document}

\maketitle

\begin{abstract}
In this study, we investigate generalized quasi-Einstein structure for \NCPM s.  Firstly, we deal with elementary properties and examine, existence, and characterizations of \GQE \ \NCPM. Secondly, the generalized quasi-constant curvature of \NCPM s are studied and it is proven that a \NCPM \  with generalized quasi-constant curvature is a \GQE \ manifold. Normal metric contact pair manifolds satisfying cyclic Ricci tensor and the Codazzi type of Ricci tensor are considered and its shown that a  \GQE\ \NCPM \ does not satisfy Codazzi type of Ricci tensor. Finally, we work on \NCPM s \ satisfying certain curvature conditions related to $ \mathcal{M}- $projective, conformal, and concircular curvature tensors. We show that a \NCPM\ with generalized quasi-constant curvature is  locally isometric to the Hopf manifold $  S^{2n+1}(1) \times S^1 $.
\end{abstract}
\begin{keywords*}
	\GQE,\ Contact pairs, generalized quasi-constant curvature, \\$ \mathcal{M}- $projevtice curvature tensor
\end{keywords*}
\section{Introduction}
\quad An Einstein manifold, which comes from the solution of Einstein fields equation, is defined as Ricci tensor $ Ric $ of type $ (0, 2) $ is non-zero and proportional to the metric tensor. Since Einstein manifolds have important differential geometric properties as well as their physical applications, they are studied by geometers in a wide perspective. A generalization of Einstein manifolds is quasi-Einstein manifolds. A Riemannian manifold $ M $ is called quasi-Einstein if the Ricci curvature tensor has the following form:
\begin{equation*}
Ric(X_1,X_2)=\lambda g(X,Y)+\beta\omega(X_1)\omega(X_2)
\end{equation*}
for all $ X_1,X_2 \in \Gamma(TM) $, where $ \lambda, \beta $ are scalars and $ \omega $ is a $ 1- $form. In contact geometry $ \eta- $Einstein manifolds could be seen as a special case of quasi-Einstein manifolds. Quasi-Einstein manifolds arose during the study of exact solutions of the
Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker space-times are quasi-Einstein manifolds. For more details about quasi-Einstein manifolds see \cite{de2004quasi,ghosh2006certain,tripathi2007n}.
  \par
  The generalization of quasi-Einstein manifolds has been presented in different perspectives. One of them was given by Chaki in \cite{chaki2001generalized} and another was presented by Catino \cite{catino2012generalized} with to generalize the concepts of the Ricci soliton, the Ricci almost soliton and quasi-Einstein manifolds. The third definition of \GQE\ manifolds were given by De and Ghosh \cite{de2004generalized} as follow,
  \begin{equation*}
  Ric(X_1,X_2)=\lambda g(X,Y)+\beta\omega(X_1)\omega(X_2)+\mu\eta(X_1)\eta(X_2)
  \end{equation*}
  where $ \omega, \mu $ are two non-zero $ 1- $ forms and $   \lambda, \beta$
 are certain non-zero scalars \cite{chaki2000quasi}. The unit and orthogonal vector fields $ \xi_1 $ and $ \xi_2 $ corresponding to the $ 1- $forms $ \omega  $ and $\eta $   are defined by
  $ g(X_1,\xi_1)=\omega(X_1), \ g(X_2,\xi_2)=\eta(X_1) $, respectively  \cite{de2004generalized}. The geometric properties of \GQE \ manifolds have been studied in \cite{de2004generalized,sular2012characterizations,de2011existence,prakasha2014some,de2015generalized}.
  A \GQE\ manifold , in addition to its geometrical features, it also has physical applications in general relativity \cite{mallick2016class,guler2016study,de2019generalized}. Complex $ \eta-$Einstein manifolds are a special case of \GQE\ manifolds (see \cite{turgutvanlicomplexeta2017}). \par

  In \cite{bande2005contact} Bande and Hadjar defined a new contact structure on a differentiable $ (m=2p+2q+2)-$dimensional manifold $ M $ with two $ 1- $forms $ \alpha_1,\alpha_2 $.  This structure was firstly studied by Blair, Ludden and Yano \cite{blair1974geometry} as the name of bicontact manifolds. Bande and Hadjar considered a special type of $ f- $structure with complementary frames related to these contact forms and they obtained associated metric. A differentiable manifold with this structure is called metric contact pair (MCP) manifold . Riemannian geometry of MCP manifolds is given in \cite{bande2010normal,bande2009contact}. \par
This paper is on applications of \GQE\ manifolds in contact geometry. We consider the \NCPM\ admits \GQE\ structure and call this type of manifold as \GQE\ \NCPM s. After presenting definitions and basic properties, we examine the existence of such manifolds. Also, we present a characterization of \GQE \ \NCPM s.\ Moreover, we consider the notion of generalized quasi-constant curvature for \NCPM s and we obtain some results on the sectional curvature.  We investigate a generalized quasi-Einstein normal metric contact pair manifold under some conditions for Ricci tensor.
Finally, we study on $ M-$projective curvature tensor, conformal curvature tensor and concircular curvature tensor on a \NCPM\ and we gave some results under flatness and symmetry conditions.

\section{Preliminaries}
\quad Contact pairs were defined by Bande and Hadjar \cite{bande2005contact} in 2005, for details see \cite{bande2005contact,bande2009contact,bande2010normal}. In this section we give some fundamental facts about contact metric pair manifolds.

\begin{definition}
	A pair of two $  1-$forms, $ (\alpha_1,\alpha_2) $ on an $ (m=2p+2q+2) $ -dimensional differentiable manifold $ M $, is called contact pair of type $ (p,q) $ if we have
	\begin{itemize}
		\item $ \alpha_1 \wedge (d\alpha_1)^p\wedge\alpha_2 \wedge (d\alpha_2)^q\neq0 $,
		\item  $ (d\alpha_1)^{p+1}=0 $ and $ (d\alpha_2)^{q+1}=0, $
	\end{itemize}
	where $ p,q $ are positive integers \cite{bande2005contact}.
\end{definition}
For $ 1- $forms $ \alpha_1 $ and $ \alpha_2 $, we have two integrable subbundle of $ TM $; $ \mathcal{D}_{1}=\{X:\alpha_1(X)=0, X\in \Gamma(TM)\} $ and  $ \mathcal{D}_{2}=\{X:\alpha_2(X)=0, X\in \Gamma(TM)\} $. Then we have two characteristic foliations of $ M $, denoted $ \mathcal{F}_1=\mathcal{D}_1 \cap kerd\alpha_1$ and $ \mathcal{F}_2= \mathcal{D}_2 \cap kerd\alpha_2$ respectively. $ \mathcal{F}_1 $ and $ \mathcal{F}_2 $ are $( 2p+1) $ and $ (2q+1)-$dimensional contact manifolds with contact form induced by $ \alpha_2 $ and $ \alpha_1 $. Thus we can define $ (2p+2q)- $dimensional horizontal subbundle $ \mathcal{H}=ker\alpha_1\cap ker\alpha_2$.
To a contact pair $ (\alpha_1,\alpha_2) $ of type $ (p,q) $ there are associated two commuting vector fields $ Z_1 $ and $ Z_2 $, called Reeb vector fields of the pair, which are uniquely determined by the following equations:
\begin{eqnarray*}
	&\alpha_1(Z_1)=\alpha_2(Z_2)=1, \ \alpha_1(Z_2)=\alpha_2(Z_1)=0\\&
	i_{Z_1}d\alpha_1 =i_{Z_1}d\alpha_2=i_{Z_2}d\alpha_2=0
\end{eqnarray*}
where $ i_X $ is the contraction with the vector field X. In particular, since the Reeb vector fields
commute, they determine a locally free $ \mathbb{R}^2 $-action, called the Reeb action. \par
The tangent bundle of $ (M,(\alpha_1,\alpha_2)) $ can be split into as follow. For the two subbundle of $ TM $
\begin{eqnarray*}
	T\mathcal{G}_i=ker d\alpha_i\cap ker \alpha_1 \cap ker \alpha_2, \ i=1,2
\end{eqnarray*}
and we can write
\begin{equation*}
T\mathcal{F}_i=T\mathcal{G}_i\oplus \mathbb{R}Z_1.
\end{equation*}
Therefore we get $ TM=T\mathcal{G}_1\oplus T\mathcal{G}_2\oplus \mathbb{R}Z_1\oplus \mathbb{R}Z_2$. The horizontal subbundle can be written as  $ \mathcal{H}=T\mathcal{G}_1\oplus T\mathcal{G}_2 $. Also, we write  $ \mathcal{V}= \mathbb{R}Z_1\oplus \mathbb{R}Z_2$, we call $ \mathcal{V} $ is  vertical subbundle of $ TM. $ \par
Let $ X $ be an arbitrary vector field on $ M $. We can write $ X=X^{\mathcal{H}}+X^{\mathcal{V}} $, where $ X^{\mathcal{H}}, X^{\mathcal{V}} $ horizontal and vertical component of $ X $ respectively. We call a vector field $ X $ as horizontal vector field if $ X \in \Gamma(\mathcal{H})$, and vertical vector field if $ X \in \Gamma(\mathcal{V})$. \par
An almost contact structure is related to contact form and it is defined with tensorial viewpoint. Many geometric properties of contact manifolds examined by this contact structure. Similarly an almost contact pair structure has been defined as follow by Bande and Hadjar\cite{bande2009contact}.
\begin{definition}
	An almost contact pair structure on a $ (m=2p+2q+2)- $dimensional differentiable manifold $ M $ is a triple $ \alpha_1, \alpha_2, \phi $, where $ (\alpha_1, \alpha_2) $ is a contact pair and $ \phi $ is a $ (1,1) $ tensor field such that:
	\begin{eqnarray}
	\phi^2=-I+\alpha_1\otimes Z_1+\alpha_2\otimes Z_2
	,\ \ \phi Z_1=\phi Z_2=0.
	\end{eqnarray}
	The rank of $ \phi $ is $ (2p+2q) $ and $ \alpha_i(\phi)=0 $ for $ i=1,2 $ \cite{bande2009contact}.
\end{definition}
The endomorphism $\phi $ is said to be decomposable, i.e $ \phi=\phi_1+\phi_2 $, if $ T\mathcal{F}_i $ is invariant under $ \phi $. If $ \phi $ is decomposable then  $ (\alpha_i, Z_i, \phi) $ induce an almost contact structure on $ \mathcal{F}_j$ for $ i\neq j$ \cite{bande2005contact}. The decomposability of $ \phi $ do not satisfy for every almost contact pair structure. An example was given in \cite{bande2005contact}, has an almost contact pair structure but $ \phi $ is not decomposable. In this study we assume that $ \phi $ is decomposable.
\begin{definition}
	Let $ (\alpha_1,\alpha_2,Z_1,Z_2,\phi) $ be an almost contact pair structure on a manifold $ M $. A Riemannian metric $ g $ is called
	\begin{enumerate}
		\item [$ \bullet $] compatible if $ g(\phi X_1,\phi X_2)=g(X_1,X_2)-\alpha_1(X_1)\alpha_1(X_2)-\alpha_2(X_1)\alpha_2(X_2) $ for all $ X_1,X_2 \in TM $,
		\item [$ \bullet $] associated if $ g(X_1,\phi X_2)=(d\alpha_1+d\alpha_2)(X_1,X_2) $ and $ g(X_1,Z_i)=\alpha_i(X_1) ,$ for $ i=1,2 $ and for all $ X_1,X_2 \in \Gamma(TM) $
	\end{enumerate}
\end{definition}
\noindent
$ 4- $tuple $ (\alpha_1,\alpha_2,\phi, g) $ is called a metric almost contact pair on a manifold $ M $ and g is an associated metric with respect to contact pair structure $ (\alpha_1,\alpha_2,\phi) $. We recall $ (M,\phi,Z_1,Z_2,\alpha_1,\alpha_2,g) $ is a metric almost contact pair manifold.\par
We have following properties for a metric almost contact pair manifold $ M $ \cite{bande2005contact}:
\begin{eqnarray}
&g(Z_i,X)=\alpha_i(X), \ \  g(Z_i,Z_j)=\delta_{ij} \\& \nabla_{Z_i}Z_j=0, \ \nabla_{Z_i}\phi=0
\end{eqnarray}
and for every $ X $ tangent to $ \mathcal{F}_i \,\ i=1,2$ we have
\begin{equation*}
\nabla_ X Z_1=-\phi_1  X , \ \ \nabla_ X Z_2=-\phi_2  X
\end{equation*}
where $ \phi=\phi_1+\phi_2 $. Normality of metric almost contact pair manifold is given by Bande and Hadjar \cite{bande2010normal}. For a metric almost contact pair manifold, we have two almost complex structure:
\begin{equation*}
J=\phi-\alpha_2\otimes Z_1+\alpha_1\otimes Z_2,\ \ T=\phi+\alpha_2\otimes Z_1-\alpha_1\otimes Z_2.
\end{equation*}
\begin{definition}
	A metric almost contact pair manifold is said to be normal if $ J $ and $ T $ are integrable \cite{bande2010normal}.
\end{definition}
\begin{theorem}
	Let $ (M,\phi,Z_1,Z_2,\alpha_1,\alpha_2,g) $ be a normal metric contact pair manifold then we have
	\begin{equation}\label{Phicovariantderivation}
	g((\nabla_{X_1}\phi)X_2,X_3)=\sum_{i=1}^{2}(d\alpha_i(\phi X_2,X_1)\alpha_i(X_3)-d\alpha_i(\phi X_3,X_1)\alpha_i(X_2))
	\end{equation}
	where $ X_1,X_2,X_3 $ are arbitrary vector fields on $ M $ \cite{bande2009contact}.
\end{theorem}

We  use the following statements for the Riemann curvature;
\begin{eqnarray*}
	&R(X_1,X_2)X_3=\nabla_{X_1}\nabla_{X_2}X_3-\nabla_{X_2}\nabla_{X_1}X_3-\nabla_{[X_1,X_2]}X_3,\\& R(X_1,X_2,X_3,X_4)=g(R(X_1,X_2)X_3,X_4).
\end{eqnarray*}
for all $ X_1,X_2,X_3,X_4 \in \Gamma(TM) $. Curvature properties of \NCPM s\ were given in \cite{bande2015bochner}. For  $ X_1,X_2, X_3 \in \Gamma(M) $ and $ Z=Z_1+Z_2 $ Reeb vector field, we have
\begin{eqnarray}\label{R(X,Z)Y}
R(X_1,Z)X_2&=&-g(\phi X_1,\phi X_2)Z,
\\
\label{R(X,Y,Z,V)}
R(X_1,X_2,Z,X_3)&=&d\alpha_1(\phi X_3,X_1)\alpha_1(X_2)+d\alpha_2(\phi X_3, X_1)\alpha_2(X_2)\\&&-d\alpha_1(\phi X_3,X_2)\alpha_1(X_1)-d\alpha_2(\phi X_3,X_2)\alpha_2(X_1) \notag,\\
\label{R(X,Z)Z}
R(X_1,Z)Z&=&-\phi^2X_1.
\end{eqnarray}
Let take an orthonormal basis of $ M $ by
\begin{equation*}
S=\{e_1,e_2,...,e_p,\phi e_1,\phi e_2,...,\phi e_p,e_{p+1},e_{p+2},...,e_{p+q}, \phi e_{p+1},\phi e_{p+2},...,\phi e_{p+q} , Z_1,Z_2\}.
\end{equation*}
Then for all $ X_1\in \Gamma(TM) $ we get the Ricci curvature of $ M $ as
\begin{equation*}
Ric(X_1,Z)=\sum_{i=1}^{2p+2q}{d\alpha_1(\phi E_i, E_i)\alpha_1(X)+d\alpha_2(\phi E_i, E_i)\alpha_2(X)}.
\end{equation*}
where $ E_i\in S , \ \ 1\leq i \leq 2p+2q+2$. So, we obtain
\begin{eqnarray}
&Ric(X_1,Z)=0,\ \ \text{for } X_1\in \Gamma(\mathcal{H}) \label{Ric(X_1,Z)},\\
&Ric(Z,Z)=2p+2q.\label{Ric(Z,Z)}\\
&Ric(Z_1,Z_1)=2p , \ Ric(Z_2,Z_2)=2q , \ Ric(Z_1,Z_2)=0.
\end{eqnarray}
\section{Existence and Characterization of Generalized Quasi-Einstein Normal Metric Contact Pairs}
\begin{definition}
	Let $ M $ be a \NCPM. \ $ M $ is called \GQE \ \NCPM \ if the Ricci curvature of $ M $ has following form;
	\begin{equation*}
	Ric(X_1,X_2)=\lambda g(X_1,X_2)+\beta \alpha_1(X_1)\alpha_1(X_2)+\mu \alpha_2(X_1)\alpha_2(X_2)
	\end{equation*}
	for functions  $ \lambda, \beta, \mu $ on $ M $ and all $ X_1,X_2 \in \Gamma(TM) $.
\end{definition}
If we set $ X_1=X_2=Z_1 $ and $ X_1=X_2=Z_2 $, respectively, we obtain $ \beta=2p-\lambda$ and $ \mu=2q-\lambda $. Thus the Ricci curvature of \GQE \ \NCPM \ is given by
\begin{equation}\label{GQEexpression}
Ric(X_1,X_2)=\lambda g(X_1,X_2)+(2p-\lambda) \alpha_1(X_1)\alpha_1(X_2)+(2q-\lambda) \alpha_2(X_1)\alpha_2(X_2)
\end{equation}
for all $ X_1,X_2 \in \Gamma(TM) $ and the scalar curvature is
\begin{equation}\label{scalar}
scal=2(\lambda+1)(p+q).
\end{equation}
Let $ X $ be an arbitrary vector field on $ M $. We can write $ X=X^{\mathcal{H} }+X^{\mathcal{V}}$. Since the Ricci curvature is a linear tensor we have
\begin{equation*}
Ric(X_1,X_2)=Ric(X^{\mathcal {H}}_1,X^{\mathcal {H}}_2)+Ric(X^{\mathcal {V}}_1,X^{\mathcal {V}}_2).
\end{equation*}
With take consider the decomposition of tangent bundle mentioned above (see \cite{unal2019some} for details), we get
\begin{equation*}
Ric(X^{\mathcal {V}}_1,X^{\mathcal {V}}_2)= 2p\alpha_1(X_1)\alpha_1(X_2)+2q\alpha_2(X_1)\alpha_2(X_2).
\end{equation*}
Thus, we reach following useful result.
\begin{proposition}\label{horizontaliseinstein}
	A \NCPM\ is \GQE\ manifold if and only if the horizontal bundle is Einstein, that is for a  function $ \lambda $ on M, we have $ Ric(X^{\mathcal {H}}_1,X^{\mathcal {H}}_2)=\lambda g(X^{\mathcal {H}}_1,X^{\mathcal {H}}_2) $.
\end{proposition}
In \cite{de2004generalized} De and Ghosh gave a theorem for the existence of a \GQE\ Riemannian manifold.
\begin{theorem}
	If the Ricci tensor $ Ric $ of a Riemannian manifold satisfies the relation
	\begin{eqnarray}\label{existence}
	Ric(X_2,X_3)Ric(X_1,X_4)-Ric(X_1,X_3)	Ric(X_2,X_4)&=&\gamma[g(X_2,X_3)g(X_1,X_4)\\&&-g(X_1,X_3)g(X_2,X_4)]\nonumber
	\end{eqnarray}
	where $ \gamma $ is a non-zero scalar, then the manifold is a generalized quasi Einstein manifold \cite{de2004generalized}.
\end{theorem}
Assume that (\ref{existence}) is satisfied on a \NCPM \ $ M $. By setting $ X_1=X_4=Z $ and $ X_2,X_3 \in \Gamma(\mathcal{H}) $, and from (\ref{Ric(X_1,Z)}) we have
\begin{equation*}
Ric(X_2,X_3)=\frac{2\gamma}{m-2}g(X_2,X_3).
\end{equation*}
Thus, with consider Proposition \ref{horizontaliseinstein} $ M $ is a \GQE\ manifold. Using (\ref{scalar}), we get $ \gamma=\frac{1}{2}(scal-m+2) $ and finally we state;
\begin{corollary}
	Let $ M $ be \NCPM\ with scalar curvature $ scal\neq m-2 $. If we have the relation
	\begin{eqnarray*}
		Ric(X_2,X_3)Ric(X_1,X_4)-Ric(X_1,X_3)Ric(X_2,X_4)&=& \frac{1}{2}(scal-m+2)[g(X_2,X_3)g(X_1,X_4)\\&&-g(X_1,X_3)g(X_2,X_4)]\
	\end{eqnarray*}
	on $ M $ for all $ X_1,X_2,X_3,X_4\in \Gamma(TM) $, then $ M $ is a \GQE\ manifold.
\end{corollary}
The sectional curvature of a Riemann manifold gives us many important geometric properties.

The characterizations of Einstein \cite{chen2000characterizations,dumitru2007einstein},  quasi-Einstein \cite{bejan2007characterization} and \GQE   \ \cite{sular2012characterizations,dumitru2012characterization}  manifolds have been obtained by using the sectional curvature of subspaces of tangent bundle. We take into consideration Theorem 2.2. in \cite{sular2012characterizations}, and thus we obtain following theorem. Firstly we should to give some fundamental facts.\par
Let $ \pi $ be a plane section in $ T_QM $ for any $ Q\in M $. The sectional curvature of $ \pi $ is given as $ Sec(\pi) = Sec(u\wedge v) $, where $ u,v $ orthonormal vector fields . For any
$( p+q )$-dimensional subspace $ \mathcal{L} \subset T_QM, \ 2 \leq p+q \leq m$, its scalar curvature $ scal(\mathcal{L})$ is
denoted by
\begin{equation}
scal(\mathcal{L})=\sum_{1\leq i,j \leq p+q}Sec(E_i\wedge E_j)
\end{equation}
\noindent
where $ {E_1, ..., E_n} $ is any orthonormal basis of $\mathcal{L}$ \cite{chen2000some}. When $ \mathcal{L}=T_QM $, the
scalar curvature  is just the scalar curvature  $ scal(Q) $ of $ M $ at $
Q \in M$.

\begin{theorem}
	An $ (m=2p+2q+2)- $dimensional \NCPM\ is a \GQE \ manifold if and only if there exist a function $ \lambda $ on $ M $ satisfying
	\begin{eqnarray*}
		scal(P)+p+q-\lambda&=&\sec(P^{\bot}), \ \ Z_1,Z_2 \in T_QP^{\bot}\\
		scal(N)+p+q&=&\sec(N^{\bot}), \ \ Z_1,Z_2 \in T_QN^{\bot}\\
		scal(R)+q-p&=&\sec(P^{\bot}), \ \ Z_1\in T_QR,Z_2 \in T_QR^{\bot}
	\end{eqnarray*}
	where $ (p+q+1)-$plane sections $ P,R $ and $ (p+q)-$plane section $ N $; $ P^{\bot},N^{\bot} $ and $ R^{\bot} $ denote the orthogonal complements of $ P,N $ and $ R $ in $ T_QM $, respectively.
\end{theorem}

\section{Generalized Quasi-constant Curvature of Normal Metric Contact Pair Manifolds}
\quad The notion of quasi-constant curvature was defined by Chen and Yano \cite{chen1972hypersurfaces}. De and Ghosh generalized this notion for a Riemannian manifold. Let $ M $ be a \NCPM. \ $ M $ is called a \NCPM \ of generalized quasi-constant curvature if the Riemannian curvature tensor of $ M $ satisfying;
\begin{eqnarray}
R(X_1,X_2,X_3,X_4)&=&A[g(X_2,X_3)g(X_1,X_4)-g(X_1,X_3)g(X_2,X_4)]\label{gen.quasiconscurv}\\&& B[g(X_1,X_4)\alpha_1(X_2)\alpha_1(X_3)-g(X_1,X_3)\alpha_1(X_2)\alpha_1(X_4)\notag\\&&+g(X_2,X_3)\alpha_1(X_1)\alpha_1(X_4)-g(X_2,X_4)\alpha_1(X_1)\alpha_1(X_3)]\notag\\&& C[g(X_1,X_4)\alpha_2(X_2)\alpha_2(X_3)-g(X_1,X_3)\alpha_2(X_2)\alpha_2(X_4)\notag\\&&+g(X_2,X_3)\alpha_2(X_1)\alpha_2(X_4)-g(X_2,X_4)\alpha_2(X_1)\alpha_2(X_3)]\notag
\end{eqnarray}
for all $ X_1,X_2,X_3,X_4 \in \Gamma(TM)$, where $ A,B $ and $ C $ are scalar functions. \par
\begin{proposition}
	Let $ M $ be a \NCPM \ of generalized quasi-constant curvature. Then, we have followings;
	\begin{enumerate}
		\item [$ \bullet $] the sectional curvature of horizontal bundle is $ A $,
		\item [$ \bullet $] the sectional curvature of plane section spanned by $ X\in \Gamma(\mathcal{H}) $ and $ Z $ is $ 2A+B+C $,
		\item [$ \bullet $] the sectional curvature of plane section spanned by $ X\in \Gamma(\mathcal{H}) $ and $ Z_1,Z_2$ is $ A+B$ and $ A+C $, respectively.
	\end{enumerate}

\end{proposition}
\begin{proof}
	Let take $ X_1=X_4=X , X_2=X_4=X^{\prime}$, where $ X,X^{\prime} $ unit and mutually orthogonal horizontal vector fields. Then from (\ref{gen.quasiconscurv}) we get
	\begin{eqnarray*}
		sec(X,X^{\prime})&=&A[g(X^{\prime},X^{\prime})g(X,X)-g(X,X^{\prime})g(X^{\prime},X)]\\&=&A
	\end{eqnarray*}
	For $ X_1=X_4=X , X_2=X_4=Z$ for unit horizontal vector field $ X $ we get
	\begin{eqnarray*}
		sec(X,Z)&=&A[g(Z,Z)g(X,X)-g(X,X^{\prime})g(X^{\prime},X)-g(X,Z)g(Z,X)]\\&&+B(g(X,X)\alpha_1(Z)\alpha_1(Z)+Cg(X,X)\alpha_2(Z)\alpha_2(Z)\\&=&2A+B+C
	\end{eqnarray*}
	By following same steps one can obtain other results.
\end{proof}
From above proposition we get
\begin{corollary}
	\NCPM \ of generalized quasi-constant curvature, we have
	\begin{equation*}
	sec(X,Z)=sec(X,Z_1)+sec(X,Z_2)
	\end{equation*}
	for any horizontal and unit vector field $ X $.
\end{corollary}
We can also characterize a \GQE \ \NCPM \  of generalized quasi-constant curvature. We get following result.
\begin{theorem}
	A \NCPM \ of generalized quasi-constant curvature is a \GQE \ manifold with coefficients $\lambda_1=A(m-1)+B+C, \ \lambda_2=B(m-2)$, and $ \lambda_3=C(m-2)$.
\end{theorem}
\begin{proof}
	Let $ M $ be \NCPM \ of generalized quasi-constant curvature.  Take an orthonormal basis of $ M $ as
	\begin{eqnarray*}
		S=\{e_1,e_2,...,e_p,\phi_1e_1,\phi_1e_2,...,\phi_1e_p, e_{p+1} ,e_{p+2},...,e_{p+q}, \phi_2e_{p+1} ,\phi_2e_{p+2},...,\phi_2e_{p+q}, Z_1,Z_2\}.
	\end{eqnarray*}
	By taking sum of (\ref{gen.quasiconscurv}) from $ i=1 $ to $ i=2p+2q+2 $ for $ X_2=X_3=E_i \in S$, we obtain
	\begin{eqnarray*}
		\sum_{i=1}^{2p+2q+2}R(X_1,E_i,E_i,X_4)&=&\sum_{i=1}^{2p+2q+2}\{A[g(E_i,E_i)g(X_1,X_4)-g(X_1,E_i)g(E_i,X_4)]\\&& B[g(X_1,X_4)\alpha_1(E_i)\alpha_1(E_i)-g(X_1,E_i)\alpha_1(E_i)\alpha_1(X_4)\notag\\&&+g(E_i,E_i)\alpha_1(X_1)\alpha_1(X_4)-g(E_i,X_4)\alpha_1(X_1)\alpha_1(E_i)]\notag\\&& C[g(X_1,X_4)\alpha_2(E_i)\alpha_2(E_i)-g(X_1,E_i)\alpha_2(E_i)\alpha_2(X_4)\notag\\&&+g(E_i,E_i)\alpha_2(X_1)\alpha_2(X_4)-g(E_i,X_4)\alpha_2(X_1)\alpha_2(E_i)]\}\notag.
	\end{eqnarray*}
	For $ 1\leq i \leq 2p+2q $ since  $ \alpha_j(E_i)=0 $, $j =1,2 $ and $ \sum_{i=1}^{2p+q+2}g(X_1,E_i)g(E_i,X_2)=g(X_1,X_2)$ we get
	\begin{eqnarray*}\label{genconstantresult}
		Ric(X_1,X_4)&=&[A(m-1)+B+C]g(X_1,X_4)+B(m-2)\alpha_1(X_1)\alpha_1(X_4)\\&&+C(m-2)\alpha_2(X_1)\alpha_2(X_4)\notag
	\end{eqnarray*}
	which completes the proof.
\end{proof}
\section{Normal Metric Contact Pair Manifold Satisfying Cyclic Ricci Tensor}
\quad A Riemann manifold satisfies cyclic Ricci tensor if we have
\begin{equation*}
(\nabla_{X_1}Ric)(X_2,X_3)+(\nabla_{X_2}Ric)(X_3,X_1)+(\nabla_{X_3}Ric)(X_1,X_2)=0
\end{equation*}
for all $ X_1,X_2,X_3 \in \Gamma(TM) $. De and Mallick proved that a \GQE\ Riemann manifold satisfies cyclic Ricci tensor if generators of the manifolds are Killing \cite{de2015generalized}. \par
The characteristic vector fields of a \NCPM\ $ Z_1,Z_2 $ are Killing vector fields \cite{bande2013symmetry}. Thus, by easy computations, we get
\begin{eqnarray}
(\nabla_{X_1}\alpha_1)X_2+(\nabla_{X_2}\alpha_1)X_1=0 \label{covdervalpha1},\ \  (\nabla_{X_1}\alpha_2)X_2+(\nabla_{X_2}\alpha_2)X_1=0.
\end{eqnarray}
Let $ M $ be a \GQE\ \NCPM. \ It is well known that
\begin{equation*}
(\nabla_{X_1}Ric)(X_2,X_3)=\nabla_{X_1}Ric(X_2,X_3)-Ric(\nabla_{X_1}X_2,X_3)-Ric(X_2,\nabla_{X_1}X_3).
\end{equation*}
Then, from (\ref{GQEexpression}) we obtain
\begin{eqnarray}\label{nablaRic}
(\nabla_{X_1}Ric)(X_2,X_3)&=&X_1[\lambda]g(\phi X_2,\phi X_3)\\&&+(2p-\lambda)[\left((\nabla_{X_1}\alpha_1)X_3\right)\alpha_1(X_3)+\left((\nabla_{X_1}\alpha_1)X_2\right)\alpha_1(X_2)] \notag \\&&(2q-\lambda)[\left((\nabla_{X_1}\alpha_1)X_3\right)\alpha_2(X_2)+\left((\nabla_{X_1}\alpha_1)X_2\right)\alpha_2(X_3)]\notag
\end{eqnarray}
where $ X_1[\lambda] $ is the derivation of $ \lambda $ in the direction of $ X_1 $. Thus from (\ref{covdervalpha1}) we obtain
\begin{eqnarray*}
	(\nabla_{X_1}Ric)(X_2,X_3)+(\nabla_{X_2}Ric)(X_3,X_1)+(\nabla_{X_3}Ric)(X_1,X_2)&&\\=
	X_1[\lambda]g(\phi X_2,\phi X_3)+X_2[\lambda]g(\phi  X_3,\phi X_1)+X_3[\lambda]g(\phi X_1, \phi X_2).
\end{eqnarray*}
As a consequence , we can state following theorem.
\begin{theorem}\label{cyclicRicciTen.Theo}
	Let $ M $ be \GQE \ \NCPM. \ If $ \lambda  $ is constant then $ M $ satisfies cyclic Ricci tensor.
\end{theorem}
A Riemann manifold satisfies Codazzi type of Ricci tensor if
\begin{equation*}
(\nabla_{X_1}Ric)(X_2,X_3)-(\nabla_{X_2}Ric)(X_1,X_3)=0
\end{equation*}
for all $ X_1\,X_2 $ vector fields on $ M $. In \cite{de2015generalized}
it has been proven that if a \GQE\ Riemann manifold satisfies Codazzi type of Ricci tensor, then the associated 1-forms are closed. \par
Suppose that Ricci tensor $ Ric $ of a \NCPM\ $ M $ is Codazzi type. Then, from (\ref{covdervalpha1}) and (\ref{nablaRic}) we obtain
\begin{eqnarray*}
	&&(2p-\lambda)[((\nabla_{X_1}\alpha_1)X_2-(\nabla_{X_2}\alpha_1)X_1)\alpha_1(X_3) \\&&+ ((\nabla_{X_1}\alpha_1)X_3\alpha_1(X_2)+(\nabla_{X_2}\alpha_1)X_3\alpha_1(X_1))]\\&&+(2q-\lambda)[((\nabla_{X_1}\alpha_1)X_2-(\nabla_{X_2}\alpha_1)X_1)\alpha_2(X_3) \\&&+ ((\nabla_{X_1}\alpha_1)X_3\alpha_2(X_2)+(\nabla_{X_2}\alpha_1)X_3\alpha_2(X_1))]=0.
\end{eqnarray*}
Let take $ X_3=Z_1 $, then we get
\begin{equation*}
(2p-\lambda)((\nabla_{X_1}\alpha_1)X_2-(\nabla_{X_2}\alpha_1)X_1)=0
\end{equation*}
which implies $ \lambda=2p $ or $ (\nabla_{X_1}\alpha_1)X_2-(\nabla_{X_2}\alpha_1)X_1=0 $. If $ \lambda=2p $ the manifold is not \GQE ,\ so this case is not possible. In the other case we obtain
\begin{equation*}
0=(\nabla_{X_1}\alpha_1)X_2-(\nabla_{X_2}\alpha_1)X_1=d\alpha_1(X_1,X_2)=0
\end{equation*}
and so $ \alpha_1 $ is closed. Similarly by choosing $ X_3=Z_2  $ we obtain $ \alpha_2$ is closed. As we know contact pairs $ (\alpha_1,\alpha_2) $ are not closed. So our assumption is not valid. Finally we conclude that
\begin{theorem}
	A \GQE\ \NCPM \ do not satisfy Codazzi type of Ricci tensor.
\end{theorem}
\section{Normal Metric Contact Pair Manifold Satisfying Certain curvature Conditions }
\quad  Curvature tensors give us many geometric properties of contact manifolds. Some properties of \NCPM\ satisfying certain conditions of curvature tensors were given in \cite{bande2015bochner,unal2019some}. In this section we examine the $ \mathcal{M}- $projective curvature tensor $ \mathcal{W} $, conformal curvature tensor $ \mathcal{C} $ and concircular curvature tensor $ \mathcal{Z} $ on a \NCPM. \par
Let $ M $ be an $ (m=2p+2q+2)- $dimensional \NCPM. \ $ \mathcal{M}- $projective curvature tensor of $ M $ is given by ,
\begin{eqnarray}\label{m-projective}
\mathcal{W}( X_1,X_2)X_3&=&R( X_1,X_2)X_3-\frac{1}{2(m-1)}[Ric(X_2,X_3)X_1\\&&-Ric(X_1,X_3)X_2+g(X_2,X_3)QX_1  -g(X_1,X_3)QX_2]\notag
\end{eqnarray}
for $ X_1,X_2,X_3 \in \Gamma(TM)$, where $ Q $ is Ricci operator is given by $ Ric(X_1,X_2)=g(QX_1,X_2) $  \cite{pokhariyal1970curvature}.  $ \mathcal{M}- $projective curvature tensor on manifolds with different structures studied by many authors \cite{ayar2019m,prakasha2020M,zengin2014m}.\par
From (\ref{m-projective}), we have
\begin{eqnarray}
\mathcal{W}( X_1,Z)Z&=&\frac{m}{2m-1}X_1-\frac{1}{m-1}QX_1 \label{M(X,Z)Z}\\
\mathcal{W}( X_1,X_2)Z&=&R(X_1,X_2)Z \label{M(X,Y)Z}\\
\mathcal{W}( X_1,Z)X_2&=&\left[\frac{(2m-1)(m-2)}{2(m-1)}g(X_1,X_2)+\frac{1}{2(m-1)}Ric(X_1,X_2)\right]Z\label{M(X,Z)Y}
\end{eqnarray}
for $ X_1,X_2,X_3 \in \Gamma(\mathcal{H}) $. Also, since $ Ric(X_1,X_2)=g(QX_1,X_2) $, where $ Q $ is the Ricci operator, we have
\begin{eqnarray}
\mathcal{W}( X_1,X_2, X_3 , X_4)&&=R( X_1,X_2, X_3 , X_4)-\frac{1}{2(m-1)}[Ric(X_2,X_3)g(X_1,X_4)\label{m-projec}\\&&-Ric(X_1,X_3)g(X_2,X_4)+g(X_2,X_3)Ric(X_1,X_4)\notag \\&& -g(X_1,X_3)Ric(X_2,X_4)]\notag.
\end{eqnarray}
for all $X_1,X_2,X_3 \in \Gamma(M)  $.
$ M $  is called $ \mathcal{M}- $projective flat if $ \mathcal{W} $ vanishes identically on $ M $.
\begin{theorem}
	A \GQE \ \NCPM\ is $ \mathcal{M}- $projective flat if and only if it is of generalized quasi-constant curvature.
\end{theorem}
\begin{proof}
	Suppose that $ M $ is a \GQE \ manifold. Then, from (\ref{GQEexpression}) and (\ref{m-projective}) we have
	\begin{eqnarray}\label{mprojectflat}
	\mathcal{W}( X_1,X_2, X_3 , X_4)&=&R(X_1,X_2,X_3,X_4)\\&&-\frac{\lambda}{m-1}[g(X_2,X_3)g(X_1,X_4)-g(X_1,X_3)g(X_2,X_4)]\notag\\&&- \frac{2p-\lambda}{2(m-1)}[g(X_1,X_4)\alpha_1(X_2)\alpha_1(X_3)-g(X_1,X_3)\alpha_1(X_2)\alpha_1(X_4)\notag\\&&+g(X_2,X_3)\alpha_1(X_1)\alpha_1(X_4)-g(X_2,X_4)\alpha_1(X_1)\alpha_1(X_3)]\notag\\&&- \frac{2q-\lambda}{2(m-1)}[g(X_1,X_4)\alpha_2(X_2)\alpha_2(X_3)-g(X_1,X_3)\alpha_2(X_2)\alpha_2(X_4)\notag\\&&+g(X_2,X_3)\alpha_2(X_1)\alpha_2(X_4)-g(X_2,X_4)\alpha_2(X_1)\alpha_2(X_3)]\notag.
	\end{eqnarray}
	Thus it is seen that $ M $ is $ \mathcal{M}- $projective flat if and only if  $ M $ is of generalized quasi-constant curvature with coefficients $ A= \frac{\lambda}{m-1}, \ B=\frac{2p-\lambda}{2(m-1)}$  and $ C=\frac{2q-\lambda}{2(m-1)}$.
\end{proof}
The Riemann manifolds satisfying $ R(X_1,X_2).R=0 $ are called semi-symmetric, where $ R(X_1,X_2) $ acts on $ R $ as a derivation. Semi-symmetric contact manifolds were studied  by Perrone \cite{contactRiemanPerrone}.  Similarly, if $ \mathcal{W}(X_1,X_2).R=0$ then $ M $ is called $ \mathcal{M}- $projective semi-symmetric.  $ \mathcal{W}(X_1,X_2).R$ is defined as
\begin{eqnarray}\label{M.Rgeneral}
(\mathcal{W}(X_1,X_2).R)(X_3,X_4)X_5&= \mathcal{W}(X_1,X_2).R(X_3,X_4)X_5-R(\mathcal{W}(X_1,X_2),X_3,X_4)X_5\\&
-R(X_3,\mathcal{W}(X_1,X_2)X_4)X_5-R(X_3,X_4)\mathcal{W}(X_1,X_2)X_5\notag
\end{eqnarray}
for all $ X_1,X_2,X_3,X_4,X_5\in \Gamma(TM) $.  Also, we have
\begin{eqnarray}\label{M.Ricgeneral}
(\mathcal{W}(X_1,X_2).Ric)(X_3,X_4)=-Ric(\mathcal{W}(X_1,X_2),X_3)-Ric(X_3,\mathcal{W}(X_1,X_2)X_4).
\end{eqnarray}
If $ \mathcal{W}(X_1,X_2).Ric=0 $ then $ M $ is called $ \mathcal{M}- $projective Ricci semi-symmetric.
\begin{theorem}
	A \NCPM \ is $ \mathcal{M}- $projective semi-symmetric if and only if $ M $ is \GQE \ manifold.
\end{theorem}
\begin{proof}
	From (\ref{M.Rgeneral}) and using (\ref{M(X,Z)Z}), (\ref{M(X,Y)Z}) and (\ref{M(X,Z)Y}) we obtain
	\begin{eqnarray*}
		(\mathcal{W}(X_1,Z).R)(X_3,X_4)X_5&=&KR(X_1,X_3,X_4,X_5)Z+LR(X_3,X_4,X_5,QX_1)Z\\&&-(Kg(X_1,X_3)+LRic(X_1,X_3))g(X_4,X_5)Z\\&&+(Kg(X_1,X_5)+LRic(X_1,X_5))R(X_3,X_4)Z
	\end{eqnarray*}
	where $ K=\frac{(2m-1)(m-2)}{2(m-1)} $ and $ L= \frac{1}{2(m-1)}$. \par
	Let take $ X_1,X_3,X_4,X_5 $ horizontal vector fields and $ X_4=Z $, then from (\ref{R(X,Z)Y}),  (\ref{R(X,Y,Z,V)}), (\ref{R(X,Z)Z}), (\ref{Ric(X_1,Z)}) and (\ref{Ric(Z,Z)}), then, we get
	\begin{eqnarray*}
		(\mathcal{W}(X_1,Z).R)(X_3,Z)X_5=-(Kg(X_1,X_5)+LRic(X_1,X_5))X_3.
	\end{eqnarray*}
	Thus, we conclude that $ (\mathcal{W}(X_1,Z).R)(X_3,X_4)X_5=0 $ if and only if horizontal bundle of  $ M $ is Einstein. Take into consideration of  Proposition \ref{horizontaliseinstein}, we obtain
	\begin{eqnarray}\label{GQEinproof}
	Ric(X_1,X_5)=-\frac{K}{L}g(X_1,X_5)
	+(2p+\frac{K}{L})\alpha_1(X_1)\alpha_1(X_5)+(2q+\frac{K}{L})\alpha_2(X_1)\alpha_2(X_5)
	\end{eqnarray}
	So, the manifold is \GQE .
\end{proof}
\begin{theorem}
	An $ (m=2p+2q+2) $-dimensional \NCPM \ admitting an $ \mathcal{M}- $projective curvature
	tensor and a nonzero Ricci tensor $ Ric $ satisfies the equality $ \mathcal{W}.Ric =0 $ if and only $ M $ is \GQE \ manifold.
\end{theorem}
\begin{proof}
	For $ X_1,X_3,X_4 \in \Gamma(TM) $ from (\ref{M.Ricgeneral}) we get
	\begin{eqnarray*}
		(\mathcal{W}(X_1,Z).Ric)(X_3,X_4)&=&-Ric([Kg(X_1,X_3)+LRic(X_1,X_3)]Z,X_4)\\&&-Ric(X_3,[Kg(X_1,X_4)+LRic(X_1,X_4)]Z).
	\end{eqnarray*}
	Let take $ X_1,X_4 $ horizaontal vector fields and $ X_3=Z $ from (\ref{Ric(X_1,Z)}), (\ref{Ric(Z,Z)}), we obtain
	\begin{eqnarray*}
		(\mathcal{W}(X_1,Z).Ric)(X_3,X_4)=-(2p+2q)[Kg(X_1,X_4)+LRic(X_1,X_4)].
	\end{eqnarray*}
	Therefore,  $ (\mathcal{W}(X_1,Z).Ric)(X_3,X_4)=0 $ if and only if horizontal bundle is Einstein. From Proposition (\ref{horizontaliseinstein}) we get (\ref{GQEinproof}), which completes the proof.
\end{proof}
Conformal $ \mathcal{C} $ and concircular $ \mathcal{Z} $ on a $(m=2p+2q+2)$-dimensional normal contact metric pair manifold are given by:
\begin{eqnarray*}

	\mathcal{C}( X_1,X_2) X_3 &=&R( X_1,X_2) X_3\\
	&&+\frac{scal}{(
		m-1) (m-2)}\left( g(X_2,X_3)X_1-g(X_1,X_3)X_2\right)  \label{conformal} \notag\\
	&&+\frac{1}{m-2}( g( X_1,X_3) QX_2-g( X_2,X_3) QX_1\notag\\
	&&+Ric(
	X_1,X_3) X_2-Ric( X_2,X_3) X_1) , \notag\\

	\mathcal{Z}( X_1,X_2) X_3&=&R( X_1,X_2) X_3-\frac{scal}{	m (m-1) }[ g(X_2,X_3)X_1-g(X_1,X_3)X_2] \notag
	\label{concircular} \\
\end{eqnarray*}
where  $ X_1,X_2,X_3 \in \Gamma(TM) $. Conformal and concircular curvature tensors on contact manifolds have been studied in \cite{unal2018concircular,unal2019some,turgut2017conformal}.

Blair, Bande and Hadjar \cite{bande2015bochner} studied on conformal flatness of \NCPM s and they proved following theorem.
\begin{theorem}
	A conformal flat \NCPM \ is locally isometric to the Hopf manifold $  S^{2q+1}(1) \times S^1 $ \cite{bande2015bochner}.
\end{theorem}
Thus we get following results, for a \GQE \ \NCPM.
\begin{theorem}
Let $ M $ be a \GQE \ \NCPM. If $ M $ is of generalized quasi-constant curvature  with coefficients $ A=-\frac{\lambda m-m+2}{(m-1)(m-2)} , \ B=\frac{2p-\lambda}{m-2}$ and $ C= \frac{2q-\lambda}{m-2}$, then it is  locally isometric to the Hopf manifold $  S^{2q+1}(1) \times S^1 $.
\end{theorem}
\begin{proof}
	Let $ M $ be \GQE \ \NCPM. \ Then, we have
	\begin{eqnarray*}
		\mathcal{C}( X_1,X_2, X_3 , X_4)&=&R(X_1,X_2,X_3,X_4)\\&&+\frac{scal}{(m-1)(m-2)}[g(X_2,X_3)g(X_1,X_4)-g(X_1,X_3)g(X_2,X_4)]\\&&-\frac{2\lambda}{m-2}[g(X_2,X_3)g(X_1,X_4)-g(X_1,X_3)g(X_2,X_4)]\notag\\&& -\frac{2p-\lambda}{m-2}[g(X_1,X_4)\alpha_1(X_2)\alpha_1(X_3)-g(X_1,X_3)\alpha_1(X_2)\alpha_1(X_4)\notag\\&&+g(X_2,X_3)\alpha_1(X_1)\alpha_1(X_4)-g(X_2,X_4)\alpha_1(X_1)\alpha_1(X_3)]\notag\\&& -\frac{2q-\lambda}{m-2}[g(X_1,X_4)\alpha_2(X_2)\alpha_2(X_3)-g(X_1,X_3)\alpha_2(X_2)\alpha_2(X_4)\notag\\&&-g(X_2,X_4)\alpha_2(X_1)\alpha_2(X_3)]\notag.
	\end{eqnarray*}
	Suppose that, $ M $ is of generalized quasi-constant curvature  with coefficients $ A=-\frac{\lambda m-m+2}{(m-1)(m-2)} , \ B=\frac{2p-\lambda}{m-2}$ and $ C= \frac{2q-\lambda}{m-2}$  . Then, we get $ \mathcal{C}=0 $  which means that $ M $ is conformal flat. Thus, from Theorem 6.4 $ M $ is locally isometric to the Hopf manifold $  S^{2q+1}(1) \times S^1 $.
\end{proof}
By using the definition of $ \mathcal{M}- $projective curvature tensor and conformal curvature tensor, we have
\begin{eqnarray}\label{relationWMC}
\mathcal{C}( X_1,X_2) X_3&=&\frac{2(m-1)}{m-2}\mathcal{W}( X_1,X_2) X_3-\frac{m}{m-2}R( X_1,X_2,)X_3\\&&+\frac{scal}{(m-1)(m-2)}[g(X_2,X_3)X_1-g(X_1,X_3)X_2].\notag
\end{eqnarray}
If $ M $ is a $ \mathcal{M}- $projective flat \NCPM.\ , then, from (\ref{relationWMC}) , $M $ is conformal flat if and only if
\begin{eqnarray*}
	R( X_1,X_2,)X_3=\frac{scal}{	m (m-1) }[ g(X_2,X_3)X_1-g(X_1,X_3)X_2],
\end{eqnarray*}
which means $ M $ is concircular flat. Finally, we conclude that
\begin{theorem}
	Let $ M $ be $ \mathcal{M}- $projective flat \NCPM . \ If $ M $ is also concircular flat then it is locally isometric to Hopf manifold $  S^{2q+1}(1) \times S^1 $.
\end{theorem}

config: arxiv


TITLE: Prove that if $a$ and $b$ are natural numbers, then $a < b$ if and only if $a^2 < b^2$.
QUESTION [0 upvotes]: Where the natural numbers are defined as the positive integers.
My attempt at a proof:
Assume $a < b$. Then there exists a natural number $x$ such that $a + x = b$.
$(a + x)^2 = b^2$
$a^2 + (2ax + x^2) = b^2$
Since $(2ax + x^2)$ is a natural number, $a^2 < b^2$.
Is this correct? Does my proof satisfy the iff part?

REPLY [4 votes]: Does my proof satisfy the iff part?

No, it does not. From $a^2<b^2$ you can only deduce that $b^2=a^2+x$ for some natural $x$. It isn't clear how to deduce from that that $x$ has to be of the form $2ax+x^2$ (in order to use the short multiplication formula). Another problem is the implication
$$(a+x)^2=b^2\ \Rightarrow\ a+x=b$$
While true in naturals you would have to give a proof.

So here's an easier approach.
Since $0<a$, $0<b$ and $a<b$ then
$$a^2=aa<ab<bb=b^2$$
The other implication is actually very easy. Assume that $a^2<b^2$ and assume that $b\leq a$ (note that the order on naturals is total). If $b=a$ then obviously $a^2=b^2$. If $b<a$ then the first part implies that $b^2<a^2$. Both lead to contradiction. Hence the only possibility is $a<b$.
Note that the assumption that $a,b$ are naturals is not needed. It is enough if both are positive reals.

REPLY [0 votes]: write $$a^2<b^2$$ as $$0<(b-a)(b+a)$$

REPLY [0 votes]: You have shown (correctly), that $a<b$ implies $a^2<b^2$.
It remains to show that $a^2<b^2$ implies $a<b$.
To this end show that $b^2-a^2>0$ implies $b>a$. Observe that $b^2-a^2=(b-a)(b+a)$.
Can you take it from here ?

REPLY [0 votes]: We have to prove two things:

$a < b \implies a^2 < b^2$, and,
$a^2< b^2 \implies a<b$.

For the first part, note that: $$ \begin{align} a< b \implies a^2 < ab \\ \implies ab < b^2 \\ \implies a^2< b^2 \end{align}$$
For the second part, note that: $$\begin{align} a^2< b^2 \implies a^2-b^2<0 \\ \implies (a-b)(a+b) < 0\end{align}$$ and since $(a+b) > 0$, we have: $$(a-b) < 0 \implies a< b$$
The two parts done!


TITLE: Mathematical induction question: why can we "assume $P(k)$ holds"?
QUESTION [16 upvotes]: So I see that the process for proof by induction is the following
(using the following statement for sake of example:
$P(n)$ is the formula for the sum of natural numbers $\leq n$:  $0 + 1 + \cdots +n = (n(n+1))/2$ )

Show that the base case is trivially true: $P(0)$: let $n = 0$. Then $0 = (0(0+1))/2$, which is true.
Show that if $P(k)$ holds then $P(k+1)$ also holds. Assume $P(k)$ holds for some unspecified value of $k$. It must be then shown that $P(k+1)$ holds.

This is the part I don't get: 'Assume $P(k)$ holds'
We are just assuming something is true and then 'proving' that something else is true based on that assumption. But we never proved the assumption itself is true so it doesnt seem to hold up to me. Oviously proof by induction works so am I viewing the process incorrectly?

REPLY [1 votes]: Here is a very prosaic way to think about it. Imagine a pond has lily pads lined up forever in a straight line.  Also, imagine you have a frog, who is guaranteed to hop on the next lily pad if he hops on any lily pad.
What is true if you place the frog on the first pad?  Well, he will hop on the second. Since he hops on the second, he will visit the third.  This process will keep going on.   You can see that the frog will eventually visit all of the pads.
This is how the principle of induction works.
So there are two basic steps to an induction argument. Show it works for some first value. This it the base case.   (Place Mr. Frog on the first pad)
Now show that the truth of the argument for a positive integer $n$ guarantees it  is true for $n + 1$; this is the induction step. Then it is true for all positive integers beyond the first one you showed it was true for in the base case.


TITLE: confidence in a sample size
QUESTION [1 upvotes]: Firstly, I am not a math expert so I have limited knowledge and language here. bare with me :)
Currently, I am gathering metrics about servers that exist in a datacenter. the situation is that there maybe 100 servers of the same model that i need to sample. Yet I can only collect metrics on 50 of them for reasons that are not relevant. One metric happens to be power. So I collect power consumption on 50 servers and average the power data to come up with a number like 500 watts for example. So the average power consumption of a particular model of server is 500 watts. There are a few questions:

Is there anyway to know that the sampling I have is significant enough to be considered?
How confident/probable am I that the next server I install of the same model type will be around 500 watts?

I've been reviewing statistical significance, confidence interval, tolerance interval and the like but I want to be sure this is the right way to go. Maybe I'm completely off target on my research. So I am looking for some information to at least point me in the correct direction.

REPLY [1 votes]: First off, all those confidence intervals usually assume that you take random samples from the population. It sounds to me like your are not selecting your servers randomly to collect measurements, but rather, "for reasons you think are not relevant (but maybe are)".
So before you compute any confidence intervals, you should either be certain that the servers you sample are no different (on average) than those you cannot sample---so look at those irrelevant reasons. Or, if you cannot, you need to control for that difference somehow.
In the first case, where you think sampled and nonsampled servers don't differ, you can indeed compute confidence intervals fairly easily. First, you compute the sample mean as usual, then you compute the standard error of the mean (using finite sample correction, since you want to sample about 50% of your population) and on this basis construct confidence intervals (CIs) using the t-distribution.
CIs can be used to answer your 2nd question. Suppose your standard error is 15W. Then you can construct a 95% confidence interval as
$$CI=\bar{x}\pm 1.96 SE=500W \pm 1.96*15W.$$
The 1.96 factor comes from the fact that, if the measured values are normally distributed (which they are for sufficient sample size), then 95% of the values lie within 1.96 of the standard error above and below the sample mean. As I mentioned, if the sample size is relatively small, you should instead use the t-distribution (then the factor will be $>1.96$, i.e., you estimate the values are more spread out). To sum up, after this calculation, you can choose confidence level $l$ yourself (in the example: $l=95$) to calculate the range in which $l$% of the server measurements will be. If you did everything right, not only do you know your confidence ($l$), but can also quantify "around" with the CIs. All of this, of course, relies on random sampling.
I don't understand your first question. Significant sampling? If you sample 50% of your servers, that would be a lot and you need not worry about small sample size. The more important thing about sampling, as stressed above, is that it is random, to rule out differences between the sampled and nonsampled units. We recently had a thread discussing what can happen with nonrandom sampling.