\documentclass[10pt]{article} %\parindent=0cm \parskip 2ex \setlength{\textheight}{190mm} \setlength{\textwidth}{125mm} \setlength{\parindent}{0cm} \usepackage{amssymb} \usepackage{amsthm} \usepackage{epsf,epsfig,amsfonts,amsmath} %\usepackage[english,russian]{babel} %\usepackage{cancel} \usepackage{graphicx} %\usepackage{mathcomp} \textwidth=14.7cm \textheight=18.1cm \usepackage{a4wide} \textwidth=1.04\textwidth \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{remark}{Remark} \newtheorem{proposition}{Proposition} \newtheorem{corollary}{Corollary} \newtheorem{example}{Example} \newtheorem{exercise}{Exercise} \newtheorem{Q}{Question} \newtheorem{definition}{Definition} %\newtheorem{proof}{��������������} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\avg}{G} \def\trace{\mbox{\rm trace}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\oC}{\overline{C}} \newcommand{\og}{\overline{g}} \newcommand{\oF}{\overline{F}} \newcommand{\gf}{G} \newcommand{\gF}{G} \newcommand{\ol}{\overline} \newcommand{\R}{\mathbb{R}} \newcommand{\rn}{\mathbb{R}^n} \newcommand{\weg}[1]{} % <<<<<<<<<<<<<<<<< %------------------------------------ \def\mg{\hbox{$(M,g)$}} \def\na{\hbox{$\nabla$}} \def\d{\hbox{$\delta$}} \def\la{\hbox{$\lambda$}} \def\La{\hbox{$\Lambda$}} \def\RR{\hbox{$\cal R$}} \def\s{\hbox{$\sigma$}} \def\be(#1){\begin{equation}\label{#1} } \def\ee{\end{equation}} \def\re(#1){\eqref{#1}} \def\xx{\hbox{$\{x^1,x^2,\dots,x^n\}$}} %----------------------------- % \begin{document} \title{On generalized semisymmetric Riemannian manifolds} \author{Josef Mike\v s, Sergey E. Stepanov} \date{} \maketitle \begin{abstract} There exist several generalizations of the concept of semisymmetric Riemannian manifolds. In the present paper, we consider some special types of generalized semisymmetric Riemannian manifolds and we express the conditions for non-existence of these manifolds in terms of curvature operator and sectional curvature. As applications of our theory we show non-existence results for semisymmetric, Ricci-semisymmetric and birecurrent Riemannian manifolds. \end{abstract} %---------------------------------------------------------------------------------------% {\bf Keywords:} generalized semisymmetric manifold, Ricci-semisymmetric, semisymmetric manifolds, sectional curvature. %---------------------------------------------------------------------------------------% {\bf subclass}: 53B20 %---------------------------------------------------------------------------------------% % \def\a{\alpha} \def\b{\beta} \def\G{\Gamma} \def\l{\ell} \def\epsilon{\varepsilon} \def\s{\sigma} \def\vn{$V_n$} \def\vnn{$\bar V_n$} \def\nad#1#2{\buildrel{#1} \over{#2}\!\!\strut} \def\pod#1#2{\mathrel{\mathop{#2}\limits_{#1}}\strut} \def\tran{; transl.~from } \def\uu{\accent23u} % % \section{Introduction}\label{section:1} An $n$-dimensional $(n\geq3)$ Riemannian manifold \mg\ with the Levi-Civita connection \na\ is a {\it semisymmetric manifold} if its Riemannian curvature tensor $R$ satisfies the following condition \be(1.1) R(X,Y)\circ R=0. \ee Here \ $\circ$ \ denotes tensor derivations and $X,Y$ are arbitrary vector fields on the manifold $M$. Semisymmetric manifolds have been investigated by E.~Cartan and are the generalization of symmetric Riemannian manifolds in the sense that the curvature tensor of any locally symmetric Riemannian manifolds satisfies \re(1.1). However, there exist examples of semisymmetric but not locally symmetric spaces. There are many papers where authors consider these manifolds (see, for example \cite{ref:2}, \cite{ref:3}, \cite{ref:6}, \cite{ref:8}, \cite{ref:11}, \cite{ref:12}, \cite{ref:15}, \cite{ref:19}, \cite{ref:20}). These manifolds have been locally by classified by Z.I.\ Szab\'o (see \cite{ref:20}). The author proves that for every Riemannian semisymmetric space there exists an everywhere dense open subset $U$ such that, around every point of $U$, the space is locally isometric to a space which is the direct product of symmetric spaces, 2-dimensional Riemannian spaces, elliptic, hyperbolic, Euclidean and K\"ahlerian cones, and spaces foliated by $(n{-}2)$-dimensional Euclidean spaces (i.e.\ Riemannian spaces $(M,g)$ with index of nullity $\nu(p) = n - 2$ for any $p\in M$). The theory of Riemannian semisymmetric manifolds has been presented in the monograph \cite{ref:2}. In the present paper, we consider some special types of generalized semisymmetric Riemannian manifolds with a non-degenerate curvature operator or nonzero sectional curvature. As applications of our theory we show non-existence results for semisymmetric, Ricci-semisymmetric and birecurrent Riemannian manifolds. % \section{$T$-semisymmetric Riemannian manifolds}\label{section:2} %\subsection{Preliminaries} There are several generalizations of the concept of semi-symmetric Riemannian manifolds. For example (see \cite{ref:10}, \cite{ref:11}, \cite{ref:15}, \cite{ref:13}, \cite{ref:14}), a Riemannian manifold \mg\ is called a \textit{T-semisymmetric manifold} if \mg\ admits a tensor field $T$ of the type $(p,q)$ such that the following condition \be(1.2) R(X,Y)\circ T=0 \ee is true for the Riemannian curvature tensor $R$ of \mg, where $R(X,Y)$ acts as a derivation on~$T$ and for any vector fields $X, Y$ on $M$. We recall that at each point $x$ of $M$, the quadratic form $g_x$ induces a canonical isomorphism $T_xM\to T^*_xM$ and more generally, a canonical isomorphism between any spaces $T^{(q,p)}M$ and $T^{(r,h)}M$ for $r + h = p + q$. These isomorphisms correspond to lowering (resp.~raising) indices in classical tensor notation. For example, we have the following $T_{i_1\cdots i_qk_1\cdots k_p}=g_{k_1j_1}\cdots g_{k_pj_p}T_{i_1\cdots i_q}^{j_1\cdots j_p}$ by local expressions $T=(T_{i_1\cdots i_q}^{j_1\cdots j_p})$ and $g=(g_{ij})$ for the tensor $T$ and the metric tensor $g$ respectively. This fact and also that $\nabla$ is natural connection of metric $g$ (i.e. $\nabla g = 0$) allows us to consider only covariant tensors $T$. In addition, note that $g$ defines a positive define quadratic form on the space $T^{(q,0)}_xM$ of covariant tensors of type $(q,0)$ and consequently on any subspace of $T^{(q,0)}_xM$ such as the space $\La^qM$ of skew-symmetric and space $S^qM$ of symmetric covariant tensors of type $(q, 0)$. %\subsection{Orthogonal decomposition of skew and symmetric tensors} \bigskip On the other hand, there is a well-known point-wise orthogonal decomposition (see \cite[p.~45]{ref:1}) $$T^{(2,0)}M=T^*M \otimes T^*M=S^2M \oplus \La^2M,$$ from which we obtain the following orthogonal decomposition $T=ST+\La T$ for any tensor field $T$ of type $(2, 0)$ where $ST=Pr_{S^2M}T$ and $\La T=Pr_{\La^2M}T$ are orthogonal projections on the tensor spaces $S^2M$ and $\La^2M$, respectively. Next, we note that if any Riemannian manifold \mg\ is a $T$-semisymmetric manifold with a tensor field $T$ of the type $(2, 0)$ then \mg\ is a $ST$ and $\La T$-semisymmetric manifold. Moreover, the converse proposition is true. % \section{Statement of the main results}\label{section:3} \subsection{Conditions for non-existence of $T$-semisymmetric manifold} In accordance with above results, we consider two special types of $T$-semisymmetric manifolds. Firstly, it is a $T$-semisymmetric $n$-dimensional Riemannian manifold with a covariant skew-symmetric tensor field of type $(2, 0)$, i.e. $T\in\La^2M$. In this case the following lemma is true. \begin{lemma} % Lemma 1. Let \mg\ be an n-dimensional $(n\geq3)$ Riemannian manifold with a positive (negative) define Riemannian curvature operator \RR: $\La^2M\to\La^2M$, then \mg\ can not be a T-semisymmetric Riemannian manifold with a nonzero covariant skew-symmetric tensor field $T\in S^2M$. \end{lemma} Lemma 1 implies in particular that for a Riemannian manifold of constant positive (negative) sectional curvature, or for a conformally flat Riemannian manifold with positive definite (negative definite) Ricci tensor, we have $T = 0$. Secondary, we consider a $T$-semisymmetric manifold with a covariant symmetric tensor field of type $(2,0)$, i.e. $T\in S^2M$. In this case we will prove the following lemma. \begin{lemma} %Lemma 2. Let \mg\ be a T-semisymmetric n-dimensional $(n\geq2)$ Riemannian manifold with a covariant symmetric tensor field $T\in S^2M$ and $\{e_1,\dots,e_n\}$ be an orthonormal basis at an arbitrary point $x\in M$ which consists of eigenvectors of T such that $T(e_i,e_j)=\la_i\d_{ij}$ for the eigenvalue $\la_i$ and Kronecker delta $\d_{ij}$. If the sectional curvature $sec(e_i\wedge e_j)\neq0$ then $\la_i=\la_j$. In particular, if \mg\ has nonzero sectional curvatures at each point of \mg, then $T=\la\cdot g$ for some smooth scalar function $\la$. \end{lemma} Next, from Lemma 1 and Lemma 2 we conclude that the following theorem holds. \begin{theorem} %Theorem 1. Let \mg\ be a T-semisymmetric n-dimensional $(n\geq3)$ Riemannian manifold with a tensor field T of the type $(2,0)$ and a positive (negative) Riemannian curvature operator \RR: $\La^2M\to \La^2M$, then $T=\la\cdot g$ for some smooth scalar function $\la$. \end{theorem} For the proof of this theorem, we note that from positive or negative definiteness of the Riemannian curvature operator \RR\ of \mg\ we can conclude that the sectional curvature of \mg\ is positive or negative definite, respectively. In dimension 3, the converse is true. % \subsection{Condition for non-existence of Ricci-semisymmetric manifolds} In particular, if $T=Ric$ for the Ricci tensor $Ric$ of \mg\ then \mg\ is called a \textit{Ricci-semisymmetric manifold} which was defined in (\cite{ref:10}, \cite{ref:16}) by the following condition $R(X,Y)\circ Ric=0$. Remark here, that though $R(X,Y)\circ R=0$ implies $R(X,Y)\circ Ric=0$, but the converse is not true, in general. The main structure theorem of the paper \cite{ref:16} says that a smooth Riemannian manifold is Ricci-semisymmetric if and only if it is locally a product of two-dimensional Riemannian manifolds, Einstein spaces and semi-Einstein spaces. Some results by Z.I.~Szab\'o are used in the proof of this theorem. On the other hand, we have following corollary. \begin{corollary} % Corollary 1. Let \mg\ be a Ricci-semisymmetric n-dimensional $(n\geq3)$ Riemannian manifold and $\{e_1,\dots,e_n\}$ be an orthonormal basis at an arbitrary point $x\in M$ which consists of the Ricci principal directions such that $Ric(e_i,e_j)=\la_i\d_{ij}$ for the eigenvalue $\la_i$ and Kronecker delta $\d_{ij}$. If the sectional curvature $sec(e_i\wedge e_j)\neq0$ then $\la_i=\la_j$. In particular, a Ricci-semisymmetric Riemannian manifold \mg\ with nonzero sectional curvatures at each point of \mg\ is an Einstein manifold $($i.e.~$Ric={const}\cdot g)$ and, moreover, in dimension 3 is a Riemannian manifold of constant curvature. \end{corollary} Geometrical interpretation of the Ricci principal directions has been present in the monograph \cite{ref:4}. % \section{$T$-birecurrent Riemannian manifold}\label{section:4} Lichnerowicz has defined (see \cite{ref:16}) a \textit{recurrent of second order} (or briefly a \textit{birecurrent}) Riemannian manifold by the equation $\nabla^2R=a\otimes R$, where $R$ is the Riemannian curvature tensor and $a$ is a covariant tensor field of order 2. He proved that if a birecurrent \mg\ is compact and the scalar curvature does nowhere vanish it is recurrent in the ordinary sense: $\nabla R=b\otimes R$ where $b$ is a 1-form on \mg. Wakakuwa in \cite{ref:21} proved the same result for non-compact irreducible Riemannian manifold \mg\ with dimension $n\geq3$. There are several generalizations of the concept of birecurrent Riemannian manifold. For example (see \cite{ref:17} and \cite{ref:5}), a tensor field $T$ of type $(p,q)$ is called a birecurrent tensor if it satisfies the equation \be(1.3) \nabla^2T=a\otimes T. \ee The author of \cite{ref:16} has proved that for any covariant birecurrent tensor $T$ its associated tensor field $a$ is a symmetric tensor field. Therefore using the standard commutation formulas (Ricci identities) $$ (\nabla^2T)(X,Y)-(\nabla^2T)(Y,X)=R(X,Y)\circ T $$ we obtain from \eqref{1.3} the equation \eqref{1.2}. Hence if a Riemannian manifold \mg\ admits a birecurrent tensor $T$ then \mg\ is a $T$-semisymmetric manifold. Therefore, from the Lemma 2 we obtain automatically the following corollary. \begin{corollary} % Corollary 2. Let T be a birecurrent covariant symmetric tensor field of the type $(2,0)$ on an \hbox{n-dimen}\-sio\-nal \hbox{$(n\geq2)$} Riemannian manifold \mg\ and $\{e_1,\dots,e_n\}$ be an orthonormal basis at an arbitrary point $x\in M$ which consists of eigenvectors of T such that $T(e_i,e_j)=\la_i\d_{ij}$ for the eigenvalue $\la_i$ and Kronecker delta $\d_{ij}$. If the sectional curvature $sec(e_i\wedge e_j)\neq0$ then $\la_i=\la_j$. In particular, if \mg\ has nonzero sectional curvatures at each point of \mg, then $T=\la\cdot g$ for some smooth scalar function~$\la$. \end{corollary} On the other hand, from the Theorem 1 we obtain another corollary. \begin{corollary} %Corollary 3. Let T be a birecurrent covariant tensor field of the type $(2,0)$ on an n-dimensional \hbox{$(n\geq3)$} Riemannian manifold \mg\ with positive (negative) Riemannian curvature operator \RR:\linebreak $\La^2M\to \La^2M$, then \hbox{$T=\la\cdot g$} for some smooth scalar function~$\la$. \end{corollary} % \section{Proof of the results}\label{section:5} %\subsection{Proof of Lemma 1} {\it %\large %\ref{section:5}.1. Proof of Lemma 1.} Let \xx\ be a local coordinate system on the chart $(U,\phi)$. We denote by $g=(g_{ij})$, $R=(R^h_{ijk})$, $Ric=(R_{ij}:=R^k_{ikj})$ and $T=(T_{i_1i_2\cdots i_q})$, respectively, the metric, the curvature, the Ricci tensors and a covariant tensor field $T$ of type $(q,0)$ with respect to the local coordinate system \xx. Then the formula \eqref{1.2} becomes \be(2.1) T_{li_2\cdots i_q} R^l_{i_1jk}+T_{i_1li_3\cdots i_q} R^l_{i_2jk}+ \cdots + T_{i_1i_2\cdots i_{q-1}l} R^l_{i_qjk}=0 \ee by local expression for the covariant tensor field $T$ and the curvature tensor $R$. Let $T$ be a skew-symmetric tensor field $T\in\La^2M$ then from \eqref{2.1} we obtain \be(2.2) T_{lk} R^l_{ist}+T_{il} R^l_{kst}=0. \ee If we transvect \eqref{2.2} with $g^{kt}T^{is}$ we get (see \cite[Section 4 of Chapter III]{ref:23}, \cite[p.~70]{ref:22}) \be(2.3) F_2(T):=R_{ij}T^i_{\,k}T^{jk}-\frac12 R_{ijkl}T^{ij}T^{kl}=0. \ee where $F_2$: $\La^2M\otimes\La^2M\to\R$ is the well known (see \cite[p.~53]{ref:1}; \cite[p.~211]{ref:18}) quadratic form from the classic formula of Bochner-Weitzenb\"ock $g(\Delta T,T)=g(\nabla^*\nabla T,T)+F_q(T,T)$ for $q = 2$. Here $\Delta$ is the \textit{Hodge Laplacian} and $\nabla^*\nabla$ is the \textit{rough (Bochner) Laplacian}. Moreover, \eqref{2.3} is equals to the following equations (see \cite[pp.~220,221]{ref:18}) \be(7) \sum_\a r_\a \|[\theta_\a,T]\|^2=0 \ee where $r_\a$ are the eigenvalues and $\theta_\a$ the duals of the eigenvectors of the standard symmetric Riemannian curvature operator \RR: $\La^2M\to\La^2M$ given by the identity (see \cite[p.~36]{ref:19}) $$ g(\RR(X\wedge Y), V\wedge W)=R(X\wedge Y,V\wedge W)=R(X,Y,V,W) $$ for arbitrary vector fields $X, Y, V$ and $W$. Furthermore, we say that \mg\ has \textit{positive} (\textit{negative}) \textit{Riemannian curvature operator} if all eigenvalues of \RR\ are positive (negative). Next we suppose that \mg\ has the positive (negative) Riemannian curvature operator, then $r_\a > 0$ ($r_\a < 0$) and the formula \re(7) implies that $[\theta_\a,T]=0$ for all $\a$. In this case $T$ must vanish (see \cite[p.~221]{ref:18}). This completes the proof of Lemma 1. \bigskip % {\it %bf\large %\ref{section:5}.2. Proof of Lemma 2.} The formula \eqref{1.2} becomes \be(2.4) T_{lk} R^l_{ist}+T_{il} R^l_{kst}=0. \ee by local expression for the symmetric tensor field $T$ of type $(2,0)$ and the curvature tensor $R$. Let $\{e_1,\dots,e_n\}$ be an orthogonal basis in $T_xM$ at an arbitrary point $x\in M$ such that $T(e_i,e_j)=\la_i\ \d_{ij}$ for the Kronecker delta $\d_{ij}$. Then we can rewrite the identities \eqref{2.4} as \be(2.5) (\la_i-\la_j)\cdot sec(e_i\wedge e_j)=0 \ee where $sec(e_i\wedge e_j)=g(\RR(e_i\wedge e_j),e_i\wedge e_j)$ is called a \textit{sectional curvature of two-plane} (see \cite[p.~36]{ref:18}). If we suppose that $sec(e_i\wedge e_j)\neq0$ then from \eqref{2.5} we obtain $\la_i=\la_j$. In particular, if sectional curvatures of all two-planes $\pi\subset T_xM$ at each point $x\in M$ are not zero then from \eqref{2.5} we obtain $\la_1=\cdots=\la_n=\la$. Hence Lemma 2 is proved. \smallskip % {\it %\bf\large %\ref{section:5}.2. Proof of Corollary 1.} The Corollary 1 follows from Lemma 2 automatically. If $n = 3$, then \mg\ is Einstein if and only if it has constant sectional curvature too (see \cite[p.~44]{ref:1}). {\bf Remark.} We note that if $n = 2$, then at each point $x$ in $M$, we have $Ric = \frac12 s \cdot g$ for the scalar curvature~$s$, and the condition $R(X,Y)\circ Ric=0$ is always satisfied. % ------------------------------------------------------------------------ {\bf Acknowledgments.} The paper was supported by grant P201/11/0356 of The Czech Science Foundation.\\ Josef Mike\v s\\ Dept. of Algebra and Geometry, Faculty of Science, Palacky University,\\ 17. listopadu 12, 77146 Olomouc, Czech Republic\\ {\it E-mail}: josef.mikes@upol.cz Sergey E. 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