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Generalized Einstein manifolds. (English) Zbl 1172.53029

Kowalski, Oldřich (ed.) et al., Differential geometry and its applications. Proceedings of the 10th international conference on differential geometry and its applications, DGA 2007, Olomouc, Czech Republic, August 27–31, 2007. Hackensack, NJ: World Scientific (ISBN 978-981-279-060-6/hbk). 57-64 (2008).
A Riemannian manifold \((M,g)\) is called quasi-Einstein, provided it carries a globally defined unit vector field \(\xi\) (which we refer to as the generator) and its dual form \(\eta\) w.r.t \(g\) \(\eta(\cdot)= g(\xi,.)\) such that \(T=\beta \eta\otimes\xi\), where \(\beta\) is a function on \(M\) and \(T\) is the energy-momentum tensor. The authors show that, if the dual form \(\eta\) of \(\xi\) is harmonic, then a quasi-Einstein manifold \(M\) is \(\eta\)-Einstein if and only if its scalar curvature is constant. They also show that a locally decomposable Riemannian manifold \(M=M_1\times M_2\) with \(M_i\) endowed by the unit form \(\eta_i\), for \(i=1,2\), has both components of quasi-constant sectional curvature if and only if its \((0,4)\)-tensor field \(R\) is of the form
\[ \begin{aligned} R(X,Y,Z,V)&= \alpha(U(X,Y,Z,V)+ U(PX,PY,Z,V)) \\ +&\beta(U(PX,PY,PZ,V)+ U(PX,PY,Z,PV))\\ +&\gamma(S(X,Y,Z,V)+ S(PX,PY,PZ,PV)+ S(PX,PY,Z,V)+ S(X,Y,PZ,PV))\\ +&\nu(S(X,Y,Z,PV)+ S(X,Y,PZ,V) + S(PX,PY,Z,PV)+ S(PX,PY,PZ,V)),\end{aligned} \]
for any \(X\), \(Y\), \(Z\), \(V\in\Gamma(TM)\), where \(\alpha, \beta, \gamma, \nu\) are functions on \(M\),
\[ \begin{aligned} U(X,Y,Z,V)&= g(X,V)g(Y,Z) -g(Y,V)g(X,Z),\\ S(X,Y,Z,V)&= g(X,V)\eta(Y)\eta(Z)- g(X,Z)\eta(Y)\eta(V)+ g(Y,Z)\eta(X)\eta(V)- g(Y,V)\eta(X)\eta(Z)\end{aligned} \] and \(\eta=(\eta_1+ \eta_2)/2\).
For the entire collection see [Zbl 1154.53003].

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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