On \({\varphi}\)-Einstein contact Riemannian manifolds
Let \(M\) be a Riemannian manifold which is equipped with an almost contact metric structure \((\varphi,\xi,\eta,g)\). Define the \(\varphi\)-Ricci tensor \(\rho^\varphi\) of \(M\) by \(2\rho^\varphi(X,Y) =\) symmetric part of trace \(R(X,\varphi Y)\varphi\) and a tensor \(g^\varphi\) by \(g^\varphi(X,Y) = g(\varphi X, \varphi Y)\). The manifold \(M\) is said to be \(\varphi\)-Einstein if \(\rho^\varphi = \lambda g^\varphi\) for some \(\lambda \in {\mathbb R}\). The authors prove some general properties of \(\varphi\)-Einstein manifolds and study in more detail such manifolds in dimension \(3\). Every real hypersurface in a complex space form inherits an almost contact metric structure from the metric and complex structure of the ambient space. The authors investigate the \(\varphi\)-Einstein condition on such real hypersurfaces, and derive partial classifications. In the final part of this paper the authors investigate the \(\varphi\)-Einstein condition on unit tangent sphere bundles.
scientific article; zbMATH DE number 5728333
scientific article