On the curvature of contact metric manifolds
Let \(M^{2m+1}\) be a contact metric manifold and \((\omega, \xi, g, \varphi)\) its contact metric structure where \(\omega\) is a contact 1- form, \(\xi\) the characteristic vector field, \(g\) an associated Riemannian metric and \(\varphi\) a tensor field of type \((1,1)\). The author first studies the geometry of a contact metric manifold with parallel Ricci tensor. One of his results is the following: If \((M^{2m+1}, \omega, \xi, g)\) has parallel Ricci tensor and moreover \(Q\xi= f\xi\), \(f>2m-2\), where \(Q\) is the Ricci operator, then \((M^{2m+1}, g)\) is Einstein. Thereafter he studies contact metric manifolds with non-vanishing sectional curvature \(K(\xi, X)\) and \(K(\xi, X)= K(\xi, \varphi X)\) everywhere and for all \(X\perp \xi\). Also conformally flat contact metric manifolds are studied. Finally some results on the \(k\)-nullity distribution of contact metric manifolds are given.
scientific article; zbMATH DE number 785157
scientific article