Contact Riemannian manifolds with constant \(\varphi\)-sectional curvature
Let \(M\) be a \((2n+1)\)-dimensional Riemannian almost contact manifold with a structure \((\varphi, \xi, \eta, g)\). \(M\) is a contact manifold if there exists a 1-form \(\omega\) on \(M\) such that \(\omega \wedge (d\omega)^n\neq 0\). \(M\) is normal if \([\varphi,\varphi]+ 2d\omega\otimes\xi=0\), where \([\phi,\phi]\) is the Nijenhuis tensor for \(\phi\). A manifold \(M\) is a Sasakian manifold if there exists a normal contact structure on \(M\). In this paper, the author studies properties of a \((\kappa,\mu)\)-contact Riemannian manifold \(M\) for which the characteristic vector field \(\xi\) belongs to the so-called \((\kappa, \mu)\)-nullity distribution, i.e., the curvature operator \(R(X,Y)\) of the manifold satisfies the condition \( R(X,Y)\xi = \kappa[\eta(Y)X -\eta(X)Y]+ \mu [\eta(Y)h X- \eta(X)h Y]\), where \(\kappa\), \(\mu\) are constants and \(2h\) is the Lie derivative of \(\varphi\) in the direction \(\xi\). It is proved that, if the \(\varphi\)-sectional curvature at a point \(p\in M\) is independent of a \(\varphi\)-section at \(p\), then it is constant. Second, the author proves that a non-Sasakian \((\kappa,\mu)\)-contact Riemannian manifold \(M\) is of constant \(\varphi\)-sectional curvature if and only if \(\mu=\kappa+1\). In this case an explicit expression for the curvature tensor \(R\) is given. As an application of the second theorem the author shows that the tangent sphere bundle of a Riemannian manifold of constant sectional curvature \(c\) is of constant \(\varphi\)-sectional curvature if and only if \(c=2\pm \sqrt 5\).
constant \(\varphi\)-sectional curvature
\((\kappa,\mu)\)-contact Riemannian manifold
scientific article; zbMATH DE number 1054586