Curvature of indefinite almost contact manifolds
The authors investigate the curvature properties of indefinite almost contact manifolds \((M,\varphi,\xi,\) \(\eta,g).\) Formally, \((\varphi,\xi,\eta,g)\) is an almost contact metric structure on the differentiable manifold \(M\) [cf., e.g., \textit{D. E. Blair} [Contact manifolds in Riemannian geometry. Lect. Notes Math. 509 (1976; Zbl 0319.53026)] but the metric \(g\) is not assumed to be positive definite. The investigation focuses on the so-called \(C(\alpha)\)-manifolds introduced by \textit{D. Janssens} and \textit{L. Vanhecke} [Kodai Math. J. 4, 1-27 (1981; Zbl 0472.53043)], i.e., those satisfying the condition \[ R(X,Y,Z,W-R(X,Y,\varphi Z,\varphi W)= \] \[ \alpha\bigl(g(Y,Z)g(X,W)-g(X,Z)g(Y,W)-g(Y,\varphi Z)g(X,\varphi W) +g(X,\varphi Z)g(Y,\varphi W)\bigr) \] for any vector fields, \(\alpha\) being a constant. By means of the study of the Jacobi operator along spacelike, timelike and null geodesics, spaces of constant curvature are characterized as well as spaces of pointwise constant \(\varphi\)-sectional curvature. There is an essential difference on the behaviour of the Jacobi operator along null and non-null geodesics. This motivates the definition of the \(\varphi\)-isotropic \(C(\alpha)\)-manifolds as those satisfying the condition \(R(U,\varphi U)\varphi U=c_UU\) (\(c_U=\text{const.}\)) for any null vector \(U\) tangent to the contact distribution. A local classification of \(\varphi\)-isotropic \(C(\alpha)\)-manifolds with parallel and diagonalizable Ricci tensor is obtained.
scientific article; zbMATH DE number 1000346
scientific article