On the index form of a geodesic in a pseudoriemannian almost-product manifold
A Riemannian almost product manifold is a triple (M,g,P) where g is a Riemannian metric and P a (1,1)-tensor field such that \(P^ 2=identity\) and \(g(PS,PT)=g(S,T)\) for any vector fields S,T\(\in \chi M\). Giving such a P, it is equivalent to prescribe two orthogonal distributions V (the vertical distribution) and H (the horizontal distribution). Further, such a P induces a pseudo-Riemannian metric \(\phi\) defined by \(\phi (S,T)=\neq g(PS,T)\) and it is proved that the converse is also true. Under the hypothesis that both, V and H are involutive and if \(\nabla\) and \({\tilde \nabla}\) are the symmetric connections making g and \(\phi\) parallel, the authors compute the components of the curvature tensor \(\tilde R\) of \({\tilde \nabla}\) in terms of those of the curvature tensor R of \(\nabla\). Some properties are also discussed if V is totally geodesic.
scientific article; zbMATH DE number 4024229