XXII.—Tensor Fields and Connections on Cross-Sections in the tangent bundle of a differentiable manifold

Source abstract

Synopsis Tensor fields and linear connections in an n-dimensional differentiable manifold M can be extended, in a natural way, to the tangent bundle T ( M ) of M to give tensor fields of the same type and linear connections in T ( M ) respectively. We call such extensions complete lifts to T ( M ) of tensor fields and linear connections in M . On the other hand, when a vector field V is given in M, V determines a cross-section which is an n -dimensional submanifold in the 2 n -dimensional tangent bundle T ( M ). We study first the behaviour of complete lifts of tensor fields on such a cross-section. The complete lift of an almost complex structure being again an almost complex structure, we study especially properties of the cross-section as a submanifold in an almost complex manifold. We also study properties of cross-sections with respect to the linear connection which is the complete lift of a linear connection in M and with respect to the linear connection induced by the latter on the cross-section. To quote a typical result: A necessary and sufficient condition for a cross-section to be totally geodesic is that the vector field V in M defining the cross-section in T ( M ) be an affine Killing vector field in M .