On the multiplication of tensor fields

Abstract

Let M be a paracompact n-dimensional manifold of class Ck+1, T(x) the tangent space of the point xeM and T(x) * the dual space. A real valued (p+r)-linear function 4(x) with p arguments in T(x) and r arguments in T(x) * is called a tensor of type (p, r) at the point x. The tensors of type (p, r) at x form a linear space Tp(x). The products of two tensors DE Tp(x) and 'E Ta(x) is defined by (Vk) (X; 61 .. * +q) t*1 . . . *8