Elementary Differential Geometry

Abstract

We consider an N-dimensional Riemannian manifold M and let g_i be a basis at the point P_i(u¹,…,u^N) and g_j be another basis at the other point P_j(u¹,…,u^N). Note that each such basis may only exist in a local neighborhood of the respective points, and not necessarily for the whole space. For each such point we may construct an embedded affine tangential manifold. The N-tuple of coordinates are invariant in any chosen basis; however, its components on the coordinates change as the coordinate system varies. Therefore, the relating components have to be taken into account by the coordinate transformations.