Tangent Bundle TM. Vector, Fiber, Tensor and Tensor Densities, and Associate Bundles

Abstract

Let us return, encouraged by what was said in Chapter 15, to the theory of vector bundles! Since the notion of tangent bundle TM of differentiable manifold is perhaps the most fundamental notion of modern analysis (which is today to large extent the analysis on manifolds) and the since tangent bundle is the ‘mother of vector bundles’ and, more generally, fiber bundles, we will present once again the construction of the bundle TM → M without defining tangent spaces T _x M. This construction can be extended — virtually without changes — to the case of general fiber bundles.