Differential Geometry Basics

. This is a very terse development of differential geometry, written for the author's benefit. The point is to gather all of the basic constructions together with consistent sign and other conventions. 1 Conventions M is a smooth (C 1 ) manifold. All maps, curves, etc. are smooth unless otherwise specified. E, F and G are vector bundles over M . We use Penrose's abstract notation [6], as described in [1], with shape symbols for E, F and G being ~, 4 and ./, respectively. We use capital greek indices for E and capital roman indices for F , the indices appearing as superscripts. We use subscripts for E and F . We will not need index symbols for G. A section of a tensor product of copies of TM and T M will sometimes be called a world-tensor. A metric is a (not necessary positive definite) nondegenerate symmetric element of S \Pi\Pi . Our notational and sign conventions follow Penrose. 2 The Exterior Derivative of a Function If X 2 S \Pi then X is a derivation on functio..