Notation In general, we follow the notation used in the text Advanced Calculus by Loomis and Sternberg, Addison-Wesley Publishing Co., Reading, Mass. 1968, with some slight changes. The letters X, Y, Z, W9 M will be used for differentia- ble manifolds usually M will denote a general differentiable manifold, and X a manifold carrying some additional structures. Points on these manifolds will be denoted by lower case letters such as x, y, z, etc. Local coordinates will be written as (x 1 ,... , x").j If M is a manifold and T*M is its cotangent bundle, then the local coordinates on T*M associated with the local coordinates (x 1 ,.. . ,xn) on M will be (x 1 ,... , xn, £j,.:.., £n) and, sometimes (ql,...,qn9pl,... ,pn). The funda- mental one form on the cotangent bundle will be written as a, so that, in terms of these local coordinates, a = £}dxx 4- + £ndxn or a = pxdqx + + pndqn. We will use bold face greek letters, usually £, ij, or £ to denote either tangent vectors or vector fields. The tangent space to a manifold M at a point x will be denoted by TMX, and so a typical element of this tangent space will be £ E TMX. If /: M N is a smooth map between differentiable manifolds, its differential at the point x will be denoted by dfx, so that dfx: TMX - TNf^. The "pull back" of a differential form 9 on N by/will be denoted by f*0. Thus, for example, if 0 is a linear differential form on N, and 0 E T* N is its value at y EL N, then for £ E TMX we have where ( ,) gives the pairing between vectors and covectors. We will use the symbol D* to denote the Lie derivative with respect to the vector field £. Thus, if £7 is a Worm, D^l is its Lie derivative with respect to £ if 1 7 is another vector field, D^q = [J, r\] is the Lie bracket if u is a function, we frequently write %u for D^u. In an oriented Riemannian or psuedoriemmanian manifold of dimension n, we have the "star operator", denoted by *, mapping k forms into n k forms. Thus, for example, if S is an oriented surface in Euclidean three space, R3 and u is a smooth function on R3, we write jj * du for the expression Jj (du/dn)dS that occurs in many of the older analysis texts. xv
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