int - The volume form 1 Orientation on vector spaces and...

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The volume form. 1. Orientation on vector spaces and manifolds. (1) Let V be a vector space of dimension n 1. Recall that an orientation on V is specified by choosing a particular basis e 1 , . . . , e n of V . If v 1 , . . . , v n is any other basis of V , this basis is declared positive if the transition matrix A has det ( A ) > 0 and negative if det ( A ) < 0. Here A = ( v j i ) ij , where v i = n j =1 v j i e j . Note that if v 1 = e 1 , . . . v n = e n , then A = I n is the identity matrix and hence e 1 , . . . , e n is a positive basis. (2) Let M n be an n -manifold. An orientation of M can be viewed as a continuous (with respect to varying p ) choice of orientations on all the tangent spaces M p , where p M . Let ( U i , φ i ) i be an orienting atlas on M (that is the Jacobians of all the transition maps φ i φ - 1 j have positive determinants). For p M this atlas defines an orientation on M p by declaring that the basis ∂x 1 i | p , . . . , ∂x n i | p is a positive basis of M p , where p ( U i , φ i = ( x 1 i , . . . , x n i )). Conversely, suppose that we have specified a a continuous (with respect to varying p ) choice of orientations on all the tangent spaces M p , where p M . We can construct an orienting atlas on M as follows. Start with any atlas ( U i , φ i = ( x 1 i , . . . , x n i )) i with connected sets U i . For every U i choose p U i and check whether ∂x 1 i | p , . . . , ∂x n i | p is a positive basis of M p . If yes, keep ( U i , φ i ) without change. If not, interchange x 1 i and x 2 i in φ i . Do this for every i . The resulting collection of charts is an orienting atlas on M . 2. The volume form on a vector space. Let V be a vector space of dimension n 1 with a chosen orientation. Let , be a positive-definite inner product on V . The volume form on V corresponding to this choice of an orientation and to , is the unique n -form ω on V such that for any positive basis v 1 , . . . , v n of V we have
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