By Leonard Lovering Barrett
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Additional info for An Introduction to Tensor Analysis
Then f'can(C2) = can(C1) . In particular, for (M1, C1) = (M2,C2) = (M, C), can(C) is invariant under the group of con]ormal transformations of (M, C). Yq The most important example of a scalar positive, fiat conformal structure is the conformal class Cs of the standard metric gs of the sphere S u. 2) will be (cp. 5 Let M be a connected and closed manifold of dimension n >_3 with a scalar positive, flat conformal structure C. If (M, C) is conformally diffeomorphic to (Sn, Cs), then can(C) vanishes identically.
2), one deduces v~#Tr : ( T - 1 ) * 7 . T . #T = (c+, o = (T-l)" (c+ 1 IT' o'~1~
6) /M L (/M 6(I9,q)ut (p) d#[g](p)) u2(q)d#[g](q) *'~/M (/M G(p,q)ul (p) d/~[g](p)) Lu2(q)d#[g](q) = /M (/M G(p'q)Lu2(q)d#[g](q)) uI(p) d#[g](p) = Cn [ JM u2(P)Ul (P) d#[gl(p) , which gives L(/MG(p,q)ul(p)d/~[g](p) ) With = Cnul(q) . 7) leads to L (/M G(p, q)Lu(p) d/~[g](p)) = cnLu(q) . 2 The conformal Laplacian 21 Since L is invertible, it follows M G(P, q)Lu(p) d#[g](p) = ¢nu(q) for all u E C ~ ( M ) . 6). 9 Let M be closed and let g be a Riemannian metric on M which is conformally equivalent to a Riemannian metric with positive scalar curvature.
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