# An Introduction to Tensor Analysis by Leonard Lovering Barrett

By Leonard Lovering Barrett

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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~ (T-l)* # r Since 5 (FT) = 5(F) and -~1, the statement follows. 9. 6) for the conformal diffeomorphism fT : ( ~ ( r ) / r , v r ) -+ ( ~ ( r T ) / r ~ , c r ~ ) induced by the transformation T. 4. 6), one has to normalize Nayatani's metric in order to construct an L2-metric on moduli spaces of flat conformal structures by means of it (cp. 2). We summarize that the Riemannian metrics gKob and gr presented in this section differ strictly from our canonical metric can(C).

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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An Introduction to Tensor Analysis by Leonard Lovering Barrett
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