By Wraith, Gavin

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**Example text**

F; point Xo of M x Sn, let (X S) be a system of normal coordinates with (Xl, ... , xm) coordi nates in M and (xm + 1, ... , X m + n) coordinates along Sn. We have orthogonal eigenvectors: JVj = (asasfi)v j + 2(a s fi)va s Vj + f;va s V[svj' But for each s, (asfi) . (vas Vi) = 0 since one of the factors must be zero, so that V~s v~/fi . Vj) 4. fi . Vi + fi . IVj' = fi Riccisn(vj ), we get Jrr(fi . Vj) ,. f;· Vj + fi . JVj = (fJ.. v.. i are mUltiplied by l/c. 9) COROLLARY. 8), with M = Sm (m ;:;.

PROOF. At a point x o ' let X, Y, Z be three tangent vectors, and extend Y and Z in a neighborhood in such a way that at Xo we have vxY == 0 vxZ. Then, at xo: )tain (vxfjJ*h)(Y, Z)::: vx(fjJ*h(Y, Z)) == since vdfjJ = 0, so Nowe I> = 0). ItfjJ :rmonic. rpothesis. t harmonic). == vx(dfjJ' Y, dfjJ' Z) v < xdfjJ . Y, dfjJ . Z) + (dfjJ . Y, vxdfjJ . Z) 0, that v(fjJ*h) == O. *(g, fjJ*h) so that Therefore, vS", = O. We shall also use the stress-energy tensor in subsequent sections. REMARK (ADDED IN PROOF).

Therefore, PROOF. e(O) < maxldl/l1 2 • e(¢) yEK and e(O) Thus, if 1/1: S -+ < e(I/I)' maxld(/)12. x ElvI N is any smooth map, its composition with maps ¢i: S -+ S homotopic to the identity and such that £(¢i) -+ 0 provides a sequence 1/1 a ¢I homotopic to 1/1 with r K such that there is a num < p. Thus, For the second assertion, let ¢: M -+ S be a Riemannian submersion and 1/11: S sequence of maps homotopic to the identity with £(1/11) -+ O. Then £(1/11 of spheres of 0 ¢) < maxld¢12 . fHe(I/I,)(¢(x»vgCx) < maxld¢12 .

### Algebraic theories by Wraith, Gavin

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