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By Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

ISBN-10: 0821834762

ISBN-13: 9780821834763

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Example text

Note that, in this case, K D R1212 D R11 D R22 D 1 S: 2 We observe that the previous result often goes under the name of Schur’s theorem. We are now going to show a similar fact that, in the recent literature, also goes under the same name. 65) for some 2 R. p/ of class C1 , then is constant. 67) Rjt being the components of the Ricci tensor. 68) (this equation is sometimes called Schur’s identity). 68) yields  2 m à 1 S;t D 0; and we conclude that, if m 3, the scalar curvature, and therefore , is constant.

Note that the definition of a Darboux (co)frame is equivalent to say that the vectors fEi g (locally) span f TM, the image of TM through f in TN, while the vectors fE˛ g are orthogonal to f TM and span in fact the normal bundle TM ? (sometimes denoted by NM), that is the set of (local) vector fields in N that are orthogonal to f TM. 124) where f  ˛ is the pullback of  ˛ by the map f . Indeed, for every i, . ei / D  ˛ . Ei / D 0. 36 1 A Crash Course in Riemannian Geometry ˚ « Let now Âba be the Levi-Civita connection forms of N relative to f a g.

3 The matrix of curvature 2-forms D . n/ and, if e e D eA is a (local) change of orthonormal frame with A W U ! n/-valued 2-form. A D. A 1 / ^ dA C A 1 dA A 1 ^ dA C A and this proves the proposition. A DA 1 dA ^ A dA ^ A 1 dA ^ A 1 1 1 ACA / ^ dA C A 1 1 dA/ . 0; 4/ version). 43) This identity goes under the name of first Bianchi identity. 44) should be called “Ricci identity”. Rktij C Rkijt / C Rtjki D Rktij C Rkijt C . 34) again to obtain 0 D dÂli ^ Âjl D . dRljkt C Rijkt Âli D 2 C 1 i dR ^  k ^  t 2 jkt Âli ^ dÂjl 1 i 1 R d k ^  t C Rijkt  k ^ d t 2 jkt 2 1 i dR ^  k ^  t Âli ^ .

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Affine Algebraic Geometry by Gutierrez J., Shpilrain V., Yu J.-T. (eds.)


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