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By Kapovich M.

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Let G be a group of transformations of M. Then, G is called a Lie group of transformations of M if G is a Lie group such that ¢ : G x M -+ M is smooth and if whenever X is a complete vector field on M such that the corresponding global R-transformation group ¢ has the property that the set of transformations {

SEMI-RIEMANNIAN MANIFOLDS AND HYPERSURFACES PROOF. Since T(1:)J.. , consider a complementary distribution E to T(1:)J.. in SJ... 56) that SJ.. is degenerate at least at one point of U. As in the case of null curves, define on U a null vector field 1 g(W, W) N = g(£, W) {W- 2g(£, W) £}. 1. 56) we have the following decomposition: T(M) = S EB (T(1:)J.. 58) + denotes non-orthogonal complementary sum and T(1:) = S EB T(1:)J... 59), we observe that T(1:)J.. is integrable since it is a 1-dimensional distribution on 1:.

En} defined on a coordinate neighborhood U C M and preserving their causal character along U such that {E 1 , ••. , En }p is a basis for each p E U, is called a local orthonormal frame field on M. Based on above definition we have n g( Ei, Ej) = fi 8ij (no summation in i) X= Lfig(X,Ei)Ei, i=l where {Ei} is the signature of {Ei}. 25) i=l If Ricci tensor vanishes on M we say that M is Ricci flat. 26) then M is called an Einstein manifold. 26) is not necessarily constant. 26) we deduce that M is an Einstein manifold if and only if r is constant and .

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