By Franco Tricerri; Lieven Vanhecke
The imperative subject of this ebook is the theory of Ambrose and Singer, which supplies for a attached, entire and easily attached Riemannian manifold an important and adequate for it to be homogeneous. it is a neighborhood which should be happy in any respect issues, and during this approach it's a generalization of E. Cartan's approach for symmetric areas. the most objective of the authors is to exploit this theorem and illustration thought to provide a category of homogeneous Riemannian buildings on a manifold. There are 8 periods, and a few of those are mentioned intimately. utilizing the confident facts of Ambrose and Singer many examples are mentioned with distinct cognizance to the normal correspondence among the homogeneous constitution and the teams performing transitively and successfully as isometrics at the manifold.
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Extra resources for Homogeneous structures on Riemannian manifolds
A = I, ... ,r . 9. Hence, if e is a fixed point of G (we always suppose E p- 1(u)), then there exists a unique Lie group structure on G such that e e is the identity and such that 28 B1 , ••• ,Bn,Af•···•A; are left invariant vector fields. 64)). connected subgroup of G with Lie algebra g0 . 19. Proof. Note that Let G0 be the We first prove M and G/G0 are diffeomorphic. 68) p. 0 Then rr 1 : G + M is a fibre bundle with projection map rr 1• Further, it follows from the bundle homotopy sequence (see [46, p.
Proof. Note that Let G0 be the We first prove M and G/G0 are diffeomorphic. 68) p. 0 Then rr 1 : G + M is a fibre bundle with projection map rr 1• Further, it follows from the bundle homotopy sequence (see [46, p. 377]) and from the fact that M is simply connected, that the fibres are connected. is continuous, they are also closed. • ,r , (1. 69) and hence the fibres are tangent to g0 . As a consequence the fibres are maximal integral submanifolds of the involutive distribution determined by g0 • So the fibres are the classes a c"' function aG0 • The projection map rr 1 induces which is a diffeomorphism since (rr 2):: is an isomorphism at each point and rr 2 is a 1 - 1 function.
P B of ~o(V) is given by fo~ tr(~P where 'P and 1jJ are skew-symmetric endomorphisms of V. bilinear form which is negative definite. <1> p p (a,S) = -B(A .. 1 ,A 0 .. , .. P B is a symmetric Since a:: is a Killing vector field, Aa:: is skew-symmetric (see section A). 41) 1/1) o Hence we can put , a,S E is a symmetric bilinear form on g which depends on p. have The g. (1. 13. positive definite on kp. Proof. Let f3 E kp with ~p (f3,f3) = 0. Then As::IP = 0. 2 that s:: is identically zero. 14. ~ p E = 0.