2. MODEL GEOMETRIES 5 on TxMn, one obtains a ^-invariant scalar product gx by averaging gx un- der the action of Qx = G*. Then since G is transitive, there exists for any y G Mn an element j y x G G such that ^yx (y) = x. So one may define a scalar product gy for all V,W e TyMn by 9v (V, W) = ( 7 * ) (V, W) = gx ((7yx)„ V, (7,,), W) . To see that gy is well defined, suppose that j y x is another element of Q such that 7 ^ {y) x. Then 7 ^ o 7 ^ G £ x , so by £/x-invariance of gx, we have (7^fe) (V, W) = flx ( ( 7 ^ o 7 " ^ ( 7 , ^ V, (yyx o 7 ^ ( 7 ^ w ) = 9, ( ( 7 ^ ) , V, ( 7 ^ ) , W) = {{iyxY gx) (V, W). Because the action of Q is smooth, one obtains in this way a smooth Rie- mannian metric g on Mn. Because the action of Q is transitive, g is complete by Proposition 1.8. D To complete the connection between homogeneous models and model geometries, we want to establish the converse of the construction above. Namely, we want to show that every homogeneous model {M.n,g) may be regarded as a model geometry (Ain,G,G*) We begin by recalling some classical facts about the set Isom (.M71,5) of isometries of a Riemannian manifold. Isom(.Mn, #) is clearly a subgroup of the diffeomorphism group 2) (Mn). Moreover, it is clear that (Mn,g) is homogeneous if and only if Isom (A^n, g) acts transitively on Mn. The following facts are classical. (See [100] and [87].) PROPOSITION 1.10. Let (Mn,g) be a smooth Riemannian manifold. (1) Isom (Mn,g) is a Lie group and acts smoothly on Mn. (2) For each x G Mn, the isotropy group h (Mn, g) = {^e Isom (Mn, g) : 7 (*) = x} is a compact subgroup of Isom. (M.n, g). (3) For each x G Mn, the linear isotropy representation \x:Ix(Mn,g)^0(T x Mn,g(x)) defined by Xx (7) = 7* : TxMn - TxMn is an infective group homomorphism onto a closed subgroup of the orthogonal group O (TxAin, g (x)). In particular, Ix (Mn, g) * Xx (Ix (Mn, g)) C O (TxMn, g (x)) is compact.
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