1547671870-The_Ricci_Flow__Chow

2. MODEL GEOMETRIES 5

on TxMn, one obtains a Yx-invariant scalar product gx by averaging §x un-
der the action of Yx ".::=: g*. Then since g is transitive, there exists for any

y E Mn an element 'Yyx E g such that 'Yyx (y) = x. So one may define a

scalar product gy for all V, W E TyMn by

gy (V, W) ~ b;xgx) (V, W) = gx (hyxt V, ('Yyxt W).

To see that gy is well defined, suppose that 'Y~x is another element of g such

that "f~x (y) = x. Then "f~x o "f;} E Yx, so by Yx-invariance of gx, we have

('Y;xgx) (V, W) = gx ( ('Y~x^0 'Y;x^1 )J'Yyx) V, ('Y~x o 'Y;x^1 )J'Yyx) w)

= gx ( b~xL V, b~xL w) = (b~xr gx) (V, W).

Because the action of g is smooth, one obtains in this way a smooth Rie-

mannian metric g on Mn. Because the action of g 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 (Mn,g) may be

regarded as a model geometry (Mn, 9, 9*). We begin by recalling some

classical facts about the set Isom (Mn, g) of isometries of a Riemannian

manifold. Isom (Mn, g) is clearly a subgroup of the diffeomorphism group

'.D (Mn). Moreover, it is clear that (Mn, g) is homogeneous if and only if

Isom (Mn, g) acts transitively on Mn. The following facts are classical. (See

[100] and [87].)

PROPOSITION 1.10. Let (Mn, g) be a smooth Riemannian man4old.

( 1) Isom (Mn, g) is a Lie group and acts smoothly on Mn.

(2) For each x E Mn, the isotropy group

Ix (Mn,g) ~ b E Isom (Mn,g): "( (x) = x}

is a compact subgroup ofisom(Mn,g).

(3) For each x E Mn, the linear isotropy representation

defined by

Ax ( "() = "(* : TxMn --* TxMn

is an injective group homomorphism onto a closed subgroup of the

orthogonal group 0 (TxMn, g ( x)). In particular,

Ix (Mn, g) ".::=:Ax (Ix (Mn, g)) ~ 0 (TxMn, g (x))

is compact.