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(TxMn,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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