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K G : K ;! O(p). The slices are points since the action is transitive. 7. Lemma. Let M be a proper Riemannian G-manifold, x 2 M . Then the following three statements are equivalent: (1) x is a regular point. (2) The slice representation at x is trivial. (3) Gy = Gx for all y 2 Sx for a su ciently small slice Sx . Proof. Clearly, (2) () (3). To see (3) =) (1), let Sx be a small slice at x. Then U := G:S is an open neighborhood of G:x in M , and for all g:s 2 U we have Gg:s = gGs g;1 = gGx g;1 . Therefore G:x is a principal orbit.

R1 x is G-equivariant. Therefore, S has only nitely many orbit types: those of S n;1 and the 0-orbit. 17. Theorem. If M is a proper G-manifold then the set Msing=G of all singular G-orbits does not locally disconnect the orbit space M=G (that is to every point in M=G the connected neighborhoods remain connected even after removal of all singular orbits). Proof. As in the previous theorem, we will reduce the statement to an assertion about the slice representation. 10, there is a G-invariant Riemann metric on M .

4) If H is a set of isometries, then Fix(H ) = fx 2 M : '(x) = x for all ' 2 H g is also a totally geodesic submanifold in M . 2. De nition. Let M be a proper Riemannian G-manifold, x 2 M . The normal bundle to the orbit G:x is de ned as Nor(G:x) := T (G:x)? Let Nor" (G:x) = fX 2 Nor(G:x) : j X j < "g, and choose r > 0 small enough for expx : TxM Br (0x) ;! M to be a di eomorphism onto its image and for expx (Br (0x)) \ G:x to have only one component. Then, since the action of G is isometric, exp de nes ; a di eomorphism from Norr=2(G:x) onto an open neighborhood of G:x, so exp Norr=2 (G:x) =: Ur=2(G:x) is a tubular neighborhood of G:x.

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