5. Transformation Groups 83 G˜2 ◦ ◦ ◦
5.3.19. Chevalley Theorem. LetW be a finite Coxeter group of rankkon Rk. Then there existk W-invariant polynomialsu1, . . . , uksuch that the ring ofW-invariant polynomials on Rkis the polynomial ringR[u1, . . . , uk].
Exercises.
1. Classify rank 1 and2 Coxeter groups directly by analytic geometry and standard group theory.
2. SupposeW is a rank3finite Coxeter group on R3. (i) Show thatW leaves S2invariant,
(ii) Describe the fundamental domain ofW on S2 forW =A3, B3.
84 Part I Submanifold Theory is a totally geodesic submanifold ofM.
Proof. This follows from the fact that T Fx is the eigenspace of the linear mapdϕx with respect to the eigenvalue1.
In section 5.2 we used the existence of slices for PFG-manifolds to prove the existence of G-invariant metrics. We will now see that conversely the existence of slices for PF Riemannian actions is easy.
LetN be an embedded closed submanifold of a Riemannian manifoldM. For r > 0 we let Sr(x) = {expx(u)| x ∈ N, u ∈ ν(N)x, u < r}, and νr(N) = {u ∈ ν(N)x | x ∈ N, u < r}. If exp maps νr(N) diffeomorphically onto the open subsetUr = exp(νr(N)), thenUr is called a tubular neighborhood ofN. SupposeM is a PF RiemannianG-manifold and N = Gp. Then there exists anr > 0such thatexppis diffeomorphic on the r-ballBrofT Mpandexpp(Br)∩Nhas only one component (or, equivalently, dM(p, N\expp(Br)) ≥r). ThenUr/2is a tubular neighborhood ofN = Gp inM.
5.4.3. Proposition. LetM be a Riemannian PFG-manifold. Letr > 0be small enough thatUr = exp(νr(Gx))is a tubular neighborhood ofGxinM. LetSx denoteexpx(νr(Gx)x). Then
(1)Sgx = gSx,
(2)Sx is a slice atx, which will be called the normal slice atx.
Proof. (1) is a consequence of Proposition 5.4.1. Sinceνr(Gx)is a tubular neighborhood, Sx andSy are disjoint ifx = y. So if gSx∩Sx = ∅, thenSgx = Sx andgx= x.
LetM be aG-manifold. The differential of the actionGx defines a linear representationιof Gx on T Mx called the isotropy representation atx. Now suppose that M is a RiemannianG-manifold. Thenιis an orthogonal repre-sentation, and the tangent spaceT(Gx)x to the orbit ofxis an invariant linear subspace. So the orthogonal complementν(Gx)x, i.e., the normal plane ofGx inM atx, is also an invariant linear subspace, and the restriction of the isotropy representation ofGx toν(Gx)x is called the slice representation atx.
5.4.4. Example. Let M = Gbe a compact Lie group with a bi-invariant metric. LetG×Gact onGby(g1, g2)·g =g1gg2−1. ThenMis a Riemannian G × G-manifold (in fact a symmetric space), Ge is the diagonal subgroup {(g, g)| g ∈ G}, and the isotropy representation ofGe Gon T Ge = G is just the adjoint action as in Example 5.1.4 (3).
5.4.5. Example. LetM = G/K be a compact symmetric space, and G = K+Pis the orthogonal decomposition with respect to−b, wherebis the Killing form on G. Then T MeK = P and GeK = K. Let ad denote the adjoint representation ofGonG. Then ad(K)(P) ⊆ P. So it gives a representation
5. Transformation Groups 85 of K on P, which is the isotropy representation ofM at eK. For example, M = (G×G)/Ggives Example 5.4.4.
5.4.6. Remark. The set of all isotropy representations for non-compact sym-metric spaces is the same as the set of all isotropy representations for compact symmetric spaces.
5.4.7. Proposition. LetM be a Riemannian PFG-manifold, andx ∈ M. ThenGxis a principal orbit if and only if the slice representation atxis trivial.
Proof. Let S denote the normal slice at x. Then Gy ⊆ Gx for all y ∈S. SoGxis a principal orbit if and onlyGy = Gx for ally ∈S, i.e.,Gx
fixesS. Then the result follows from Proposition 5.4.1.
5.4.8. Corollary. LetM be a RiemannianG-manifold,xa regular point, and Sx the normal slice atxas in Proposition 5.4.3. ThenGy = Gx for all y ∈ Sx.
5.4.9. Corollary. LetMbe a RiemannianG-manifold,Gxa principal orbit, andv ∈ν(Gx)x. Thenˆv(gx) =dgx(v)is a well-defined smooth normal vector field ofGxinM.
Proof. Ifgx= hx, theng−1h∈Gx. By Proposition 5.4.7,d(g−1h)x(v) = v, which implies thatdgx(v) =dhx(v).
5.4.10. Definition. LetMbe a RiemannianG-manifold, andNan orbit ofM. A sectionuofν(N)is called an equivariant normal field ifdgx(u(x)) =u(gx) for allg ∈Gandx∈ N.
5.4.11. Corollary. Let M be a Riemannian G-manifold, Gx a principal orbit, and{vα}an orthonormal basis forν(Gx)x. Letvˆα be the equivariant normal field defined byvαas in Corollary 5.4.9. Then{ˆvα}is a global smooth orthonormal frame field onGx. In particular, the normal bundle ofGxinM is trivial.
5.4.12. Proposition. LetMbe a RiemannianG-manifold,Nan orbit inM, andvan equivariant normal field onN. Then
(1) Av(gx) = dgx ◦ Av(x)◦ dgx−1 for all x ∈ N, where Av is the shape operator ofN with respect to the normal vectorv,
(2) the principal curvatures ofN alongvare constant, (3){exp(v(x))|x∈ N}is again aG-orbit.
86 Part I Submanifold Theory
Proof. Sincedgx(T Nx) =T Ngxandgis an isometry, (1) follows. (2) is a consequence of (1). Sincev(gx) =dgx(v(x)), (3) follows from Proposition 5.4.1.
5.4.13. Corollary. Let Nn(c) be the simply connected space form with constant sectional curvaturec,Ga subgroup of Iso(Nn(c)),MaG-orbit, and v an equivariant normal field on M. Then {Y(v(x)) | x ∈ M} is again a G-orbit, whereY is the endpoint map ofM inNn(c).
We will now consider the orbit types of PF actions.
5.4.14. Proposition. IfM is a PFG-manifold, then there exists a principal orbit type.
Proof. By Remark 5.2.3 all of the isotropy subgroups ofMare compact.
It follows that there exists an isotropy subgroup,Gx, having minimal dimension and, for that dimension, the smallest number of components. By Theorem 5.2.6, there exists a slice S atx. ThenGS is an open subset, andGs ⊆ Gx for all s ∈ S. By the choice ofxit follows that in factGs = Gx for alls ∈ S. But thenGgs =gGsg−1 =gGxg−1, so(Gx)is a principal orbit type.
5.4.15. Theorem. If M is a PF G-manifold, then the set Mr of regular points is open and dense.
Proof. Openness follows from the existence of slice. To prove dense-ness, we proceed as follows: LetU be an open subset ofM, x∈ U, andS a slice at x. Choose y ∈ GS ∩U so that Gy has smallest dimension and, for that dimension, the smallest number of components. Let S0 be a slice at y, andz∈ GS0∩U∩GS. It follows form Corollary 5.1.13 (2) that there exists g ∈ Gsuch that Gz ⊆ gGyg−1. Since the dimension of Gy is less than or equal to the dimension of Gz, we conclude that Gz andGy in fact have the same dimension, and then since the number of components ofGy is less than or equal to the number of components ofGz,Gz = gGyg−1. This proves that Gyis a principal orbit.
5.4.16. Theorem. IfM is a PFG-manifold then given a pointp∈M there exists aG-invariant open neighborhoodU containingpsuch thatU has only finitely manyG-orbit types.
Proof. By Theorem 5.2.7 we may assume thatM is a PF Riemannian G-manifold. Let S be the normal slice at p. Then S is of finite dimension,
5. Transformation Groups 87 and Gp is a compact group acting isometrically on S so, by 5.1.13(4), it will suffice to prove this theorem for Riemannian G-manifolds of finite dimension n. We prove this by induction. Forn = 0the theorem is trivial. Suppose it is true for all properG-manifolds of dimension less thannand letM be a proper RiemannianG-manifold of dimensionn, p ∈ M, andS the normal slice atp. By 5.1.13(4) again, it will suffice to prove that locallyS has only finitely many orbit types. If dim(S) < n, then this follows from the induction hypothesis, so assume that dim(S) = n. Then by Proposition 5.4.1 theGp-actionρonS is an orthogonal action on T Mp = Rn with respect to geodesic coordinates.
Now Gp leaves Sn−1 invariant and, by the induction hypothesis, locally Sn−1 has only finitely many orbit types. But then because Sn−1 is compact, it has finitely many orbit types altogether. Now note that, in a linear representation, the isotropy group (and hence the type of an orbit) is constant on any line through the origin, except at the origin itself. Soρhas at most one more orbit type onS than on Sn−1, and hence only finitely many orbit types.
5.4.17. Theorem. IfM is a PF G-manifold, then the set M˜s = Ms/Gof singular orbits does not locally disconnect the orbit spaceM˜ = M/G.
Proof. Using the slice representation as in the previous theorem, it suffices to prove this theorem for linear orthogonalG-action on Rn. We proceed by induction. Ifn= 1, then we may assume thatG= O(1) =Z2. It is easily seen that R/G is the half line {x| x ≥ 0} with 0 as the only singular orbit.
So {0} does not locally disconnect R/G. SupposeG ⊆ O(n). Applying the induction hypothesis to the slice representation of Sn−1, we conclude that the set of singular orbits of Sn−1 does not locally disconnect Sn−1/G. But Rn/G is the cone over Sn−1/G. So the set of singular orbits of Rn does not locally disconnect Rn/G.
5.4.18. Corollary. IfM is a connected PFG-manifold, then (1)M/Gis connected,
(2)M has a unique principal orbit type.
5.4.19. Corollary. SupposeM is a connected PF G-manifold. Let m = inf{dim(Gx)|x ∈ M}, andkthe smallest number of components of all the dimensionmisotropy subgroups. Then an orbitGx0is principal if and only if Gx0 has dimensionmandkcomponents.
88 Part I Submanifold Theory