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Hyperpolar Homogeneous Foliations on Symmetric Spaces of Noncompact Type

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Hyperpolar Homogeneous Foliations on Symmetric Spaces of Noncompact Type HYPERPOLAR HOMOGENEOUS FOLIATIONS ON SYMMETRIC SPACES OF NONCOMPACT TYPE JURGEN¨ BERNDT Abstract. A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jos´eCarlos D´ıaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in [1], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SLr+1(R)/SOr+1. 1. Isometric actions on the real hyperbolic plane The special linear group a b G = SL2(R) = a, b, c, d ∈ R, ad − bc = 1 c d acts on the upper half plane H = {z ∈ C | =(z) > 0} by linear fractional transformations of the form a b az + b · z = . c d cz + d By equipping H with the Riemannian metric dx2 + dy2 dzdz¯ ds2 = = (z = x + iy) y2 =(z)2 we obtain the Poincar´ehalf-plane model of the real hyperbolic plane RH2. The above action of SL2(R) is isometric with respect to this metric. The action is transitive and the stabilizer at i is cos(s) sin(s) K = s ∈ R = SO2. − sin(s) cos(s) 2000 Mathematics Subject Classification. Primary 53C12, 53C35; Secondary 53S20, 22E25. Key words and phrases. Riemannian symmetric spaces of noncompact type, homogeneous foliations, hyperpolar foliations, polar foliations, horospherical decompositions. 1 The subgroup K is a maximal compact subgroup of G, and we can identify RH2 with the homogeneous space G/K = SL2(R)/SO2 in the usual way. There is a unique decomposi- tion of every matrix in SL2(R) into three matrices of the form a b cos(s) sin(s) exp(t) 0 1 u = (s, t, u ∈ ). c d − sin(s) cos(s) 0 exp(−t) 0 1 R We define an abelian subgroup A of G by exp(t) 0 A = t ∈ R 0 exp(−t) and a nilpotent subgroup N of G by 1 u N = u ∈ R . 0 1 Then G is diffeomorphic to the manifold product KAN. The decomposition G = KAN is known as an Iwasawa decomposition of G. Each of three subgroups K, A and N acts isometrically on RH2. • The orbits of K consist of the single point i, which is the fixed point of the action of K on RH2, and the geodesic circles in RH2 whose center is at i. • The orbits of A are the geodesic R → RH2, t 7→ exp(2t)i and the curves in RH2 which are equidistant to this geodesic. • The orbits of N are the horospheres given by the equations =(z) = y, y ∈ R+. The action of K has exactly one singular orbit, and the orbits of A and N form a Riemannian foliation on the real hyperbolic plane. Every isometric action on RH2 by a nontrivial connected subgroup H of G is orbit equivalent to one of these three actions, that is, there is an isometry of RH2 which maps the orbits of the action of H onto the orbits of the action of either K, or of A, or of N. The real hyperbolic plane is the most elementary example of a Riemannian symmetric space of noncompact type. A joint project with Jos´eCarlos D´ıaz Ramos (Santiago de Compostela) and Hiroshi Tamaru (Hiroshima) is to generalize aspects of what we described above from RH2 to Riemannian symmetric spaces of noncompact type. 2. The problem There are several natural ways to generalize what we discussed in the previous section. The cohomogeneity of each of the three actions is equal to one. The cohomogeneity of an isometric action is the codimension of a principal orbit of the action, or equivalently, the dimension of the orbit space of the action. A cohomogeneity one action is therefore an action for which the principal orbits of the action are hypersurfaces. Cohomogeneity one actions are special cases of polar and hyperpolar actions. Let M be a connected complete Riemannian manifold and H a connected closed subgroup of the isometry group I(M) of M. Then each orbit H ·p = {h(p) | h ∈ H}, p ∈ M, is a connected closed submanifold of M. A connected complete submanifold S of M that meets each orbit of the H-action and intersects the orbit H ·p perpendicularly at each point p ∈ S is called a 2 section of the action. A section S is always a totally geodesic submanifold of M (see e.g. [6]). In general, actions do not admit a section. The action of H on M is called polar if it has a section, and it is called hyperpolar if it has a flat section. For motivation and classification of polar and hyperpolar actions on Euclidean spaces and symmetric spaces of compact type we refer to the papers by Dadok [4], Podest`aand Thorbergsson [12], and Kollross [9], [10]. If all orbits of H are principal, then the orbits form a homogeneous foliation F on M. In general, a foliation F on M is called homogeneous if the subgroup of I(M) consisting of all isometries preserving F acts transitively on each leaf of F. Homogeneous foliations are basic examples of metric foliations. A homogeneous foliation is called polar resp. hyperpolar if its leaves coincide with the orbits of a polar resp. hyperpolar action. The problem I want to discuss in this article is the classification of hyperpolar homoge- neous foliations on Riemannian symmetric spaces of noncompact type. 3. Riemannian symmetric spaces of noncompact type This section contains a brief introduction to Riemannian symmetric spaces of noncom- pact type. A basic reference is the book [7] by Sigurdur Helgason. A Riemannian symmetric space is a connected Riemannian manifold M with the prop- erty that for each point p ∈ M there exists an isometric involution sp ∈ I(M) for which p is an isolated fixed point. Geometrically, such an involution sp is the geodesic reflection of M in the point p. Every Riemannian symmetric space is homogeneous, that is, the isometry group I(M) acts transitively on M. The Riemannian universal covering space Mf of a Riemannian symmetric space M is also a Riemannian symmetric space, and each factor in the de Rham decomposition Mf = M0 × M1 × · · · × Mk is a Riemannian symmetric space as well. Here M0 is the Euclidean factor (which may be 0-dimensional), and each manifold Mi, 1 ≤ i ≤ k, is an irreducible simply connected Riemannian symmetric space. A Riemannian symmetric space M is said to be of compact type if dim M0 = 0 and each Mi is compact, and it is said to be of noncompact type if dim M0 = 0 and each Mi is noncompact. Every Riemannian symmetric space of noncom- pact type is simply connected. The rank of a Riemannian symmetric space is the maximal dimension of a flat totally geodesic submanifold. Riemannian symmetric spaces have been classified by Elie´ Cartan. The Riemannian symmetric spaces of noncompact type and with rank one are the hyperbolic spaces FHn over the normed real division algebras F ∈ {R, C, H, O}, where n ≥ 2 and n = 2 if F = O. Let M be a connected Riemannian symmetric space of noncompact type. We denote by n the dimension of M and by r the rank of M. It is well known that M is diffeomorphic to Rn. Let G = Io(M) be the connected component of I(M) containing the identity transformation of M. Then G is a connected semisimple real Lie group with trivial center. Let o ∈ M and K = {g ∈ G | g(o) = o} be the isotropy subgroup of G at o. Then K is a maximal compact subgroup of G. We identify M with the homogeneous space G/K in 3 the usual way and denote by g and k the Lie algebra of G and K, respectively. Let B : g × g → g , (X, Y ) 7→ tr(ad(X) ◦ ad(Y )) be the Killing form of g. It is well-known from Cartan’s criterion for semisimple Lie algebras that the Killing form B is non-degenerate. Let p = {X ∈ g | B(X, Y ) = 0 for all Y ∈ k} be the orthogonal complement of k with respect to B. Then g = k ⊕ p is a Cartan decomposition of g, that is, we have [k, k] ⊂ k , [k, p] ⊂ p , [p, p] ⊂ k, and B is negative definite on k and positive definite on p. The Cartan involution cor- responding to the Cartan decomposition g = k ⊕ p is the automorphism of g defined by θ(X) = X for all X ∈ k and θ(X) = −X for all X ∈ p. We can define a positive defi- nite inner product on g by hX, Y i = −B(X, θY ) for all X, Y ∈ g. As usual, we identify p with ToM, and we normalize the Riemannian metric on M so that its restriction to ToM × ToM = p × p coincides with h · , · i. Example. M = SLr+1(R)/SOr+1. The linear group GLr+1(R) can be identified with the group of all regular (r + 1) × (r + 1)-matrices with real coefficients, or equivalently, r+1 with the group of all automorphisms of R . The special linear group G = SLr+1(R) of all orientation- and volume-preserving transformations in GLr+1(R) is SLr+1(R) = {A ∈ GLr+1(R) | det(A) = 1}.
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