Submanifolds in Symmetric Spaces

Transcription

1 Submanifolds in Symmetric Spaces Jürgen Berndt University College Cork OCAMI-KNU Joint Workshop on Differential Geometry and Related Fields Submanifold Geometry and Lie Theoretic Methods Osaka City University, 30 October - 3 November 2008

2 Overview I. Euclidean spaces 1. Hypersurfaces with constant principal curvatures 2. s-representations 3. Normal holonomy 4. Polar representations II. Symmetric spaces 1. Cohomogeneity one actions: general theory 2. Cohomogeneity one actions: classifications 3. Hypersurfaces with constant principal curvatures 4. Hyperpolar and polar actions

3 General problems in submanifold geometry Find good geometric invariants for submanifolds, e.g. principal curvatures mean curvature Classify submanifolds according to these invariants, e.g. all principal curvatures vanish: classification of totally geodesic submanifolds all principal curvatures are constant: classification of isoparametric submanifolds mean curvature vanishes: classification of minimal submanifolds We discuss a particular example:

4 Surfaces in E 3 Problem. What are the surfaces in E 3 having the same shape everywhere? In other words, what are the surfaces in E 3 with constant principal curvatures? Examples: affine planes, round cylinders, round spheres Question: Are there others? Let s investigate this problem Let P E 3 be a surface with constant principal curvatures λ 1, λ 2 and unit normal field ξ

5 Surfaces in E 3 (II) Case 1: λ 1 = λ 2 =: λ (= P umbilical surface in E 3 ) λ = 0 = P totally geodesic = P open part of an affine plane in E 3 λ > 0. Consider F : P E 3, p p + 1 λ ξ p Then D p F (u) = u + 1 λ ( A ξ p u) = 0 for all u T p P; A shape operator of P. Thus F is constant, say F (p) = z E 3. Conclusion: P lies on the sphere with center z and radius 1/λ

6 Surfaces in E 3 (III) Case 2: λ 1 λ 2 Choose local unit vector fields E 1, E 2 Γ(TP) with AE i = λ i E i Then the Codazzi equation implies and hence (λ i λ j ) P E i E i, E j = ( P E i A)E i, E j = ( P E j A)E i, E i = 0 P E i E i = 0 The Codazzi equation also implies and hence λ 1 P E 2 E 1 A P E 2 E 1 = ( P E 2 A)E 1 = ( P E 1 A)E 2 = λ 2 P E 1 E 2 A P E 1 E 2 A[E 1, E 2 ] = λ 2 P E 1 E 2 λ 1 P E 2 E 1

7 Surfaces in E 3 (IV) Now use the Gauß-Codazzi equations for λ 1 λ 2 = R P (E 1, E 2 )E 2, E 1 = P E 1 P E 2 E 2, E 1 P E 2 P E 1 E 2, E 1 P [E 1,E 2 ] E 2, E 1 = E 2 P E 1 E 2, E 1 + P E 1 E 2, P E 2 E 1 1 ( P [E λ 2 λ 1,E 2 ] A)E 2, E = ( P E λ 1 λ 1 A)E 2, [E 1, E 2 ] 2 1 = (λ 2 P E λ 1 λ 1 E 2, [E 1, E 2 ] P E 1 E 2, A[E 1, E 2 ] ) 2 = 0 Thus λ 1 = 0 or λ 2 = 0. Assume λ 2 = 0 and λ 1 > 0.

8 Surfaces in E 3 (V) Define F : P E 3, p p + 1 λ 1 ξ p Then DF (E 1 ) = 0, DF (E 2 ) = ( 1 λ ) 2 E 2 = E 2 λ 1 Thus F has constant rank one and hence for each point p P there exists an open neighborhood U of p in P such that F : U E 3 is a submersion onto a one-dimensional submanifold Q P lies on the tube of radius 1/λ 1 around Q What is the geometry of Q?

9 Surfaces in E 3 (VI) Let c be an integral curve of E 2 Then α(ċ, ċ) = α(ċ, ċ), ξ c ξ c = A ξ ċ, ċ = 0 and Ṗ c ċ = ( P E 2 E 2 ) c = 0 Thus c is a geodesic in E 3 by Gauß formula, hence a straight line segment Moreover, E3 ċ ξ = A ξ ċ = 0, which shows that ξ is parallel along c Thus F c is a straight line segment as well Altogether this implies that Q is a straight line segment, and hence P lies on a round cylinder in E 3 Thus we have proved the following result:

10 Surfaces in E 3 (VII) Somigliana 1919: A connected surface in E 3 has constant principal curvatures if and only if it is an open part of a plane in E 3, or a sphere of radius r R + in E 3, or a tube of radius r R + around a line E 3 It is remarkable that the surfaces listed above are precisely the homogeneous surfaces in E 3. WHY?

11 Hypersurfaces in E n Segre 1938: A connected hypersurface in E n has constant principal curvatures if and only if it is an open part of a homogeneous hypersurface: an affine hyperplane in E n, or a tube around an affine subspace of E n of dimension k {0,..., n 2} Again, it is remarkable that the hypersurfaces listed above are precisely the homogeneous hypersurfaces in E n. WHY?

12 Main ingredients of proof Fundamental equations of submanifold theory Focal set theory

13 Fundamental equations of submanifold geometry P submanifold of Riemannian manifold M Gauss formula: M X Y = P X Y + α(x, Y ) Weingarten formula: M X ξ = A ξx + X ξ Gauss equation: R P (X, Y )Z, W = R M (X, Y )Z, W + α(y, Z), α(x, W ) α(x, Z), α(y, W ) Codazzi equation: (R M (X, Y )Z) = ( X α)(y, Z) ( Y α)(x, Z) Ricci equation: R M (X, Y )ξ, η = R (X, Y )ξ, η [A ξ, A η ]X, Y

14 Focal set theory The geometry of focal sets in Riemannian geometry can be studied using Jacobi fields P hypersurface of Riemannian manifold M p P : T p M = T p P ν p P ξ (local) unit normal vector field of M γ geodesic in M with p = γ(0) P and ξ p = γ(0) R M γ = R M (, γ) γ Jacobi operator along γ A vector field Y along γ is called a Jacobi field if Y = R M γ Y A Jacobi field Y along γ is called a P-Jacobi field if Y (0) T p P and Y (0) + A ξ Y (0) ν p P Y 0 (t) = t γ(t) is a P-Jacobi field along γ

15 Focal set theory (II) J(P, γ) = real vector space of all P-Jacobi fields along γ which are perpendicular to Y 0 dim J(P, γ) = dim P exp P : νp M normal exponential map of P P r = {exp P (rξ p ) p P} displacement of P in direction ξ at distance r R + Assume that P r is a submanifold of M A r shape operator of P r with respect to γ(r) Theorem. T γ(r) P r = {Y (r) Y J(P, γ)} and A r Y (r) = (Y (r)) T

16 Symmetric spaces A connected Riemannian manifold M is called a symmetric space if for all p M there exists an isometry s p of M such that s 2 p = id M p is an isolated fixed point of s p Examples: M = E n : s p = reflection in p M = S n E n+1 : s p = reflection in line through p and 0

17 Basics about symmetric spaces Let M be a symmetric space M is homogeneous, i.e., the group G = I (M) of isometries of M acts transitively on M The Riemannian curvature tensor of M is parallel The Riemannian universal covering space M is a symmetric space with derham decomposition where M = M 0 M 1 M k, M0 = E n0 for some n 0 Z 0 ; M 0 is called the Euclidean factor of M Mk, k 1, is a simply connected irreducible symmetric space If n 0 = 0 then M is called a semisimple symmetric space If (n 0, k) = (0, 1) then M is called a simple symmetric space

18 Basics about symmetric spaces (II) If M is semisimple, then the Lie algebra g of G = I (M) is semisimple, i.e., the Killing form B : g g R, (X, Y ) tr(ad(x ) ad(y )) is non-degenerate If M is simple, then the Lie algebra g of G = I (M) is simple, i.e., g is semisimple and the only ideals in g are {0} and g If M is simple then the sectional curvature of M is either 0 or 0 If M is simple and the sectional curvature of M is 0, then M is compact with finite fundamental group If M is simple and the sectional curvature of M is 0, then M is noncompact and simply connected

19 Basics about symmetric spaces (III) A symmetric space is said to be of compact type if it is semisimple and the sectional curvature is 0 A symmetric space is said to be of noncompact type if it is semisimple and the sectional curvature is 0 Symmetric spaces have been classified by Elie Cartan using the classification of simple Lie algebras, their involutive automorphisms, and the concept of duality between symmetric spaces of compact type and of noncompact type

20 Basics about symmetric spaces (IV) Let M be a semisimple symmetric space, o M, G = I o (M), K = {g G g(o) = o}. Then M can be identified with the homogeneous space G/K. p = {X g B(X, k) = {0}} g = k p Cartan decomposition θ : g g, X + Y X Y Cartan involution (X k and Y p) [k, k] k, [k, p] p, [p, p] k The rank of M is the dimension of a maximal abelian subspace of p. Symmetric spaces of rank one: The spheres S n, n 2 The projective spaces RP n, CP n, HP n, OP 2, n 2 The hyperbolic spaces RH n, CH n, HH n, OH 2, n 2

21 s-representations An s-representation is the isotropy representation of a simply connected semisimple symmetric space M = G/K χ : K p p, (k, X ) Ad(k)X An orbit K X = {Ad(k)X k K} of an s-representation is called a real flag manifold or R-space. The orbits of s-representations provide an important class of submanifolds in Euclidean spaces, which takes us to the concept of normal holonomy...

22 Holonomy of Riemannian manifolds M n connected Riemannian manifold Hol(M) holonomy group of M Hol 0 (M) restricted holonomy group of M General facts: Hol 0 (M) connected compact subgroup of SO n Hol 0 (M) trivial M flat Hol(M) reducible M is Riemannian product Berger 1953, Simons 1962: If Hol(M) is irreducible and does not act transitively on S n 1, then M is locally symmetric ( R = 0)

23 Normal holonomy of submanifolds M n connected Riemannian manifold P k connected submanifold of M Recall Weingarten formula X Γ(TP), ξ Γ(νP) : M X ξ = A ξx + X ξ normal connection of P -parallel translation of normal vectors ξ ν p P for fixed p P along piecewise smooth curves c : [0, 1] P with c(0) = c(1) = p induces a subgroup Φ p of O(ν p P) = O n k Φ p normal holonomy group of P at p p, q M = Φ p and Φ q are conjugate

24 Normal holonomy of submanifolds (II) -parallel translation of normal vectors ξ ν p P for fixed p P along null homotopic piecewise smooth curves c : [0, 1] P with c(0) = c(1) = p induces a Lie subgroup Φ p of SO(ν p P) = SO n k Φ p restricted normal holonomy group of P at p Φ p is the identity component of Φ p Φ p is a normal subgroup of Φ p and Φ p /Φ p is countable

25 Normal holonomy of surfaces in E 3 Let P 2 be a surface in E 3. Then Φ p = {I } P 2 orientable Φ p = Z 2 P 2 non-orientable

26 Normal holonomy of curves in E 3 c : [0, l] E 3 smooth curve with c(0) = c(l) and ċ = 1 κ curvature and τ torsion of c (T, N, B) Frenet frame field along c N = κt τb and B = τn ξ unit normal vector field along c Write ξ = cos(ω)n + sin(ω)b Differentiate ξ = ω sin(ω)n + cos(ω)n + ω cos(ω)b + sin(ω)b = κ cos(ω)t + (τ ω ) sin(ω)n (τ ω ) cos(ω)b Thus ξ = 0 ω = τ ω(t) = t 0 τ(s)ds + C

27 Normal holonomy of curves in E 3 (II) Thus Define Conclusion: ξ = 0 ω = τ ω(t) = ω = l 0 t τ(s)ds = total torsion of c Φ p = {e ikω k Z} 0 τ(s)ds + C Remark: r R c : [0, l] E 3 : l 0 τ(s)ds = r Therefore the possible normal holonomy groups of closed curves in E 3 are Z and Z k (k Z >0 )

28 Normal Holonomy Theorem Olmos 1990: Let P be a submanifold of a space form (E n, S n, RH n ), p P and Φ = Φ p. Then Φ is compact There exist a unique (up to order) orthogonal decomposition ν p P = V 0 V 1 V k of ν p P into Φ -invariant subspaces V 0,..., V k and normal subgroups Φ 0,..., Φ k of Φ such that Φ = Φ 0 Φ 1 Φ k (direct product) Φ i acts trivially on Φ j (i j) Φ 0 = {I } and Φ i (i 1) acts on V i as an irreducible s-representation The proof is based on Ambrose-Singer Holonomy Theorem and Holonomy Systems

29 Ambrose-Singer Holonomy Theorem The Ambrose-Singer Holonomy Theorem relates holonomy with curvature For c : [0, 1] P piecewise smooth curve define τ c parallel transport along c in P τ c -parallel transport along c Ambrose-Singer 1953: The Lie algebra of Φ is the subalgebra of so(ν p P) generated by all transformations of the form (τ c ) 1 R τ cx,τ cy τ c where c : [0, 1] P is a piecewise smooth curve with c(0) = p and X, Y T p P

30 Holonomy systems Holonomy systems were introduced by Simons 1962 A holonomy system [V, R, G] consists of V n-dimensional Euclidean vector space R algebraic curvature tensor on V G connected compact subgroup of SO(V ) such that x, y V : R x,y g Example: If (M, g) is a Riemannian manifold and p M, then [T p M, R p, Hol 0 (M, p)] is a holonomy system

31 The idea for proof of Normal Holonomy Theorem Use arguments given by Simons 1962 for proof of Berger Holonomy Theorem Problem: Find suitable holonomy system [ν p P,?, Φ p] Note that R p Define R p = R p : 2 T p P 2 ν p P (R p ) : 2 ν p P 2 ν p P Ricci equation: R x,y ξ, η = [A ξ, A η ]x, y R p (ξ 1, ξ 2 )ξ 3, ξ 4 = (Rp ) (ξ 1 ξ 2 ), (Rp ) (ξ 3 ξ 4 ) = tr([a ξ1, A ξ2 ], [A ξ3, A ξ4 ]) R p is an algebraic curvature tensor on ν p P (with negative scalar curvature) and [ν p P, R p, Φ p] is a holonomy system

32 Some problems The definition of the holonomy system uses the Ricci equation and works only for space forms. How can one define a holonomy system for other Riemannian manifolds M? Which irreducible s-representations can be realized as the restricted normal holonomy representation of a submanifold of a Euclidean space? Heintze-Olmos 1992: Investigated normal holonomy of orbits of s-representations: All s-representations arise as normal holonomy representations with 11 exceptions. No definite answer yet for any of the 11 exceptions

33 Complex submanifolds in complex projective spaces Console-DiScala-Olmos 2008: Let P be a full complete complex submanifold of CP n. Then the normal holonomy group does not act transitively on the unit sphere in the normal space if and only if P is the projectivized (unique) complex orbit of the s-representation of an irreducible Hermitian symmetric space of rank 3. Remark: Any such complex submanifold of CP n has parallel second fundamental form; classification by Nakagawa-Takagi 1976

34 Geometry of orbits of s-representations Every singular orbit of an s-representation is a submanifold with constant principal curvatures: The principal curvatures are constant for any parallel normal vector field along any piecewise smooth curve Every principal orbit of an s-representation is an isoparametric submanifold: submanifold with constant principal curvatures and flat normal bundle Thorbergsson 1991: The converse holds if P is connected complete irreducible and codimp 3 Heintze-Olmos-Thorbergsson 1991: A submanifold of E n has constant principal curvatures if and only if it is isoparametric or a focal manifold of an isoparametric submanifold

35 Polar representations on Euclidean spaces A polar representation on E n consists of a compact Lie group G acting isometrically on E n an affine subspace Σ of E n (called a section) with the properties x E n : (G x) Σ x Σ : Σ = T x Σ ν x (G x) Examples: SO 2 on E 2 Every s-representation is polar and every maximal abelian subspace a of p is a section. Dadok 1985: Every polar representation on E n is orbit equivalent to an s-representation

36 Summary On Euclidean spaces there is a beautiful theory of submanifolds involving Symmetric spaces Polar representations Submanifolds with constant principal curvatures Plan for what follows: Discuss some generalizations of this theory from Euclidean spaces to symmetric spaces

37 Polar actions and hyperpolar actions Let M be a connected complete Riemannian manifold and H be a connected closed subgroup of I (M) The action of H on M is said to be polar if there exists a connected complete submanifold S of M such that p M : (H p) S p S : T p S ν p (H p) Any such submanifold S is called a section of the action A polar action on M is called hyperpolar if there exists a flat section. Every section of a polar action is totally geodesic Consequence: Every polar action on E n is hyperpolar

38 Basic examples of hyperpolar actions Every s-representation induces a hyperpolar action on a Euclidean space Let k {1,..., n 1} and consider H = E k acting on E n by translations. The action of H on E n is hyperpolar and S = E n k = (E k ) is a section. The orbits form a foliation on E n by parallel affine subspaces Let M = G/K be a semisimple symmetric space. Then the action of K on M is hyperpolar Every cohomogeneity one action is hyperpolar Fundamental problem: Classification of polar actions and of hyperpolar actions on symmetric spaces

39 Transformation groups Motivation: Study of symmetries of mathematical or physical structures Origin: Galois theory Felix Klein 1872: Geometry is the study of properties which are invariant under a given transformation group

40 Transitive Actions Transitive action orbit space is a point Geometry of homogeneous spaces Fundamental property: Homogeneous spaces look the same everywhere Motivation: use algebraic methods for solving geometric problems

41 Cohomogeneity One Actions Cohomogeneity one action (C1) orbit space is one-dimensional Geometry of cohomogeneity one spaces Motivation: use ODE methods for solving geometric problems Submanifold geometry: Every homogeneous hypersurface is an orbit of a cohomogeneity one action

42 Examples of cohomogeneity one actions

43 The orbit space of a cohomogeneity one action M connected complete Riemannian manifold I (M) isometry group of M H I (M) connected closed subgroup acting on M with cohomogeneity one M/H = {H p p M} orbit space Mostert 1957 (compact case) Berard Bergery 1982 (general case): M/H = R, S 1, [0, ) or [0, 1] M/H = R, S 1 orbits form a Riemannian foliation on M M/H = [0, ), [0, 1] boundary point singular orbit interior point principal orbit

44 Finite fundamental group π 1 (M) finite M/H = S 1 Proof: use exact homotopy sequence for the fibre bundle F M M/H = S 1 :... π 1 (M) π 1 (S 1 ) Z π 0 (F ) 0... π 1 (M) = 0 and M compact M/H = [0, 1]

45 Hadamard manifolds A connected, simply connected, complete Riemannian manifold with nonpositive sectional curvature is called a Hadamard manifold M Hadamard M/H = R or [0, ) Proof: Assume there exists a singular orbit Let K H maximal compact Cartan s Fixed Point Theorem: o M : K o = o isotropy H o = K and H o singular orbit. Assume there exists second singular orbit H p h H : H p hkh 1 = hh o h 1 = H h(o) H p fixes p and h(o) H p fixes pointwise the geodesic in M from p to h(o) (Contradiction!)

46 Cohomogeneity one method M connected compact smooth manifold G compact Lie group acting on M with cohomogeneity one Assume M/G = [0, 1], M 0, M 1 singular orbits M = M \ (M0 M 1 ) = union of principal orbits M diff = (0, 1) P with P a principal orbit g = dt 2 + g t on M g t one-parameter family of G-invariant metrics on P

47 Applications metrics of holonomy G 2 or Spin 7 (Gibbons, Pope,...) hyperkähler and quaternionic Kähler structures (Dancer, Swann,...) Einstein, Einstein-Kähler and Einstein-Weyl structures (Berard-Bergery, Bonneau, Dancer, Wang...) metrics with positive or nonnegative Ricci or sectional curvature (Grove, Wilking, Ziller,...) harmonic maps, Yang-Mills equations (Urakawa,...) special Lagrangian submanifolds (Joyce, Min-Oo,...) differential topology (Atiyah-Berndt,...)

48 Cohomogeneity one actions on spheres Hsiang-Lawson 1971: Every cohomogeneity one action on a sphere is orbit equivalent to the isotropy representation of a semisimple Riemannian symmetric space of rank 2 compact simple Riemannian symmetric spaces of rank 2: SU 3 /SO 3, SU 3, SU 6 /Sp 3, E 6 /F 4 Sp 2, SO 10 /U 5, E 6 /Spin 10 U 1 SO n+2 /SO n SO 2, SU n+2 /S(U n U 2 ), Sp n+2 /Sp n Sp 2 G 2 /SO 4, G 2

49 Known classifications Compact spaces: Takagi 1973: CP n Uchida 1977: H (M, Q) = H (CP n, Q) Iwata 1978: H (M, Q) = H (HP n, Q) Iwata 1981: H (M, Q) = H (OP 2, Q) Kollross 2001: simply connected irreducible Riemannian symmetric spaces of compact type and rank 2 Noncompact spaces: Segre 1938: E n Cartan 1938: RH n

50 Euclidean spaces I o (E n ) = E n SO n Segre 1938: Every cohomogeneity one action on E n is orbit equivalent to one of the following: H = E n 1 : orbits give a foliation by parallel hyperplanes H = E k SO n k, 0 k < n 1: orbits are a totally geodesic E k and the tubes around it Proof: use Gauss-Codazzi equations and focal set theory

51 Real hyperbolic spaces I o (RH n ) = SO o 1,n Cartan 1938: Every cohomogeneity one action on RH n is orbit equivalent to one of the following: H = SO1,n 1 o : orbits give a foliation by a totally geodesic hyperplane RH n 1 and its equidistant hypersurfaces H = SO o 1,k SO n k, 0 k < n 1: orbits are a totally geodesic RH k and the tubes around it H = N KAN = SO1,n o : orbits give a foliation by horospheres Proof: use Gauss-Codazzi equations and focal set theory

52 Restricted root space decomposition M = G/K, G = I o (M), Riemannian symmetric space of noncompact type B Killing form of g, B(X, Y ) = tr(ad(x ) ad(y)) p = {X g B(X, k) = 0} g = k p Cartan decomposition θ : g g, X + Y X Y Cartan involution X, Y = B(X, θy ) positive definite inner product on g, induces Riemannian metric on M ad(x ) selfadjoint for all X p a p maximal abelian subspace a dual vector space of a

53 Restricted root space decomposition (II) g λ = {X g H a : ad(h)x = λ(h)x }, λ a 0 λ a restricted root g λ {0} Σ a set of restricted roots a abelian restricted root space decomposition g = g 0 g λ, g 0 = z k (a) a λ Σ Σ {A r, B r, C r, D r, E 6, E 7, E 8, F 4, G 2, BC r } for M simple Λ = {α 1,..., α r } set of simple roots of Σ Σ = Σ + Σ

54 Construction of Dynkin diagram Dynkin diagram: graph consisting of circles and lines, arrows circles represent simple roots angle between two simple roots is π 2, 2π 3, 3π 4, 5π 6 Join α i and α j by 0, 1, 2, 3 lines if the angle between α i and α j is π 2, 2π 3, 3π 4, 5π 6, respectively If α i > α j and α i and α j are joined by a line then draw an arrow pointing from α i to α j

55 Σ of type A r or B r α 1 α 2 α r 1 α r M = SL r+1 (R)/SO r+1 : (1,..., 1) M = SL r+1 (C)/SU r+1 : (2,..., 2) M = SL r+1 (H)/Sp r+1 : (4,..., 4) M = E 26 6 /F 4 : (8, 8) M = SO o 1,n+1 /SO n+1 = RH n+1 : (n), n 2 α 1 α 2 α r 2 α r 1 α r M = SO 2r+1 (C)/SO 2r+1 : (2,..., 2, 2) M = SOr,r+n/SO o r SO r+n : (1,..., 1, n), n 1

56 Iwasawa decomposition n = λ Σ + g λ nilpotent subalgebra of g s = a n solvable subalgebra of g with [s, s] = n g = k a n Iwasawa decomposition of g (vector space direct sum) G = KAN Iwasawa decomposition of G (diffeomorphism) M = G/K = AN = S solvable Lie group equipped with left-invariant Riemannian metric

57 The setup for cohomogeneity one actions M = G/K connected irreducible Riemannian symmetric space of noncompact type G noncompact semisimple real Lie group K maximal compact subgroup of G o M with K o = o H connected closed subgroup of G acting on M with cohomogeneity one L connected proper maximal subgroup of G with H L g, k, h, l corresponding Lie algebras Mostow 1961: l is reductive or parabolic

58 The reductive case Karpelevich 1953: L has a totally geodesic orbit S M = H has a totally geodesic orbit S S reflective geodesic reflection of M in S is an isometry totally geodesic submanifold S of M with o S and T o S = ν o S Leung 1979: Classification of reflective submanifolds in irreducible simply connected Riemannian symmetric spaces

59 The reductive case (II) Berndt-Tamaru 2004: S is a totally geodesic singular orbit of a cohomogeneity one action on M S reflective and rank S = 1, or S is one of the following totally geodesic non-reflective submanifolds: S M dim S dim M CH 2 G 2 2 /SO SL 3 (R)/SO 3 G 2 2 /SO G 2 2 /SO 4 SO o 3,4 /SO 3SO SL 3 (C)/SU 3 G C 2 /G G C 2 /G 2 SO C 7 /SO

60 Parabolic subalgebras g = k p Cartan decomposition a maximal abelian subspace of p restricted root space decomposition ( ) g = g 0 g α Λ set of simple roots for Σ α Σ Φ subset of Λ, Σ Φ = Σ span{φ} ( ) l Φ = g 0 α Σ g Φ α, n Φ = α Σ + \Σ + g α Φ l Φ reductive subalgebra, n Φ nilpotent subalgebra q Φ = l Φ n Φ parabolic subalgebra; [q Φ, n Φ ] n Φ (Chevalley decomposition) Every parabolic subalgebra of g is conjugate to q Φ for some subset Φ Λ

61 Parabolic subalgebras (II) a Φ = α Φ ker α, m Φ = l Φ a Φ m Φ reductive subalgebra, a Φ abelian subalgebra q Φ = m Φ a Φ n Φ (Langlands decomposition) [q Φ, a Φ n Φ ] a Φ n Φ k Φ = k q Φ = k m Φ [k Φ, m Φ ] m Φ, [k Φ, a Φ ] = {0}, [k Φ, n Φ ] n Φ F s Φ = M Φ o semisimple symmetric space with rank equal to Φ, totally geodesic in M, boundary component of M w.r.t. maximal Satake compactification A Φ o = E r Φ Euclidean space, totally geodesic in M L Φ o = F Φ = F s Φ Er Φ totally geodesic in M M = F s Φ Er Φ N Φ (horospherical decomposition)

62 Construction Method I: ν o (H o) m Φ, Φ H s Φ I o (F s Φ ) M Φ acting on F s Φ with cohomogeneity one h = h s Φ a Φ n Φ subalgebra of q Φ H acts on M with cohomogeneity one Rank reduction - Such a cohomogeneity one action can be constructed by a canonical extension of a cohomogeneity one action on a boundary component

63 Construction Method II: ν o (H o) a Φ Φ = : q Φ = k 0 a n minimal parabolic subalgebra M = AN solvable Lie group with left-invariant metric l a one-dimensional linear subspace s l = (a l) n subalgebra of a n S l acts on M with cohomogeneity one Construction method II produces precisely all horosphere foliations on M

64 Construction Method III: ν o (H o) n Φ Λ = {α 1,..., α r }, {H 1,..., H r } dual basis of Λ in a Φ j = Λ \ {α j }: Put q j = q Φj, n j = n Φj, etcetera n ν j = α Σ + \Σ + j,α(h j )=ν g α n j = ν>0 nν j Assume that gradation generated by n 1 j v n 1 j ; define n j,v = n j v subalgebra of n j NL o j (n j,v ) acts transitively on F j = Fj s E NL o j K (v) acts transitively on the unit sphere in v if dim v 2 Then H j,v = N o L j (n j,v )N j,v acts on M with cohomogeneity one Remark: dim v = 1 corresponds to foliation

65 Construction Method III - rank M = 1 M G K K 0 g α n RH n SO1,n o SO n SO n 1 R n 1 R n 1 CH n SU 1,n U n U n 1 C n 1 C n 1 R HH n Sp 1,n Sp 1 Sp n Sp 1 Sp n 1 H n 1 H n 1 R 3 OH 2 F4 20 Spin 9 Spin 7 O O R 7 Λ = {α}, Φ =, l Φ = g 0 = k 0 a, n Φ = n = g α g 2α q Φ = g 0 n = k 0 a n minimal parabolic subalgebra Problem: Find all k-dimensional (k 2) linear subspaces v of g α for which there exists a subgroup of K 0 acting transitively on the unit sphere in v

66 Construction Method III - rank M = 1 M G K K 0 g α n RH n SO1,n o SO n SO n 1 R n 1 R n 1 CH n SU 1,n U n U n 1 C n 1 C n 1 R HH n Sp 1,n Sp 1 Sp n Sp 1 Sp n 1 H n 1 H n 1 R 3 OH 2 F4 20 Spin 9 Spin 7 O O R 7 Berndt-Tamaru 2007: R: any linear subspace v R n 1 C: any linear subspace v C n 1 with constant Kähler angle H: some linear subspaces v H n 1 with constant quaternionic Kähler angle (no complete classification) O: any linear subspace v O of dimension k {2, 3, 4, 6, 7}

67 Constant quaternionic Kähler angle Examples of subspaces V of H n 1 with constant quaternionic Kähler angle Φ: Φ = (0, 0, 0) V quaternionic Φ = (0, 0, π/2) V = Im(H)v, v H n 1 Φ = (0, π/2, π/2) V totally complex Φ = (π/2, π/2, π/2) V totally real Φ = (ϕ, π/2, π/2) V subspace with constant Kähler angle ϕ (0, π/2) in a totally complex subspace of H n 1 Φ = (0, ϕ, ϕ) V complexification of a subspace with constant Kähler angle ϕ (0, π/2) in a totally complex subspace of H n 1

68 Construction Method III - Σ of type A r α 1 α 2 α r 1 α r M = SL r+1 (R)/SO r+1 : (1,..., 1) M = SL r+1 (C)/SU r+1 : (2,..., 2) M = SL r+1 (H)/Sp r+1 : (4,..., 4) M = E 26 6 /F 4 : (8, 8) M = SO o 1,n+1 /SO n+1 = RH n+1 : (n), n 2 Method produces cohomogeneity one action with totally geodesic singular orbit for Φ 1 and Φ r. No cohomogeneity one actions for other Φ i. Nothing new!

69 Construction Method III - Σ of type B r α 1 α 2 α r 2 α r 1 α r M = SO 2r+1 (C)/SO 2r+1 : (2,..., 2, 2) Nothing new! M = SO o r,r+n/so r SO r+n : (1,..., 1, n), n 1 Method produces three cohomogeneity one actions with non-totally geodesic singular orbit Φ1 : F 1 = SO o r 1,r 1+n /SO r 1SO r 1+n E, n 1 1 = Rr 1 R r 1+n Φ r 1 : F r 1 = SL r 1 (R)/SO r 1 SO o 1,1+n /SO 1+n E, n 1 r 1 = Rr 1 R (r 1)(1+n)

70 The Classification Berndt-Tamaru 2008: Let M be a connected irreducible Riemannian symmetric space of noncompact type. Then every cohomogeneity one action on M either has a totally geodesic singular orbit, or it is orbit equivalent to a cohomogeneity one action obtained by construction method I, II or III.

71 Cohomogeneity one actions on SL 3 (R)/SO 3 The boundary components of SL 3 (R)/SO 3 are F s 1 = F s 2 = RH 2 There are three types of cohomogeneity one actions on RH 2 = SL 2 (R)/SO 2. Write SL 2 (R) = KAN. SO2 : the canonical extension is a cohomogeneity one action with a minimal RH 3 as a singular orbit N: the canonical extension leads to a horosphere foliation (with a singular point at infinity) A: the canonical extension leads to a foliation with a minimal homogeneous hypersurface P as a leaf, where P is the canonical extension of a geodesic in RH 2 RH 2 E = F 1 = F2 is the only reflective submanifold S in SL 3 (R)/SO 3 for which S has rank one

72 Cohomogeneity one actions on SL 3 (R)/SO 3 (II) Theorem: Every cohomogeneity one action on SL 3 (R)/SO 3 has one of the following orbit structures: the reflective submanifold RH 2 E and the tubes around it a horosphere foliation the foliation given by the canonical extension of a geodesic in RH 2 to a minimal homogeneous hypersurface, and its equidistant hypersurfaces the minimal RH 3 and the tubes around it

73 The case of foliations M F = set of all homogeneous codimension one foliations on M up to isometric congruence r = rank of M Aut(DD) {I, Z 2, S 3 } automorphism group of the Dynkin diagram associated to M Berndt-Tamaru 2003: M F = (RP r 1 {1,..., r})/aut(dd)

74 The two foliations on hyperbolic spaces horosphere foliation foliation with exactly one minimal leaf S M = RH n : S = RH n 1 totally geodesic M = CH n : S = ruled real hypersurface associated to a horocycle in a totally geodesic RH 2 CH n

75 Duality and triality M F depends only on the rank and on possible duality or triality principles on the symmetric space Example: r = 8, Aut(DD) = I M F = RP 7 {1,..., 8} for the following symmetric spaces: SO17/SO C 17, Sp8 R /U 8, Sp8 C /Sp 8, SO16/U H 16, SO17/U H 17 E8 8 /SO 16, E8 C /E 8 and for the hyperbolic Grassmannians G8 (R n+16 ) (n 1), G8 (C n+16 ) (n 0), G8 (H n+16 ) (n 0)

76 Geometry of the foliations l RP r 1 horosphere foliation F l on M all leaves of F l are congruent to each other if r 2 then some foliations F l are harmonic (i.e. all leaves are minimal) and hence induce a harmonic map M R i {1,..., r} α i Λ foliation F i on M F i contains exactly one minimal leaf α i = α j F i, F j have the same principal curvatures with the same multiplicities Corollary: If r 3, then there exist noncongruent homogeneous isoparametric systems on M with the same spectral data for the second fundamental form

77 Example M = SL 4 (R)/SO 4 = x ij R, x 11 x 22 x 33 x 44 = 1 x 11 x 12 x 13 x 14 0 x 22 x 23 x x 33 x x 44 DD = α 1 α 2 α 3 F 1 x 11 0 x 13 x 14 0 x 22 x 23 x x 33 x x 44, F 2 x 11 x 12 x 13 x 14 0 x 22 0 x x 33 x x 44

78 Outline of proof I Step 1 There exists a connected closed solvable subgroup of H which acts simply transitively on each orbit of H Therefore may assume that H is solvable and acts simply transitively on each orbit Step 2 h t a n for some Iwasawa decomposition g = k a n, where t is a maximal abelian subalgebra of z k (a) Step 3 h a+n is a subalgebra of a n

79 Outline of proof II Step 4 Classify the codimension one subalgebras of a n: (a n) Rξ with ξ a or ξ RH α g α, α simple root Step 5 Investigate the orbit equivalence of the corresponding cohomogeneity one actions: Write h a+n = (a n) Rξ with ξ a + n ξ a h, h a n induce same foliation ξ g α h, h a n induce same foliation ξ RH α g α, ξ / RH α, ξ / g α h, h a n induce orbit equivalent foliations Step 6 Investigate orbit equivalence of the model foliations F l, l RP r 1, and F i, i {1,..., r}

80 Classification of homogeneous hypersurfaces in CH n Berndt-Tamaru 2005: Every homogeneous real hypersurface in CH n is congruent to one of the following real hypersurfaces: a horosphere in CH n a ruled real hypersurface generated by a horocycle in RH 2 CH n, or to one of its equidistant hypersurfaces a tube around CH k CH n for some k {0,..., n 1} a tube around RH n CH n a tube around F k CH n for some k {2,..., n 1} a tube around F k,ϕ CH n for some k {1,..., [(n 1)/2]} and ϕ (0, π/2)

81 The submanifolds F k and F k,ϕ Iwasawa decomposition su 1,n = k a n = u n a (g α g 2α ) n = g α g 2α = (2n 1)-dimensional Heisenberg algebra a = R, g 2α = Ja, g α = C n 1 v linear subspace of g α = C n 1 = s v = a (g α v) g 2α subalgebra of a n = S v o homogeneous submanifold v k-dimensional real subspace of g α = C n 1, 2 k n 1 = F k = S v o is a (2n k)-dimensional homogeneous submanifold with real normal bundle of rank k v 2k-dimensional subspace of g α = C n 1 with constant Kähler angle ϕ (0, π/2), 1 k [(n 1)/2] = F k,ϕ = S v o is a 2(n k)-dimensional homogeneous submanifold with normal bundle of rank 2k and constant Kähler angle ϕ

82 Constant principal curvatures in CH n Assume that M CH n is real hypersurface with constant principal curvatures, and define g = number of distinct principal curvatures M homogeneous = g {2, 3, 4, 5} Problems: Does g {2, 3, 4, 5} hold in general? Note: g = 1 (umbilical) is impossible Is any real hypersurface of CH n with constant principal curvatures an open part of a homogeneous hypersurface? YES for g = 2 (Montiel 1985) YES for Hopf hypersurfaces (Berndt 1989)

83 Classification of real hypersurfaces with g = 3 in CH n (Berndt Díaz-Ramos 2006, 2007): Every real hypersurface in CH n (n 2) with three distinct constant principal curvatures is an open part of a homogeneous hypersurface: a ruled real hypersurface generated by a horocycle in RH 2 CH n, or to one of its equidistant hypersurfaces a tube around CH k CH n for some k {1,..., n 2} a tube with radius r ln(2 + 3) around RH n CH n a tube with radius r = ln(2 + 3) around F k CH n for some k {2,..., n 1}.

84 Outline of proof (I) λ 1, λ 2, λ 3 principal curvatures m 1, m 2, m 3 corresponding multiplicities T λ1, T λ2, T λ3 corresponding eigenbundles TM = T λ1 T λ2 T λ3 ξ (local) unit normal vector field (Jξ is Hopf vector field on M) We can assume that M is non-hopf: RJξ T λi Step 1 There exists no point p M such that the orthogonal projections of Jξ p onto T λi (p), i = 1, 2, 3, are nontrivial. Thus we can assume that there exists a point p M such that Jξ p = b 1 u 1 + b 2 u 2 with some unit vectors u i T λi (p) and 0 b i R, i = 1, 2.

85 Outline of proof (II) Step 2 There exists a unit vector a T λ3 (p) such that Rξ p Ru 1 Ru 2 Ra is a complex subspace of T p CH n. Must hold on an open neighborhood of p vector fields U 1, U 2, A near p Remark: The integral curves of A are geodesics in M and the three vector fields A, U 1, U 2 span an autoparallel distribution Step 3 m i > 1 = 4λ 3 λ i = 1 (i {1, 2}) Thus we can assume m 2 = 1 = m 1 = 1 or m 2 = 1

86 Outline of proof (III) Step 4 Case 1: m 1 > 1 λ 1 = 3/2, λ 2 = 0, λ 3 = 3/6 T λ1 RU 1 is real J(T λ1 RU 1 ) T λ3 Case 2: m 1 = 1 1/2 ( < λ 3 < 1/2 ) λ 1,2 = 1 2 3λ 3 1 3λ 2 3

87 Outline of proof (IV) Step 5 Investigate focal sets and equidistant hypersurfaces using Jacobi field theory: There exists a (2n m 1 )-dimensional focal manifold (if m 1 > 1) or equidistant hypersurface (if m 1 = 1) F of M with totally real normal bundle νf, and a unit vector field Z tangent to the maximal holomorphic subbundle of TM such that the second fundamental form α of F is given by the trivial bilinear extension of for all ξ Γ(νF ). 2α(Z, Jξ) = ξ

88 Outline of proof (V) Step 6 Prove the following rigidity result: Let F be a (2n k)-dimensional connected submanifold in CH n, n 3, with totally real normal bundle νf. Assume that there exists a unit vector field Z tangent to the maximal holomorphic subbundle of TF such that the second fundamental form α of F is given by the trivial bilinear extension of 2α(Z, Jξ) = ξ for all ξ Γ(νF ). Then F is holomorphically congruent to an open part of the ruled minimal submanifold F k. k = 1: F 1 is the homogeneous ruled real hypersurface determined by a horocycle in RH 2 CH n

89 Open problems Let M be a real hypersurface in CH n with g distinct constant principal curvatures. Is is true that g {2, 3, 4, 5}? Classification of real hypersurfaces in CH n with 4 or 5 distinct constant principal curvatures Is every real hypersurface in CH n with constant principal curvatures an open part of a homogeneous hypersurface?

90 Polar actions and hyperpolar actions Let M be a connected complete Riemannian manifold and H be a connected closed subgroup of I (M) The action of H on M is said to be polar if there exists a connected complete submanifold S of M such that p M : (H p) S p S : T p S ν p (H p) Any such submanifold S is called a section of the action A polar action on M is called hyperpolar if there exists a flat section. Every section of a polar action is totally geodesic Consequence: Every polar action on E n is hyperpolar Problem. Classify polar/hyperpolar actions on symmetric spaces

91 Polar/hyperpolar actions - the compact case M irreducible simply connected symmetric space of compact type Dadok 1985: Classification of polar actions on spheres Podestá-Thorbergsson 1999: Classification of polar actions on projective spaces Kollross 2001, 2007: Classification of hyperpolar actions for higher rank; every polar action is hyperpolar

92 Polar/hyperpolar actions - the noncompact case M = G/K symmetric space of noncompact type F foliation on M F polar/hyperpolar if F coincides with the orbits of a polar/hyperpolar action on M Example. Consider the Chevalley decomposition q Φ = l Φ n Φ of a parabolic subalgebra q Φ of g. Then the orbits of N Φ form a polar foliation on M which is hyperpolar if and only if Φ =. The totally geodesic submanifold L Φ o is a section. Problem. Classify the hyperpolar foliations on M

93 Examples of hyperpolar foliations V linear subspace E m = F m V = {p + V p Em } hyperpolar foliation on E m F {R, C, H, O}, M = G/K = FH n g = k a n Iwasawa decomposition l n one-dimensional linear subspace s l = a (n l) subalgebra of a n = F n F = {S l p p FH n } hyperpolar foliation on FH n F 1 F n k F k FV m hyperpolar foliation on F 1 H n 1 F k H n k E m F n 1

94 Examples of hyperpolar foliations (II) M = G/K symmetric space of noncompact type q Φ = m Φ a Φ n Φ Langlands decomposition of parabolic subalgebra of g Φ orthogonal set of roots, that is, α, β Φ : α, β = 0 F Φ = F1 H n 1 F Φ H n Φ E r Φ F n 1 F 1 F n Φ F Φ F r r Φ V hyperpolar foliation on F Φ F Φ,V = F n 1 F 1 F n Φ F Φ F r r Φ V N Φ hyperpolar foliation on M = F Φ N Φ F,{0} horocycle foliation on M

95 Classification of hyperpolar foliations on symmetric spaces of noncompact type Berndt-DiazRamos-Tamaru 2008: Let M be a symmetric space of noncompact type. Every hyperpolar foliation on M is isometrically congruent to F Φ,V for some orthogonal set Φ of simple roots and some linear subspace V E r Φ.

96 Open problems For symmetric spaces of noncompact type: Is every hyperpolar foliation homogeneous? Find examples of polar/hyperpolar actions with singular orbits Find examples of polar homogeneous foliations Classify polar foliations Classify polar/hyperpolar actions with singular orbits