Equifocal Submanifolds Is Always Satisﬁed If the Dimension of the Focal Distribution Is at Least Two

July 9,1996

Submanifold Geometry in Symmetric Spaces

C.-L. Terng1 and G. Thorbergsson2

1. Introduction

The classical local invariants of a submanifold in a space form are the first fundamental form, the shape operators and the induced normal connection, and they determine the submanifold up to ambient isometry. One of the main top- ics in differential geometry is to study the relation between the local invariants and the global geometry and topology of submanifolds. Many remarkable results have been developed for submanifolds in space forms whose local invariants satisfy certain natural conditions. The study of focal points of a submanifold in an arbitrary Riemannian manifold arises from the Morse theory of the energy functional on the space of paths in the Riemannian manifold joining a fixed point to the submanifold. The Morse index theorem relates the geometry of a submanifold to the topology of this path space. The focal structure is intimately related to the local invariants of the submanifold. In the case of space forms one can go backwards and reconstruct the local invariants fromthe focal structure, so it is not too surprising that most of the structure theory of submanifolds can be reformulated in terms of their focal structure. What is perhaps surprising is a fact that became increasingly evident to the authors from their individual and joint research over the past decade: while extending the theory of submanifolds to ambient spaces more general than space-forms proves quite difficult if one tries to use the same approach as for the space forms, at least for symmetric spaces it has proved possible to develop an elegant theory based on focal structure that re- duces to the classical theory in the case of space forms. This paper is an extended report on this theory, and the authors believe that the methods developed herein provide important tools for a continuing study of the submanifold geometry in symmetric spaces. First we set up some notations. Let (N,g) be a Riemannian manifold, M an immersed submanifold of N, and ν(M) the normal bundle of M. The end point map η : ν(M) → N of M is the restriction of the exponential map exp to ν(M). If v ∈ ν(M)x is a singular point of η and the dimension of the kernel of dηv is m, then v is called a multiplicity m focal normal and exp(v) is called a multiplicity m focal point of M with respect to M in N. The focal data, Γ(M), is defined to be the set of all pairs (v, m) such that v is a multiplicity m focal normal of M. The focal variety V(M) is the set of all pairs (η(v),m) with (v, m) ∈ Γ(M). The main purpose of our paper is to study the global geometry and topology

1 Research supported in part by NSF Grant DMS 9304285 2 Research supported in part by NSF Grant DMS 9100179

1 of submanifolds in symmetric spaces whose focal data satisfy certain natural conditions. In order to explain our results, we review some of the basic relations between focal points, Jacobi fields and Morse theory. For a fixed p ∈ N, let P (N,M × p) denote the space of H1-paths γ :[0, 1] → N such that (γ(0),γ(1)) ∈ M ×{p} (a path is H1 if it is absolutely continuous and the normof its derivative is square integrable). Let 1 E : P (N,M × p) → R, E(γ)= γ(t) 2dt 0 be the energy functional. Then γ is a critical point of E if and only if γ is a geodesic normal to M at γ(0) parameterized proportional to arc length. A vector field J along the geodesic γ is in the null space of the Hessian of E if and only if J satisfies the Jacobi equation ∇γ ∇γ J − R(γ ,J)(γ )=0 with boundary conditions J(0) ∈ TMγ(0), Aγ(0)J(0) + J (0) ∈ ν(M)γ(0) and J(1)=0, where Av is the shape operator of M with respect to the normal vector v. The Morse index theoremstates that if p is not a focal point of M then E is a non-degenerate Morse function, and that the index of E at a critical point γ is the sumof the integers m such that γ(t) is a multiplicity m focal point of M with respect to γ(0) with 0 tangent space T (ν(M))v can be naturally identified with ν(M)x ⊕ TMx. It is known that if u ∈ TMx then dηv(u)=J(1), where J is the Jacobi field on γ(t) = exp(tv) with the initial conditions J(0) = u and J (0) = −Av(u). Since the initial conditions of an ordinary differential equation determine the solution at time 1, the shape operators of M and the curvature tensor of N determine the focal structure of M, and one expects a close relation between the focal data, the local and global geometry, and the topology of the submanifold. For a general Riemannian manifold N,it is difficult to make this relation precise. But if N is a symmetric space, then the curvature tensor of N is a covariant constant. Hence in the coordinates obtained from a parallel normal frame along γ, the Jacobi equation becomes J + S(J)=0, (∗) where S is a constant, self-adjoint operator whose eigenvalues can be expressed in terms of the roots of the symmetric space. This gives a precise relation between shape operators and focal data for submanifolds in symmetric spaces. Recall that an r-flat in a rank k symmetric space N = G/K is an r- dimensional, totally geodesic, flat submanifold. Let G = K + P be a Cartan decomposition. Then every flat is contained in some k-flat, and every k-flat is of the form π(g exp(A)), where g ∈ G and A is a maximal abelian subalgebra in P.IfN is a compact Lie group of rank k, then a k-flat in N is just a maximal torus. But an r-flat need not be closed in general.

2 1.1 Definition. Let M be an immersed submanifold of a symmetric space N. “refDe The normal bundle ν(M) is called (i) abelian if exp(ν(M)x) is contained in some flat of N for each x ∈ M, (ii) globally flat if the induced normal connection is flat and has trivial holonomy. Let v be a normal vector field on a submanifold M. The end point map of v is the map ηv : M → N defined by x → exp(v(x)).Ifv is a parallel normal field v on M, then Mv = ηv(M) is called the parallel set defined by v.

1.2 Definition. A connected, compact, immersed submanifold M in a sym- “refGa metric space N is called equifocal if (1) ν(M) is globally flat and abelian, and (2) if v is a parallel normal field on M such that ηv(x0) is a multiplicity k focal point of M with respect to x0 then ηv(x) is a multiplicity k focal point of M with respect to x for all x ∈ M. (Or equivalently, the focal data Γ(M) is “invariant under normal parallel translation”.) To simplify the terminology we make the following definition: ∈ 1.3 Definition. Let M be a submanifold in N, and v ν(M)x. Then t0 “refGbc is called a focal radius ( 1 is called a focal curvature)ofM with multiplicity m t0 along v if expx(t0v) is a multiplicity m focal point of M with respect to v. Thus a submanifold M with globally flat abelian normal bundle of a symmetric space N is equifocal if the focal curvatures of M along any parallel normal field are constant. A non-vanishing normal field v on M is called a focal normal field if v/ v is parallel and there exists m such that v(x) is a multiplicity m focal normal of M for all x ∈ M.Ifv is a focal normal field, then v is a smooth function on M, and the end point map ηv : M → N defined by ηv(x) = exp(v(x)) has constant rank. So the kernel of dηv defines an integrable distribution Fv with −1 ηv (y) as leaves, and Mv = ηv(M) is an immersed submanifold of N. We will F −1 call v, ηv (y) and Mv respectively the focal distribution, focal leaf and the focal submanifold defined by the focal normal field v.

1.4 Definition. A connected, compact, immersed submanifold M with a “refGb globally flat and abelian normal bundle in a symmetric space N is called weakly equifocal if given a parallel normal field v on M (1) the multiplicities of focal radii along v are constant, i.e., the focal radius functions tj are smooth functions on M and are ordered as follows:

···

and the multiplicities mj of the focal radii tj(x) are constant on M, (2) the focal radius tj is constant on each focal leaf defined by the focal normal field tjv for all j, i.e., tj is the pullback of a smooth function defined on the

focal submanifold Mtj v via ηtj v.

3 1.5 Remark. We will prove in section 5 that condition (2) on the focal radii in “refGba the deﬁnition of weakly equifocal submanifolds is always satisﬁed if the dimension of the focal distribution is at least two.

It follows fromthe definitions that a (weakly) equifocal submanifoldin a rank k symmetric space has codimension less than or equal to k, and that equifocal implies weakly equifocal. When the ambient space N is the space form Sn, Rn or Hn, equifocal and weakly equifocal hypersurfaces have been extensively studied. Note that in this case, the operator S in the Jacobi equation (*) is cI, where c is the sectional curvature of the space form. It follows that t0 is a focal curvature of multiplicity m along v if and only if fc(t0) is a principal curvature along v of multiplicity m, 1 1 where fc(t0)=t0, cot( ) or coth( ) if c =0, 1 or −1 respectively. So condition t0 t0 (2) for equifocal submanifolds and condition (1) for weakly equifocal submanifolds are equivalent to the conditions that the principal curvatures along any parallel normal field are constant and have constant multiplicities respectively. In these space forms they are referred to as isoparametric and proper Dupin hypersurfaces respectively. The study of isoparametric hypersurfaces in Sn has a long history, and these hypersurfaces have many remarkable properties (cf. [Ca], [Mü]). For example, assume that M is an isoparametric hypersurface of Sn with g distinct constant principal curvatures λ1 > ... > λg along the unit normal field v with multiplicities m1,...,mg. Let Ej denote the curvature distribution defined by λj, i.e., Ej(x) is equal to the eigenspace of Av(x) with respect to the eigenvalue λj(x). Then the focal distributions of M are the curvature distributions Ej.It n π follows fromthe structure equations of S that there exists 0 <θ< g such (j−1)π that the principal curvatures are λj = cot(θ + g ) with j =1,...,g, and the − π parallel set Mt = Mtv for g +θ

Another consequence of the structure equations is that the leaves of each Ej are standard spheres. Using topological methods, M¨unzner proved that (1) g has to be 1, 2, 3, 4 or 6, (2) mi = m1 if i is odd, and mi = m2 if i is even, n (3) S can be written as the union D1 ∪D2, where D1 is the normal disk bundle + − of M , D2 is the normal disk bundle of M and D1 ∩ D2 = M, (4) the Z2-homology of M can be given explicitly in terms of g and m1,m2;in particular, the sumof the Z2-Betti numbers of M is 2g. Recall that a hypersurface M of Sn is called proper Dupin if the principal curvatures have constant multiplicities and dλ(X)=0for X in the eigenspace Eλ corresponding to the principal curvature λ (cf. [Pi]). Note that a hypersurface

4 in Sn is proper Dupin if and only if it is weakly equifocal. It is proved in [Th1] that proper Dupin hypersurfaces have the above properties (1)-(4). Recall also that a submanifold M in Rn is called isoparametric ([Ha], [CW1], [Te1]), if ν(M) is flat and the principal curvatures along any parallel normal field are constant. It is known that the normal bundle of a compact isoparametric submanifold in Rn is globally flat. So it follows that a compact submanifold in Rn is isoparametric if and only if it is equifocal. A submanifold M in Rn is called weakly isoparametric ([Te2]) if ν(M) is flat and the multiplicities of the principal curvatures λ along a parallel normal field v are constant, and if dλ(X)=0for X in the eigenspace Eλ(v) corresponding to the principal curvature λ. It is proved in the papers quoted above that these submanifolds have many properties in common with isoparametric and proper Dupin hypersurfaces in spheres. One of the main goals of our paper is to generalize many of these results to equifocal and weakly equifocal submanifolds in compact symmetric spaces. Henceforth, we will assume that N = G/K is a compact, rank k symmetric space of semi-simple type, G = K+P a Cartan decomposition, and N is equipped with the G-invariant metric given by the restriction of the negative of the Killing formof G to P. We first state a theoremthat generalizes results on isoparametrichypersur- faces in spheres to equifocal hypersurfaces in compact symmetric spaces:

1.6Theorem. Let M be an immersed, compact, equifocal hypersurface in the “refGj simply connected, compact symmetric space N, and v a unit normal ﬁeld. Then (a) normal geodesics are circles of constant length, which will be denoted by -, - (b) there exist integers m1,m2,anevennumber2g and 0 <θ< 2g such that (1) the focal points on the normal circle Tx = exp(ν(M)x) are (j − 1)- x(j) = exp θ + v(x) , 1 ≤ j ≤ 2g, 2g

and their multiplicities are m1 if j is odd and m2 if j is even, (2) the group generated by reflections in the pairs of focal points x(j),x(j + g) on the normal circle Tx is isomorphic to the dihedral group W with 2g elements, and hence W acts on Tx, (c) M is embedded, (d) M ∩ Tx = W · x, (e) let ηtv : M → N denote the end point map defined by tv, and Mt = ηtv(M)={exp(tv(x)) | x ∈ M} denote the set parallel to M at distance t, then Mt is an equifocal hypersurface and ηtv maps M diffeomorphically onto ∈ − - Mt if t ( 2g + θ, θ), + − (f) M = Mθ and M = M− - are embedded submanifolds of codimension 2g +θ + − m1 +1,m2 +1 in N, and the maps ηθv : M → M and η − - : M → M ( 2g +θ)v are Sm1 - and Sm2 - bundles respectively, + − (g) the focal variety V(M)=(M ,m1) ∪ (M ,m2),

5 { | ∈ − - } (h) Mt t [ 2g + θ, θ] gives a singular foliation of N, which is analogous to the orbit foliation of a cohomogeneity one isometric group action on N, (i) N = D1 ∪ D2 and D1 ∩ D2 = M, where D1 and D2 are diﬀeomorphic to the normal disk bundles of M + and M − respectively. (j) let p ∈ N, t ∈ R, and let E denote the energy functional on the path space P (N,p × Mt), then the Z2-homology of P (N,p × Mt) can be computed explicitly in terms of m1 and m2 and t; moreover, (1) if p is not a focal point of M then E is a perfect Morse function, (2) if p is a focal point of M then E is non-degenerate in the sense of Bott and perfect.

Equifocal submanifolds, hyperpolar actions and infinite dimensional isoparametric submanifolds in Hilbert spaces are closely related as we will now explain. Recall that an isometric H-action on N is called hyperpolar if there exists a compact flat T in N that meets every H-orbit and meets orthogonally in every point of intersection with an H-orbit (see [Co], [PT1] and [HPTT2]). Such a T is called a flat section of the action. A typical example is the action of K on the compact rank k symmetric space N = G/K with k-flats in N as flat sections. It is proved by Bott and Samelson in [BS] that this action is variationally complete, and that if M is an orbit in N, then E : P (N,p × M) → R is a perfect Morse function and the Z2-homology of P (N,p× M) can be computed explicitly in terms of the marked affine Dynkin diagram associated to the symmetric space N. It is proved in section 2 that principal orbits of a hyperpolar action are equifocal. Another main goals of our paper is to show that, although equifocal submanifolds in N need not be homogeneous, they share the same geometric and topological properties as principal orbits of hyperpolar actions. In particular, we will show that the Z2-homology of P (N,p× M) can be similarly calculated if M is equifocal or more generally weakly equifocal. Let H0([0, 1], G) denote the Hilbert space of L2-integrable paths u :[0, 1] → G, and let φ : H0([0, 1], G) → G be the parallel transport map, i.e., φ(u)= E(u)(1), where E(u) satisfies the differential equation E−1E = u with E(0) = e. We will see in section 4 that φ is a Riemannian submersion. It is proved in [Te5] that if M is a principal orbit of a hyperpolar action on G then φ−1(M) is isoparametric in H0([0, 1], G). We prove in section 5 that this statement is still true when M is equifocal, i.e., not assuming M is homogeneous. Recall that a submanifold in Rn is called taut if all non-degenerate squared distance functions are perfect. To summarize, we state some of our main results more precisely below.

1.7 Theorem. If M is an immersed, weakly equifocal compact submanifold of “refGh a semi-simple, compact symmetric space N, then (a) for a focal normal field v, the leaf of the focal distribution Fv through x ∈ M is diffeomorphic to a taut submanifold of a finite dimensional Euclidean space,

6 (b) if p ∈ N is not a focal point of M then E : P (N,p × M) → R is a perfect Morse function, otherwise E is non-degenerate in the sense of Bott and perfect, (c) M is embedded in N. Recall that a submanifold M in N is called totally focal if η−1(V(M)) = Γ(M) (cf. [CW2]). The next theoremis an analogue of Theorem1.6 in higher codimension:

1.8 Theorem. Suppose M is a codimension r equifocal submanifold of a simply “refGi connected, compact symmetric space N. Then (a) for a focal normal ﬁeld v, the leaf of the focal distribution Fv through x ∈ M

is diffeomorphic to an isoparametric submanifold in ν(Mv)ηv (x), (b) exp(ν(M)x)=Tx is an r-dimensional flat torus in N for all x ∈ M, (c) there exists an affine Weyl group W with r +1nodes in its affine Dynkin diagram such that for x ∈ M, (1) W acts isometrically on ν(M)x, and the set of singular points of the W -action on ν(M)x is the set of all v ∈ ν(M)x such that exp(v) is a focal point of M with respect to x, ∩ · (2) M Tx = expx(W 0), (d) let Dx denote the Weyl chamber of the W -action on ν(M)x containing 0, and x = exp(Dx), then (1) expx maps the closure Dx isometrically onto the closure x, (2) there is a labelling of the open faces of x by σ1(x),...,σr+1(x) and integers m1,...,mr+1 independent of x such that if y ∈ ∂x, then y is a focal point with respect to x of multiplicity my, where my is the sum of mi such that y ∈ σi(x), (e) M is totally focal in N, (f) let v be a parallel normal field on M, then Mv is an embedded submanifold, and moreover: (1) if exp(v(x)) is not a focal point then Mv is again equifocal and the end point map ηv : M → Mv is a diffeomorphism, (2) if exp(v(x)) is a focal point then ηv : M → Mv is a fibration and −1 the fiber ηv (y) is diffeomorphic to a finite dimensional isoparametric submanifold in the Euclidean space ν(Mv)y, (3) Mv ∩ Tx = exp(W · v(x)), (g) let xo be a fixed point in M, and = xo , then (1) the parallel foliation {Mv | exp(tv(xo)) ∈ for all 0 ≤ t ≤ 1 and v is a parallel normal field} is a singular foliation on N, which is analogous to the orbit foliation of a compact group action on N, (2) let y ∈ and My the parallel submanifold of M through y, then the focal variety V(M)=∪{(My,my) | y ∈ ∂}, (h) let p ∈ N, v a parallel normal field on M, and E the energy functional on the path space P (M,p × Mv), then the Z2-homology of P (M,p × Mv) can be computed explicitly in terms of W and m1,...,mr+1; moreover,

7 (1) if p is not a focal point of M then E is a perfect Morse function, (2) if p is a focal point of M then E is non-degenerate in the sense of Bott and perfect, (i) N = ∪{x | x ∈ M}, and (1) if x = y then x ∩y = ∅, (2) if x ∩ y = ∅ then it is a closed subsimplex of both x and y, (3) given x ∈ M, {y | y ∈ M ∩ Tx} is a triangulation of Tx.

Note that Theorem1.8 (i) impliesthat we can associate to each equifocal submanifold of a simply connected, compact symmetric space N a “toric building structure on N”. This is analogous to a spherical building except that the corresponding “apartments” cover tori instead of spheres. Theorem1.6 and 1.8 are not valid if N is not simply conneced. To see this, let N be the real projective space RP n and M a distance sphere in N centered at x0. Then M is certainly equifocal. Let v be a unit normal field on M. Then there exists t0 ∈ R such that exp(t0v(x)) = x0 for all x ∈ M. Let Tx be the normal circle at a point x in M. Then Dx is an interval, and x = Tx \{x0}. Moreover, there exists t1 such that the parallel set Mt1 is the cut locus of the center x0, which is a Z2-quotient of M, i.e., a projective hyperplane. Notice that the focal − variety of M consists of only one point (x0,n 1) and Mt1 is not diffeomorphic to M. In fact Mt1 has the same dimension as M and satisfies all the conditions in the definition of an equifocal submanifold except that the normal bundle does not have trivial holonomy. Although a parallel manifold Mv of M in a simply connected symmetric space N is either equifocal or a focal submanifold, this need not be the case if N is not simply connected. In fact, in this case the parallel set of an equifocal submanifold in N is still an embedded submanifold, but there are three types of parallel submanifols. The third type is, as in the example just given, a parallel submanifold which satisfies all the conditions in the definition of an equifocal submanifold except that the normal bundle does not have trivial holonomy. This is analogous to the three types of orbits an action of a compact Lie group can have: principal, singular and exceptional orbits. A submanifold M in N is called curvature adapted if for any v ∈ ν(M)x the operator Bv(u)=R(v, u)(v) leaves TMx invariant, and Bv commutes with the shape operator Av (cf. [BV]). Wu in [Wu1] defined a submanifold M in N to be hyper-isoparametric if it has globally flat abelian normal bundle, is curvature adapted and the principal curvatures along any parallel normal field are constant. Note that M is hyper-isoparametric if and only if M is curvature adapted and equifocal. Wu independently obtained some of our results by using the method of moving frames. But an equifocal submanifold in general is neither curvature adapted nor has constant principal curvatures. For example there are many such equifocal hypersurfaces in CPn (cf. [Wa]). We would like to make some comments on the methods we use to prove many of our geometric results on equifocal submanifolds. Since a symmetric space that is not a real space formhas morethan one root space, the operator S in the Jacobi

8 equation (*) has more than one eigenvalue and the shape operators and S need not commute in general. Thus there is no simple formula relating the focal points to the principal curvatures. This makes manipulation of the structure equations (the main technique in studying the geometry of submanifolds in space forms) much more complicated. So in this paper, we abandon many of the standard tools used in the study of submanifold geometry in space forms, and instead we study directly the relation between focal points of a submanifold and lifts of the submanifold under certain Riemannian submersions. In fact, the following two theorems are key steps in proving the results stated above: → 1.9 Theorem. Let π : G G/K be the natural Riemannian fibration of “refGk the symmetric space G/K, M a submanifold of G/K with globally flat abelian ∗ −1 normal bundle, and M a connected component of π (M). Let v ∈ ν(M)x, x∗ ∈ M ∗ π(x∗)=x, and v∗ the horizontal lift of v at x∗. Then (1) exp(v) is a multiplicity m focal point of M in G/K with respect to x if and only if exp(v∗) is a multiplicity m focal point of M ∗ with respect to x∗ in G, (2) ν(M ∗) is globally flat and abelian, (3) M is equifocal (weakly equifocal resp.) in G/K if and only if M ∗ is equifocal (weakly equifocal resp.) in G.

0 G → 1.10 Theorem. Let φ : H ([0, 1], ) G denote the parallel transport map, “refGl M ∗ a compact submanifold in G with globally ﬂat and abelian normal bundle, −1 ∗ ∗ ∗ ∗ and M˜ a connected component of φ (M ). Let v ∈ ν(M )x∗ , x˜ ∈ Mφ˜ (˜x)=x , and v˜ the horizontal lift of v∗ at x˜. Then (1) exp(v∗) is a multiplicity m focal point of M ∗ in G with respect to x∗ if and only if exp(˜v) is a multiplicity m focal point of M˜ with respect to x˜ in H0([0, 1], G), (2) ν(M˜ ) is globally ﬂat, (3) M ∗ is equifocal (weakly equifocal resp.) in G if and only if M˜ is isoparametric (weakly isoparametric resp.) in H0([0, 1], G).

So Theorems 1.9 and 1.10 allow us to study the geometry of an equifocal submanifold M in G/K by studying the geometry of the isoparametric submanifold M˜ = φ−1(π−1(M)) in the Hilbert space V = H0([0, 1], G). Although M˜ is infinite dimensional, the ambient space is a flat space form and most of the techniques and results in finite dimensional space forms are still valid (cf. [Te3], [PT2]). This paper is organized as follows: we give examples of equifocal submanifolds in symmetric spaces in section 2, derive an explicit relation between focal points and shape operators of submanifolds in compact Lie groups in section 3, prove that the parallel transport map φ is a Riemannian submersion and study the geometry of φ in section 4, prove Theorem1.10 in section 5, prove Theo- rem1.6, 1.7 and 1.8 and 1.9 in section 6, prove the existence of inhomogeneous

9 equifocal hypersurfaces in compact Lie groups and inhomogeneous isoparametric hypersurfaces in Hilbert spaces in section 7, and state some open problems in section 8. The authors would like to thank the Max Planck Institute in Bonn. This research started during the authors’ visit there in the fall of 1991 in an eﬀort to ﬁnd inhomogeneous isoparametric hypersurfaces in Hilbert spaces. It took us awhile to see how this simple problem actually ties together many interesting subjects in submanifold geometry.

2. Examples of equifocal submanifolds

The main result of this section is the following theorem:

2.1 Theorem. Let H be a closed subgroup of G × G that acts on G isometri- “refCb cally by · −1 (h1,h2) g = h1gh2 . If the action of H on G is hyperpolar, then the principal H-orbits are equifocal submanifolds of G. More generally, if G/K is a compact symmetric space and H is a closed subgroup of G that acts hyperpolarly on G/K, then the principal H-orbits are equifocal. PROOF. First notice that if G/K is a compact symmetric space and H a closed subgroup of G that is hyperpolar on G/K, then the subgroup H × K of G × G is hyperpolar on G, see [HPTT2]. If π : G → G/K is the natural fibration, then the H-orbits in G/K lift to H × K-orbits in G. We will prove in section 6, that a submanifold of G/K is equifocal if and only if its lift to G is equifocal. It is therefore enough to prove the theoremfor hyperpolar actions on a Lie group G. Let M be a principal orbit in G. Then the induced action on the normal bundle of M is trivial and every normal vector extends to an equivariant normal vector field. It is clear fromthe definition of hyperpolar actions that the normal bundle of M is abelian. It is proved in [PT1] that every equivariant normal field of M is parallel. Hence the normal bundle of M is globally flat. It is therefore left to prove that if v is an equivariant normal field of M such that ηv(x0) is a multiplicity k focal point of M with respect to x0, then ηv(x) is a multiplicity k focal point of M with respect to x for all x ∈ M. This follows immediately from the following observations. First note that hyperpolar actions are variationally complete; see [Co]. It is proved in [BS], see also Proposition 2.7 in [Te4], that variational completeness implies that (i) the set of focal points of M in G is exactly the set of singular points with respect to the H-action, (ii) if y is a multiplicity k focal point of M with respect to x then k is equal to the difference of the dimensions of the isotropy subgroups, i.e., k = dim(Hy) − dim(Hx).

10 2.2 Remark. Let M be a principal orbit of an isometric H-action on G.It“refCba is proved in [HPTT2] that if exp(ν(M)x) is contained in some ﬂat, then the H- action is hyperpolar. In particular, this implies that if the normal bundle of M is abelian then M is equifocal.

2.3 Examples. The following are examples of hyperpolar actions on a Lie “refEx group G or a symmetric space G/K (cf. [HPTT2]): (a) H = G(σ)={(g, σ(g)) | g ∈ G}, where σ : G → G is an automorphism, is hyperpolar on G. (b) H = K1 × K2, where Ki is the ﬁxed point set of some involution σi of G, is hyperpolar on G. Consequently K1 is hyperpolar on the symmetric space G/K2. (c) the action of H = ρ(K) × SO(n − 1) on SO(n), where ρ : K → SO(n) is the isotropy representation of a rank two symmetric space, (d) any cohomogeneity one action on G/K.

2.4 Remark. It is true in real space forms that a hypersurface is equifocal if “refExx and only if it has constant principal curvatures. This is not true in more general ambient spaces as an example of Wang [Wa] shows. In his example Wang starts with an inhomogeneous isoparametric hypersurface in an odd dimensional sphere which he shows to be the lift under the Hopf map of a hypersurface M in a complex projective space. This hypersurface M is equifocal in our terminology. He then shows that the principal curvatures of M cannot be constant. Since a curvature adapted, equifocal hypersurface must have constant principal curvatures, these hypersurfaces are not curvature adapted. They are not homogeneous either because their lifts are not homogeneous. We will discuss related examples of inhomogeneous equifocal hypersurfaces in the last section of this paper.

3. Relation between focal points and shape operators

Let G be a compact, semi-simple Lie group, G its Lie algebra, , an Ad- invariant inner product on G, and ds2 the bi-invariant metric on G defined by , . The main result of this section is to give a necessary and sufficient condition for a point in G to be a focal point of a submanifold with abelian and globally flat normal bundle in G. It is known that the Levi-Civita connection for ds2 is given by

1 ∇ Y = [X, Y ] X 2 if X, Y are left invariant vector ﬁelds, and

1 ∇ Y = − [X, Y ] X 2 11 if X, Y are right invariant vector ﬁelds. The curvature tensor is 1 R(X, Y )Z = [X, [Y,Z]], 4 where X, Y , and Z are left invariant vector ﬁelds. To simplify the notation, we will use the convention that for g ∈ G and y ∈G, gy =(Lg)∗(y) and yg =(Rg)∗(y), where Lg and Rg are left and right translations by g respectively.

3.1 Proposition. If M is a submanifold in G with abelian normal bundle, then “refFi −1 −1 [x ν(M)x, G] ⊂ x TMx.

In particular, R(ξ,TGx)ξ ⊂ TMx for ξ ∈ ν(M)x. −1 PROOF. Set A = x ν(M)x. Since the inner product on G is Ad-invariant and A is abelian, we have

[G, A], A = G, [A, A] = G, 0 =0,

−1 i.e., [G, A] ⊂ x TMx. The second part of the proposition follows since 1 R(ξ,X)ξ = [ξ,[X, ξ]]. 4

The following is an elementary fact concerning focal points and Jacobi ﬁelds.

3.2 Proposition. Let M be a submanifold of N, x(t) a smooth curve in M, “refFt v(t) a normal field of M along x(t), and η : ν(M) → N the end point map. Then dηv(0)(v (0)) = J(1), where J is the Jacobi field along x(t) satisfying the initial condition J(0) = x(0) − ∇⊥ ∇⊥ and J (0) = Av(0)(x (0)) + x(0)v, where is the normal connection of the submanifold M. Let a ∈G, γ(t)=xeat a geodesic in G, and J(t) a Jacobi field along γ. Denote the parallel transport map along γ from γ(a) to γ(b) by Pγ(a, b). Set −1 Y (t)=γ(0) Pγ(t, 0)J(t). Then the Jacobi equation for J gives rise to the following equation for Y (cf. [Mi]): 1 Y − ad(a)2Y =0. 4 Let T be a maximal abelian subalgebra of G that contains a, and G = T + Gα α∈+

12 2 2 the root space decomposition with respect to T , where ad(a) (zα)=−α(a) zα for zα ∈Gα. Let D1(a) and D2(a) be the operators deﬁned as follows: for ∈T ∈G z = p0 + α pα with p0 ,pα α, we set α(a) D (a)(z)=p + cos p , 1 0 2 α α 2 α(a) D (a)(z)=p + sin p , 2 0 α(a) 2 α α

−1 where λ sin(λ) is deﬁned to be 1 if λ =0. Notice that D1(a) and D2(a) depend only on a, but not on the choice of the maximal abelian subalgebra T containing a. One can also describe D1 and D2 in terms of the curvature tensor − 1 2 of G. For this we note that the operator, Ra(z)=R(a, z)a = 4 ad(a) (z),isa nonnegative symmetric operator, and D (a)(z) = cos( R (z)), 1 a −1 D2(a)(z)= Ra (z) sin( Ra(z)).

3.3 Theorem. Suppose M is a submanifold in G with abelian normal bundle “refFa −1 and a ∈ x ν(M)x. Then −1 −1 (1) the operator (D1(a) − D2(a)x Axax) maps x TMx to itself, where Axa is the shape operator of M at x along xa, (2) xea is a focal point of M of multiplicity m with respect to x if and only if −1 −1 the operator (D1(a) − D2(a)x Axax) on x TMx is singular with nullity m.

Part (1) of this theoremis obvious, but to prove part (2) we need the following Lemma.

3.4 Lemma. Let x(t) be a curve in M, and v(t)=x(t)a(t) a parallel normal “refFd ﬁeld along x(t). Then −1 { − −1 } −1 x0 Pγ0 (1, 0)dη(x0,v0)(v0)= D1(a) D2(a)x0 Av0 x0 (x0 x0), where 0 as an index refers to t =0and Pγ0 (1, 0) denotes the parallel transport sa0 map along the geodesic γ0(s)=x0e from γ0(1) to γ0(0). PROOF. Let γ(s, t)=x(t)esa(t) be a variation of normal geodesics of M, and ∂γ ∂γ S = ,T= . ∂s ∂t Then S(0,t)=x(t)a(t), and J(s)=T (s, 0) is a Jacobi ﬁeld along the geodesic sa(0) γ0(s)=x(0)e with J(0) = x (0). By Proposition 3.2, we have dηv(0)(v (0)) = J(1),

13 and J (0) = −Av(0)(x (0)) since v(t) is parallel. As above set

−1 Y (s)=γ0(0) Pγ0 (s, 0)J(s).

Then Y satisfies the differential equation 1 Y − ad(a)2Y =0. 4 Since the operator ad(a)2 is in diagonal formwith respect to the root space decompositon, this differential equation can be solved explicitly. In fact, the solution for the initial value problem Y (0) = p0 + α pα and Y (0) = q0 + α qα is α(a) 2 α(a) Y (t)=p + tq + p cos t + q sin t . 0 0 α 2 α α(a) 2 α Clearly −1 −1 Y (0) = x(0) J (0) = −x(0) Av(0)(x (0)). But Y (0) = x(0)−1x(0),sowehave

−1 x0 Pγ0 (1, 0)dηv0 (v0)=Y (1) α(a) 2 α(a) = p + q + p cos + q sin 0 0 α 2 α α(a) 2 α { − −1 } −1 = D1(a) D2(a)x0 Av0 x0 (x0 x0) where 0 as an index on x, x v, and v refers to t =0.

3.5 Proof of Theorem 3.3. “refPf It is obvious that we can choose a basis for T (ν(M))(x,xa), which consists of vectors of the form v(0) as in Lemma 3.4 and σ(0) with σ(t)=x(a + tb), ∈ −1 a b x ν(M)x. Since ν(M) is abelian, we have d expxa(xb)=xe b =0. The theoremnow follows fromLemma3.4.

For a ∈ ν(M)x, ν(M)x ⊕ TMx can be naturally identified with T (ν(M))a via the map → (b, u) vb(0) + vu(0), where vb is defined by vb(s)=a + sb and vu(s) is the parallel normal field M M along the geodesic γ(s) = exp (su) with vu(0) = a (here exp denotes the exponential map of M). It follows fromthe proof of Theorem3.3 that the differential dη : T (ν(M))a → TGη(a) is of the form I 0 0 D1(a) − D2(a)Aa

14 under the identiﬁcations

T (ν(M))a " ν(M)x ⊕ TMx,

TGη(a) " ν(M)x ⊕ TMx,

∈ ⊕ where in the second line (b, u) ν(M)x TMx is identified with Pγ0 (0, 1)(b + u) and γ0 is defined by γ0(s) = exp(sa). In particular, this implies that the kernel of dηa is a subset of TMx under the the above identifications (notice that we have very strongly used that the normal bundle ν(M) is abelian). So we have:

3.6Proposition. Let M be a submanifold of G with abelian and globally flat “refGo normal bundle, and v a parallel normal vector field. Then y0 = exp v(x0) is a focal point of M with respect to x0 if and only if the differential of the end point map of v, ηv : M → G, x → exp v(x), is not injective at x0. Moreover, the multiplicity of y0 as a focal point is equal to the dimension of the kernel of d(ηv)x0 . Theorem 3.3 is valid for any symmetric space. In fact, let N be a symmetric space, G = Iso(N), and M a submanifold of N with abelian normal bundle. Let ∈ G K P ∈P x0 M, K = Gx0 , = + the Cartan decomposition, and a normal to M at x0. Let A be a maximal abelian subalgebra in P containing a, and P = A + Pα α∈ its root space decomposition. Define Di(a):P→Pby Di(a)(b)=b for b ∈A, and α(a) 2 α(a) D (a)(x ) = cos x ,D(a)(x )= sin x , 1 α 2 α 2 α α(a) 2 α for xα ∈Pα. Then the same proof as for Theorem 3.3 implies that exp(a) is a multiplicity m focal point of M with respect to x0 if and only if the kernel of the operator D1(a) − D2(a)Aa is of dimension m. So we have the same result as in Proposition 3.6 for symmetric spaces:

3.7 Proposition. Let M be a submanifold of a symmetric space N with abelian “refFu and globally flat normal bundle, and v a parallel normal vector field on M. Then y0 = exp v(x0) is a focal point of M with respect to x0 if and only if the differential of the end point map of v, ηv : M → N,x → exp v(x), is not injective at x0. Moreover, the multiplicity of y0 as a focal point is equal to the dimension of the kernel of d(ηv)x0 . The following Proposition will be useful later.

15 3.8 Proposition. Let M be a submanifold in G with globally ﬂat and abelian “refFc normal bundle, x(t) ∈ M, and a(t) ∈Gsuch that v(t)=x(t)a(t) is a normal vector ﬁeld of M along the curve x(t). Then −1 −1 (1) x(0) ∇x(0)v = a (0)+1/2[x(0) x (0),a(0)], −1 (2) v(t) is parallel if and only if a (0) ∈ x(0) TMx(0). Furthermore, if v(t) is parallel, then 1 −x(0)−1A (x(0)) = a(0) + [x(0)−1x(0),a(0)]. v(0) 2

PROOF. Let y1,...,yn be a basis for G, and Yi(g)=gyi the left invariant vector ﬁeld deﬁned by yi. Write a(t)= i fi(t)yi, then v(t)= i fi(t)Yi(x(t)). Since the metric on G is bi-invariant, we have ∇ ∇ x (0)v = fi (0)Yi(x(0)) + fi(0)( x (0)Yi)(0) i 1 = f (0)x(0)y + f (0)x(0)[x(0)−1x(0),y] i i 2 i i i 1 = x(0){a(0) + [x(0)−1x(0),a(0)]}. 2 This proves (1), and (2) follows easily from(1).

4. The geometry of the parallel transport map

Let H0([0, 1]; G) denote the Hilbert space of L2-integrable paths u :[0, 1] → G, where the L2-inner product is deﬁned by 1 u, v0 = u(t),v(t) dt. 0

Let H1([0, 1]; G) denote the group of absolutely continuous paths g :[0, 1] → G −1 −1 such that g is square integrable, i.e., g g ,gg 0 is ﬁnite. Given a subset H of G × G, we let P (G, H) denote the subset of all g ∈ H1([0, 1],G) such that (g(0),g(1)) ∈ H.IfH is a subgroup, then P (G, H) is a subgroup of H1([0, 1],G). Note that P (G, e × G) is a Hilbert manifold, and

1 T (P (G, e × G))g = {vg | v ∈ H ([0, 1]; G),v(0)=0}.

Let P (G, e × G) be equipped with the right-invariant metric deﬁned by v1g, v2g = v1,v2 0.

16 Let E : H0([0, 1]; G) → P (G, e × G) be the parallel translation in the trivial principal bundle I × G over I =[0, 1] deﬁned by the connection u(t)dt, i.e., E = E(u) for u ∈ H0([0, 1]; G) is the unique solution of

E−1E = u, E(0) = e.

Let φ : H0([0, 1]; G) → G be the parallel transport from 0 to 1, i.e.,

φ(u)=E(u)(1).

The following result is known (cf. [Te3], [Te5]): 1 0 G 4.1 Theorem. Let H ([0, 1],G) act on V = H ([0, 1]; ) by gauge transfor- “refFf mations, i.e., g ∗ u = gug−1 − gg−1, and let φ : H0([0, 1], G) → G be the parallel transport map from 0 to 1. Then (1) the action of H1([0, 1],G) on H0([0, 1]; G) is proper, Fredholm and isometric, (2) φ(g ∗ u)=g(0)φ(u)g−1(1) for g ∈ H1([0, 1],G) and u ∈ H0([0, 1]; G), −1 ∈ 1 (3) if φ(u)=x0φ(v)x1 then there exists g H ([0, 1],G) such that g(0) = x0,g(1) = x1 and u = g ∗ v,

× 0 G 4.2 Corollary. The action of P (G, e G) on H ([0, 1]; ) by gauge transfor- “refPb mations is transitive and free. PROOF. Let yˆ denote the constant path with constant value y. We ﬁrst prove that the action is transitive. Let u ∈ H0([0, 1]; G). Since G is connected and compact, there exists y ∈Gsuch that φ(u)=ey. It is obvious that φ(0)ˆ ey = φ(u). So by Theorem4.1, there exists g ∈ P (G, e × G) with g(0) = e and g(1) = e−y such that u = g ∗ 0ˆ. To prove the action is free, let g ∈ P (G, e × G) be such that g ∗ 0=ˆ 0ˆ. Then g−1g =0.Sog is constant. Since g(0) = e, g(t)=e for all t.

0 G → × 4.3 Corollary. The parallel translation E : H ([0, 1], ) P (G, e G) is an “refPc isometry. PROOF. Since the inverse of E is the map F : P (G, e×G) → H0([0, 1], G) −1 −1 −1 defined by F (g)=g g = g ∗ 0ˆ and dFg(vg)=g v g, our claimfollows. Let G be a compact Lie group acting on a smooth manifold M, and let π : M → M/G denote the orbit space map. It is known that if the action of G has only one orbit type, then the orbit space M/G has a unique differentiable structure such that π is a fibration. If moreover, M is Riemannian and the G-action is isometric, then there exists a unique metric on M/G such that π

17 is a Riemannian submersion. In fact, we have T (M/G)π(x) = dπx(TMx) and ⊥ dπx(u1),dπx(u2)π(x) = u1,u2x for u1, u2 ∈ (ker dπx) and x ∈ M. In the following we will give an infinite dimensional analogue of this fact for the free 0 isometric action of Ωe(G)=P (G, e × e) on H ([0, 1]; G). First, as a consequence of Theorem4.1 and Corollary 4.2 we have × × 4.4 Corollary. Let Ωe(G) denote the subgroup P (G, e e) of P (G, e G), “refFg and φ : V = H0([0, 1]; G) → G the parallel transport map from 0 to 1. Then (1) the gauge action of Ωe(G) on V is free, (2) the fibers of φ are exactly the orbits of Ωe(G), (3) φ is a principal Ωe(G)-bundle, (4) G is the orbit space V/Ωe(G), and φ is the orbit space map, (5) for g ∈ P (G, e × G) the map τg(u)=g ∗ u maps fibers of φ to fibers of φ, −1 (6) the differential of the map τg in (5) is d(τg)u(v)=gvg .

4.5 Theorem. Let H be the horizontal distribution for the ﬁbration φ, i.e., “refFm H(u)=the normal space of the ﬁber φ−1(φ(u)) at u, and let yˆ denote the constant path in G with constant value y. Then (1) H(0)ˆ = {yˆ | y ∈G}, (2) if u = g ∗ 0ˆ, then H(g ∗ 0)ˆ = gH(0)ˆ g−1 = {gygˆ −1 | y ∈G}, −1 −1 (3) if u = g ∗ 0ˆ with g(0) = e, then dφu(gygˆ )=yφ(u)=yg(1) , (4) φ is a Riemannian submersion, (5) φ is the natural Riemannian submersion associated to the free isometric 0 action of Ωe(G) on H ([0, 1], G).

−1 PROOF. Since F =Ωe(G) ∗ 0=ˆ {−g g | g ∈ Ωe(G)},wehave TF = {ξ | ξ ∈ H1([0, 1]; G),ξ(0) = ξ(1) = 0} 0ˆ 1 = {u ∈ H0([0, 1]; G) | u(t)dt =0}. 0 So (1) is an immediate consequence. Statement (2) follows from the facts that ∗ 0 G −1 the map τg(v)=g v is an isometry on H ([0, 1]; ), d(τg)0ˆ(v)=gvg and τg maps ﬁbers of φ at 0 to ﬁbers of φ at g ∗ 0. Note that if g(0) = e and u = g ∗ 0ˆ then φ(g ∗ 0+ˆ sgygˆ −1)=φ(g ∗ syˆ ), by Theorem4 .1 = φ(ˆsy)g(1)−1 = esyg(1)−1 = esyφ(u). −1 So dφu(gygˆ )=yφ(u), which proves (3). Then (4) and (5) follows.

∈ ∈ −1 4.6Corollary. Let v TGx, u φ (x), and v˜ the horizontal lift of v at u “refFo with respect to φ. Choose g ∈ P (G, e × G) such that u = g ∗ 0ˆ. Then v˜(u)=gvx−1g−1.

18 Let M be a submanifold in G. Using the isometry E from H0([0, 1], G) to P (G, e × G), we see that φ−1(M) is isometric to the submanifold

P (G, e × M)={g ∈ P (G, e × G) | g(0) = e, g(1) ∈ M} of P (G, e × G). Note that P (G, e × M) is like a cylinder set for the Wiener measure on the set of continuous paths g in G. Motivated by the deﬁniton of a general cylinder set, we consider below the parallel transport from a to b for any 0 ≤ a

g−1g = u, g(a)=e.

1 The proof of the following facts is the same as for φ0 = φ. 2 4.7 Theorem. Let ds be a ﬁxed bi-invariant metric on G, and G[a,b] denote the “refCc 1 2 b 0 G → Lie group G with the bi-invariant metric b−a ds . Let φa : H ([a, b], ) G[a,b] denote the parallel transport map from a to b. Then b ∗ b −1 (1) φa(g u)=g(a)φa(u)g(b) , b b −1 ∈ 1 (2) if φa(v)=x1φa(u)x2 then there exists g H ([a, b],G) such that g(a)= x1, g(b)=x2 and v = g ∗ u, b ∗ { −1 | ∈G} (3) the horizontal space of φa at g 0 is gygˆ y , ∗ ˆ b −1 − b (4) if u = g 0, then d(φa)u(gygˆ )=(b a)yφa(u), b (5) φa is a Riemannian submersion, and it is the natural Riemannian submersion associated to the free, isometric action of

1 Ωe([a, b],G)={g ∈ H ([a, b],G) | g(a)=g(b)=e}

on H0([a, b], G).

··· 4.8 Corollary. Let s :0=s0

i=n 0 G → Φs : H ([0, 1]; ) G[si−1,si], deﬁned by i=1 s 1 1 | | − Φs(u)=(φ0 (u [0,s1]),...,φsn−1 (u [sn 1, 1])),

19 0 G ⊕n 0 G and identifying H ([0, 1], ) with the direct sum i=1H ([si−1,si], ) via the linear isometry

n 0 0 f : H ([0, 1], G) → H ([si−1,si], G) deﬁned by i=1

u → (u | [0,s1],...,u| [sn−1, 1]).

Then (1) Φs is the natural Riemannian submersion associated to the free product n 0 G action of i=1 Ωe([si−1,si],G) on H ([0, 1], ), −1 −1 (2) Φs(u)=(E(u)(s1),E(u )(s1) E(u)(s2),...,E(u)(sn−1) E(u)(1)). −1 (3) if M is a submanifold of i G[si−1,si] then Φs (M) is isometric to the following submanifold of P (G, e × G):

−1 −1 {g ∈ P (G, e × G) | (g(s1),g(s1) g(s2),...,g(sn−1) g(1)) ∈ M}.

PROOF. Let [a, b] ⊂ [0, 1]. By the uniqueness of solution of ordinary b | −1 diﬀerential equations we have φa(u [a, b]) = E(u)(a) E(u)(b). So the Corollary follows.

5. The geometry of lifts of submanifolds of G to H0([0, 1]; G)

Let M be a submanifold of a Hilbert space V , and η : ν(M) → V the end point map (i.e., η(v)=x + v if v ∈ ν(M)x). Recall that M is called proper Fredholm if the restriction of the end point map to any normal disk bundle νr(M) of finite radius r is a proper, Fredholmmap. If M is proper Fredholmin V then I − Av is Fredholmfor all v ∈ ν(M), and the restriction of any squared distance function to M satisfies condition C of Palais and Smale. A proper Fredholm submanifold M of V is called isoparametric if ν(M) is globally flat and for any parallel normal field v on M the shape operators Av(x) and Av(y) are conjugate for all x, y ∈ M. We refer to [Te3] and [PT2] for more detailed geometric and topological properties of these submanifolds. A proper Fredholm submanifold M of V is called weakly isoparametric if ν(M) is globally flat, the multiplicities of the principal curvatures along a parallel normal field v are constant, and if dλ(X)=0for X ∈ Eλ(v)={X | AvX = λX} where λ is a principal curvature function λ along a parallel normal field v. Let G be a compact, connected, semi-simple Lie group equipped with a bi- invariant metric, and H a closed subgroup of G × G acting on G by (h1,h2) · −1 0 G x = h1xh2 . Let P (G, H) act on H ([0, 1]; ) by gauge transformations. It is proved in [Te5] that if the action of H on G is hyperpolar, then the action of

20 P (G, H) on H0([0, 1]; G) is polar and principal P (G, H)-orbits are isoparametric. Furthermore, using the formula φ(g ∗ u)=g(0)φ(u)g(1)−1 and the fact that the −1 a ﬁbers of φ are orbits of Ωe(G), we have φ (H ·e )=P (G, H)∗aˆ. To summarize, we have

5.1 Theorem ([Te5]). If M is a principal orbit of a hyperpolar action on G, “refCf then φ−1(M) is an isoparametric submanifold in H0([0, 1], G). By Theorem2.1, a principal orbit of a hyperpolar action is equifocal. So Theorem1.10 (3) in the introduction generalizes Theorem5.1 to equifocal submanifolds M in Lie groups which are not necessarily homogeneous. Before starting with the proof of Theorem1.10 we give a few simpleappli- cations. Recall that a proper Fredholmsubmanifold M of a Hilbert space V is taut (cf. [Te3]) if for every non-focal point a ∈ V the distance squared function 2 fa : M → R deﬁned by fa(x)= x − a is a perfect Morse function.

5.2 Proposition. Let M be a weakly equifocal immersion of a compact man- “refFhi ifold into a compact Lie group G. Then the lift M˜ of M to V is taut. PROOF. By Theorem1.10, M˜ is weakly isoparametric. One can prove exacly as in the ﬁnite dimensional case, see [Te2], [Te3] and [Th1], that an inﬁnite dimensional weakly isoparametric submanifold is taut.

5.3 Proposition. A weakly equifocal immersion of a compact manifold M into “refFhii a Lie group G is an embedding. PROOF. We know fromProposition 5.2 that M˜ is taut. It is standard that taut submanifolds are embedded (cf. [Te 3]). Hence M˜ is embedded and then it is clear that M is embedded as well.

5.4 Proposition. Let M be a weakly equifocal compact submanifold in G.If“refBc p ∈ G is not a focal point of M, then the energy functional E : P (G, p×M) → R is a perfect Morse function. PROOF. Set p = ea for a ∈G. Consider the diffeomorphism ρ : P (G, p × M) → P (G, e×M) defined by ρ(g)(t)=g(t)e(t−1)a. The path space P (G, p×M) can be naturally embedded into H0([0, 1], G) as the submanifold M˜ = φ−1(M) via the map g → F (ρ(g)) where F : P (G, e × G) → H0([0, 1], G) is the isometry defined by F (g)=g−1g as in the proof of Corollary 4.3. Using this embedding, the functional E on P (G, p × M) corresponds to the restriction of the Hilbert − 2 ˜ ˜ distance squared function fa(u)= u a 0 to M. Since M is taut (Proposition 5.2), fa is a perfect Morse function. We will now prove several lemmas needed for the proof of Theorem 1.10. The notation will be the same as in Theorem 1.10 except that we do not assume that M has a globally flat normal bundle when not explicitly stated. First as a consequence of Corollary 4.6, we have

21 ∈ × ∗ ˆ ∈ ˜ 5.5 Lemma. Suppose h P (G, e G), u = h 0 M and x = φ(u). Then “refFp

−1 −1 ν(M˜ )u = {hbx h | b ∈ ν(M)x}.

5.6Lemma. Let v be a normal vector ﬁeld on M, and v˜ the horizontal lift of “refFr v to M˜ . Then v˜ is a parallel normal ﬁeld on M˜ if and only if v is parallel on M.

PROOF. We need to show that dv˜u(T M˜ u) is contained in T M˜ u if and only if dvx(TMx) ⊂ TMx. Now let h ∈ P (G, e × G) be such that u = h ∗ 0ˆ, and gs a smooth curve in P (G, e × G) such that g0(t)=e for all t and gs ∗ u ∈ M˜ . Then x(s)=φ(gs ∗ u) ∈ M. Let d d −1 ξ = gs,ξ(u)= (gs ∗ u),a(x)=x v(x). ds s=0 ds s=0 A direct computation gives d dv˜u(ξ(u)) = v˜(gs ∗ h ∗ 0) ds s=0 d −1 −1 −1 = gshx(s)a(x(s))x(s) h gs ds s=0 −1 −1 −1 −1 −1 −1 −1 =[ξ,hx0a(x0)x0 h ]+h[x (0)x0 ,x0a(x0)x0 ]h + hx0a (0)x0 h ,

˜ { −1 | ∈ } where x0 = x(0). By lemma 5.5, ν(M)u = hx0bx0 h x0b ν(M)x0 . Given ∈ −1 −1 b x0 ν(M)x0 , because x0 ν(M)x0 is abelian and , is ad-invariant, we have −1 −1 −1 −1 −1 −1 −1 [ξ,hx0a(x0)x0 h ]+h[x (0)x0 ,x0a(x0)x0 ]h ,hx0bx0 h 0 =0. ∈ −1 By Proposition 3.8 (2), v is parallel if and only if a (0) x0 TMx0 . Hence v is parallel if and only if

−1 −1 −1 −1 hx0a (0)x0 h ,hx0bx0 h 0 =0 which holds if and only if v˜ is parallel. This completes the proof of the lemma.

As a consequence of Proposition 3.8 and the proof of Lemma 5.6 we have: ˜ 5.7 Lemma. The shape operators Av and Av˜ are related as follows: “refFFi 1 −h−1A˜ (ξ(u))h =[h−1ξh,vx−1]+ [x(0)x−1,vx−1] − A (x(0))x−1. v˜ 0 2 0 0 v 0

22 ˜ −1 5.8 Lemma. Let M be a compact submanifold of G. Then M = φ (M) is “refFhb a proper Fredholm submanifold of V = H0([0, 1]; G). Furthermore, the shape operators of M˜ are compact. ∈ ˜ ≤ PROOF. We prove properness ﬁrst. Suppose ξk ν(M)uk , ξk r, and uk + ξk → w. We want to prove that ξk has a convergent subsequence. To prove this, we choose gk ∈ P (G, e × G) such that gk ∗ 0=uk. Let xk = φ(uk), and ∈ −1 ak xk ν(M)xk be such that ξk is the horizontal lift of xkak, i.e., −1 −1 ξk = gkxkakxk gk .

Compactness of M implies that there exist a subsequence of xk (still denoted by xk) and x0 ∈ M such that xk → x0. Since the disk of radius r in G is compact, −1 → ∈ −1 by passing to a subsequence we may assume that xkakxk b ν(M)x0 x0 . But − −1 −1 −1 ∗ −1 → uk + ξk = gkgk + gk(xkakxk )gk = gk (xkakxk ) w −1 → × 0 G and xkakxk b. Since the action of P (G, e G) on H ([0, 1]; ) is proper (cf. 1 [Te3]), gk has a subsequence in H ([0, 1]; G) converging to g0. This implies that uk → u0 = g0 ∗ 0 ∈ M˜ . Hence ξk converges. To prove M˜ is Fredholm, we will show that its shape operators are compact. Using right translation if necessary, we may assume that e ∈ M. Let a ∈ ν(M)e be a non-zero normal vector. It suffices to prove that the shape operator A˜ aˆ at 0ˆ ∈ M˜ is a compact operator. Since ξ(0)ˆ = −ξ, using lemma 5.7 we get 1 A˜ (ξ)=[ξ,a]+ [x(0),a] − A (x(0)). aˆ 2 a We can thus write A˜ aˆ(ξ )=D(ξ )+B(ξ ) where 1 D(ξ)=[ξ,a] and B(ξ)= [x(0),a] − A (x(0)). 2 a It is clear that B is of finite rank. So it suffices to prove that D is compact. To see this, we let T be a maximal abelian subalgebra of G containing ν(M)e, G = T G { }∪ + α∈+ α the corresponding root space decomposition, and t1,...,tk + {xα,yα | α ∈ } an orthonormal basis of G such that tk−p+1,...,tk ∈ ν(M)e, tj ∈T, xα,yα ∈Gα and

[a, xα]=α(a)yα, [a, yα]=−α(a)xα.

Let zα = xα + iyα. Then

2πint 2πnt + eα,n = Re(zαe ),fα,n =Im(zαe ),α∈ ,n≥ 1,

sj,n = tj cos 2πnt, tj,n = tj sin 2πnt, n ≥ 1, + t1,...,tk−p,xα,yα,α∈

23 ˜ forman orthonormalbasis for T M0ˆ. Direct computation shows that D satisﬁes

D(sj,n)=D(tj,n)=D(ti)=0, α(a) α(a) D(e )= e ,D(f )= f α,n 2πn α,n α,n 2πn α,n D(xα)=−α(a)tyα,D(yα)=α(a)txα. It is now clear that D is a compact operator. . The following lemma is well-known: ˜ 5.9 Lemma ([Te3]). Let M be a proper Fredholm submanifold in the Hilbert “refFra space V , and v˜ ∈ ν(M˜ )u. Then u +˜v is a multiplicity m focal point of M˜ with respect to u if and only if 1 is a multiplicity m eigenvalue of the shape operator Av˜. → ∈ 5.10 Lemma. Let π : E B be a Riemannian submersion, v0 TBp, q = “refFrr B −1 expp (v0), and v the normal field on π (p) defined by v(x) being the horizontal −1 −1 E lift of v0 at x. Let f : π (p) → π (q) be the map defined by f(x) = exp (v(x)). Then f is a diffeomorphism. PROOF. The lemma follows from the fact that for the Riemannian submersion π, the horizontal lift γ˜(t) of a geodesic γ(t) in B is a geodesic in E.

We will use the end point maps of parallel normal vector ﬁelds v of M and v˜ of M˜ in the next lemmas. Recall that these are deﬁned to be ηv : M → G, p → exp(v(p)), and ηv˜ : M˜ → V,u → u +˜v(u), respectively.

5.11 Corollary. Let M be a weakly equifocal submanifold of G, v a normal “refHa ﬁeld on M, and v˜ the horizontal lift of v on M˜ = φ−1(M). Then (1) φ ◦ ηv˜ = ηv ◦ φ, −1 (2) φ (Mv)=M˜ v˜, ∈ ˜ −1 −1 (3) let y˜ Mv˜ and y = φ(˜y), then φ maps ηv˜ (˜y) diﬀeomorphically to ηv (y), (4) if d(ηv˜)(u)=0and u =0 then dφ(u) =0 . PROOF. By Theorem4.5 (4), φ is a Riemannian submersion, (1) follows from Lemma 5.10. (2) is a consequence of (1). Let φ(˜x)=x, x˜ +˜v(˜x)=˜y. Then

φ(˜x +˜v(˜x)) = φ(ηv˜(˜x)) = ηv(φ(˜x)) = ηv(x)=φ(˜y)=y, −1 −1 ∈ −1 which implies that φ maps ηv˜ (˜y) to ηv (y).Nowifx˜1, x˜2 ηv˜ (˜y) and φ(˜x1)= ∈ −1 −1 φ(˜x2)=x ηv (y), then since by Lemma 5.10 ηv maps φ (x) diﬀeomorphically −1 −1 to φ (y) we have x˜1 =˜x2. This argument also proves that φ maps ηv˜ (˜y) onto −1 ηv (y). So we have proved (3). To see (4), suppose d(ηv˜)(u)=0and dφ(u)=0 −1 −1 for some u ∈ TMx. Then u ∈ T (φ (x))x. But ηv˜ is diﬀeomorphic on φ (x). So u =0.

24 5.12 Lemma. Let M be a submanifold in G with globally ﬂat and abelian “refFrs normal bundle, x ∈ M, x˜ ∈ φ−1(x), and v˜ the horizontal lift of v by φ. Then v is a multiplicity m focal normal of M with respect to x if and only if v˜ is a multiplicity m focal normal of M˜ with respect to x˜. Moreover, dφx˜ maps the kernel of dηv˜ bijectively onto the kernel of dηv.

Notice that in general the eigenspace Eλ(u, v˜)=kerdηv˜ is not horizontal although dφ is injective on it; see section 7 for examples.

PROOF. We know fromProposition 3.6 that q0 = exp(v(p0)) is a focal point of M with respect to p0 if the diﬀerential of ηv : M → G is not injective. The multiplicity of the focal point is equal to the dimension of ker dηv at p0.By Lemma 5.9, u0 +˜v(u0) is a focal point of M˜ with respect to u0 if and only if the diﬀerential of the map ηv˜ : M˜ → V is not injective and its multiplicity is equal to the dimension of ker dηv˜ at u0. We therefore need to prove that the dimension of ker dηv˜ is equal to the dimension of ker dηv. But by Corollary 5.11, we have ηv ◦ φ = φ ◦ ηv˜. Hence we have the following commutative diagram.

˜ dηv˜ T Mu −→ V  0  dφ dφ −→dηv TMp0 TGq0

If X˜ ∈ ker dηv˜ is nonzero, then by Corollary 5.11 (4) we have X˜ ∈ ker dφ. Hence X = dφ(X˜ ) =0 and dηv(X)=0. It follows that the dimension of ker dηv˜ is less than or equal to the dimension of ker dηv. Now let X ∈ ker dηv be a nonzero element and let X˜ be the horizontal lift. ˜ Then it follows fromthe commutative diagramabove that dηv(X) ∈ ker dφu+˜v(u). By Lemma 5.10, there is an element Y˜ ∈ ker dφu such that dηv˜(Y˜ )=dηv˜(X˜ ). Hence dηv˜(X˜ − Y˜ )=0. We have thus proved that the dimension of ker dηv˜ is greater or equal to the dimension of ker dηv. This ﬁnishes the proof.

5.13 Proof of Theorem 1.10. “refpfa Lemma 5.12 proves (1), and Lemma 5.6 proves (2). It remains to prove (3). Let us ﬁrst assume that M ∗ is weakly equifocal. Then we know from Lemma 5.8 that M˜ is a proper Fredholmsubmanifoldof V . By Lemma 5.6, ν(M˜ ) is globally ﬂat. So to prove that M˜ is weakly isoparametric, it is therefore left to show that the multiplicities of the eigenvalues of Av˜(u) are constant and that dλ(X)=0for ∗ X ∈ Eλ(˜v)={X | Av˜X = λX}. Furthermore, if M is equifocal, we will show that λ is constant, thereby proving that M˜ is isoparametric. It follows from Lemma 5.12 that the multiplicities are constant. Let λ(u) be a ˜ principal curvature function of Av˜(u). Then there is a focal point of M in direction v˜(u) at distance λ(u)−1. Hence by Lemma 5.12, M has a focal point in direction

25 −1 v with respect to φ(u)=p, at distance f(p)=λ(u) . Notice that dφ(Eλ(˜v)) are the ﬁbers of the focal distribution Ffv where Eλ(˜v) is the eigenspace of Av˜(u) corresponding to eigenvalue λ(u). Hence df (dφX)=0for X ∈ Eλ(˜v) implies dλ(X)=0.ThusM˜ is weakly isoparametric. If M ∗ is equifocal, then f is constant. Hence λ is constant and M˜ therefore isoparametric. Notice that the above arguments can also be used to prove the other direction. We therefore have that M˜ (weakly) isoparametric implies M ∗ (weakly) equifocal. This ﬁnishes the proof of the theorem.

The following theoremis proved exactly as Theorem1.10.

5.14 Theorem. Let Φs be as in Corollary 4.8, and M a closed submanifold “refCe −1 with globally ﬂat and abelian normal bundle of i G[si−1,si]. Then Φs (M) is a (weakly) isoparametric submanifold of H0([0, 1]; G) if and only if M is a (weakly) equifocal submanifold of i G[si−1,si].

5.15 Remark. As in the finite dimensional case, one can show that if the “ref dimension of Eλ(˜v) is locally constant and at least two then dλ(u)=0for u ∈ Eλ(˜v), where λ is a principal curvature function in direction of the parallel normal vector v˜ of a proper Fredholm submanifold with flat normal bundle in a Hilbert space. As a consequence, it is only necessary in the definition of weakly equifocal submanifolds to assume that df (u)=0for u in the kernel of dηfv if f is a multiplicity one focal curvature of M along a parallel normal field v. A similar remark can of course also be made on the definition of weakly isoparametric submanifolds in Hilbert spaces. ˜ 5.16Theorem. Let M be a weakly isoparametric submanifold in a Hilbert “refHb space V , v˜ a focal normal field, and S a leaf of the focal distribution Fv˜ on M˜ . Then S is a compact Z2-taut submanifold that is contained in a finite dimensional affine subspace of V .

PROOF. Being weakly isoparametric, M˜ itself is taut with respect to Z2. It is even true that every squared distance function on M˜ is perfect in the sense of Bott. Set a = ηv˜(S). Then S is a critical manifold of the squared distance function fa centered at a. Since fa satisfies condition C ([Te3]), S is compact. To prove that S is taut we use the following fact proved by Ozawa in [Oz]: Suppose f is a perfect Morse function on M˜ in the sense of Bott, and S is a critical submanifold of f.Ifg is function on M˜ that restricts to a Morse function on S and has the property that f + δg is a perfect Morse function on M˜ in the sense of Bott for all δ, then g | S is a perfect Morse function on S. (Ozawa proves this in the finite dimensional case using the Morse lemma. Since the Morse Lemma is true for functions satisfying condition (C) (cf. [Pa]), Ozawa’s result is true in infinite dimension.) Now let fb be a squared distance function on M˜ that is a Morse function on S. An easy calculation shows that

fa + δfb =(1+δ)fz + c(δ),

26 where z =(a + δb)/(1 + δ) and c(δ) is a constant that depends only on δ. Since weakly isoparametric submanifold is taut, fa + δfb is a perfect, and non- degenerate in the sense of Bott for all δ. Ozawa’s result now implies that fb is a perfect Morse function on S. This proves that S is taut. Now we can show exactly as when the ambient space is finite dimensional, that S spans a subspace of dimension less than or equal to n(n+3)/2, where n is the dimension of S. To be more precise, let Op be the osculating space of S at p, i.e., the affine space through p spanned by the first and second partial derivatives of S at p (or equivalently the affine space spanned by the the tangent space at p and the vectors α(X, Y ) for X, Y ∈ TSp where α is the second fundamental formof S). It is clear that the dimension of Op is at most n(n +3)/2.Now let p be the nondegenerate maximum of some squared distance function. Then the tautness of S (or even the much weaker two-piece property) implies that S is contained in Op, cf. [CR]. In particular we have proved that S spans a finite dimensional affine subspace.

5.17 Proposition. Let M be a weakly equifocal submanifold in G, v a focal “refFdd normal field, and Fv the focal distribution defined by v. Then (1) Fv is integrable and its leaves are diffeomorphic to a compact, taut submanifold in a finite dimensional Euclidean space, (2) if M is equifocal, then the leaves of Fv are diffeomorphic to an isoparametric

submanifold in ν(Mv)ηv (x).

PROOF. The focal distribution Fv is integrable since it is the distribution defined as the kernel of the differential of the map ηv with constant rank. Then the leaves of Fv are exactly the fiber of ηv. Let M˜ be the lift of M to V . Then M˜ is taut by Proposition 5.2. Let v˜ be the horizontal lift of v to a normal vector field of M˜ . It follows from Lemma 5.12 that v˜ is a focal normal field since v is. Let x0 ∈ M˜ , and a = x0 +˜v(x0). Then (1) follows fromTheorem5.16 and Corollary 5.11. If M is equifocal, then by Theorem1.10 (3) M˜ is isoparametric in V = H0([0, 1], G). It follows fromthe slice theoremfor infinite dimensional −1 isoparametric submanifolds that ηv˜ (a) is an isoparametric submanifold of the finite dimensional affine subspace x0 + ν(M˜ v˜)a (cf. [Te3]). Then (2) follows from Corollary 5.11 (3).

6. Geometry of weakly equifocal submanifolds

In this section, we will give proofs for Theorem1.6, 1.7, 1.8 and 1.9. In the following, we let (G, K) be a compact symmetric pair, N = G/K the corresponding symmetric space, and π : G → N the Riemannian submersion associated to the right action of K on G, M a submanifold of N, and M ∗ = π−1(M).

27 We will need several lemmas for the proof of Theorem 1.9. We use the same notation as in Theorem 1.9 except that we do not assume that the normal bundle is globally ﬂat when not explicitly stated. The ﬁrst lemma is a simple consequence of the following facts: (i) the vertical distribution V of π is V(g)=gK, where K is the Lie algebra of K, (ii) π(gh)=π(ghk) for k ∈ K, in particular, we have dπg(gu)=dπgk(guk).

6.1 Lemma. Let H denote the horizontal distribution of the Riemannian sub- “refFec mersion π : G → N, and G = K + P the Cartan decomposition. Then (1) H(g)=gP, (2) for u ∈P and k ∈ K, the horizontal lift of dπg(gu) at gk is guk. Since π is a Riemannian submersion, the following lemma can be proved in exactly the same way as Corollary 5.11 by using Lemma 5.10. ∗ 6.2 Lemma. Let v be a normal vector field on M, and v the horizontal lift of “refFed v to M ∗. Then (1) ηv ◦ π = π ◦ ηv∗ , −1 ∗ (2) π (Mv)=Mv∗ , −1 ∗ −1 ∗ ∈ ∗ (3) π maps ηv∗ (y ) diffeomorphically onto ηv (y) for any y Mv∗ , (4) if d(ηv∗ )(u)=0and u =0 then dπ(u) =0 . ∗ 6.3 Lemma. (1) ν(M ) is abelian if and only if ν(M) is abelian, “refFea (2) if v is a normal vector field on M and ν(M) is abelian, then the horizontal lift v∗ of v to M ∗ is a parallel normal vector field on M ∗ if and only if v is parallel, (3) ν(M ∗) is globally flat and abelian if and only if ν(M) is globally flat and abelian.

−1 PROOF. (1) Let A = g ν(M)gK. Then exp ν(M)gK = π(g exp(A)) is contained in a ﬂat of N if and only if A is abelian. On the other hand, ∗ ∗ ν(M )g = gA and hence exp ν(M )g = g exp(A) is contained in a ﬂat of G if and only if A is abelian. ∇∗ ∗ (2) Let X be a vertical tangent vector. Since the covariant derivative X v only depends on v∗ along a vertical curve with tangent vector X, we may assume that both v∗ and X are right invariant. So

1 ∇∗ v∗ = − [X, v∗]. X 2 ∇∗ ∗ ∗ Now it follows fromProposition 3.1 that X v is a tangent vector of M since ∇∗⊥ ∗ the normal bundle is abelian by (1). We have therefore shown that X v =0 for every vertical tangent vector X. ∇∗⊥ ∗ It is only left to show that X v =0for every horizontal tangent vector X ∇⊥ if and only if dπX v =0.

28 ∗ ∇∗ ∗ Let X be a horizontal tangent vector of M . We decompose X v into horizontal and vertical components:

∇∗ ∗ ∇∗ ∗ ∇∗ ∗ X v =( X v )h +( X v )v.

∗ ∗ ∗ ∗ Since (∇ v )h is the horizontal lift of ∇dπX v, it follows that (∇ v )h lies in the X ∗ X tangent space of M if and only if ∇dπX v lies in the tangent space of M. But ∗ ∇∗⊥ ∗ vertical vectors are tangent to M . This implies that X v =0if and only if ∇⊥ dπX v =0 (3) This is now an immediate consequence of (1) and (2).

Now the proof of Lemma 5.12 carries over to our present sitution, so we have the following lemma.

6.4 Lemma. Let M in N be a submanifold with globally ﬂat and abelian “refFee ∗ −1 ∗ ∗ normal bundle, v ∈ ν(M)x, x ∈ π (x), v the horizontal lift of v at x , and M ∗ = π−1(M). Then v is a multiplicity m focal normal of M with respect to x if and only if v∗ is a multiplicity m focal normal of M ∗ with resptect to x∗. Morover, dπx∗ maps the kernel of dηv∗ bijectively onto the kernel of dηv.

Notice again that the kernel of dηv∗ is in general not horizontal. In the next section we will discuss examples that demonstrate this.

6.5 Proof of Theorem 1.9. “refFef Lemma 6.3 proves statement (2), Lemma 6.4 proves (1), and statement (3) can be proved in a similar manner as the proof of Theorem 1.10 (3).

Applying Theorem 1.9 to the rank one symmetric space Sn+1, we get n n+1 6.6 Corollary. Let M be an isoparametric hypersurface of S . Then “refFq (1) M is equifocal in Sn+1, (2) M ∗ = π−1(M) is an equifocal submanifold of SO(n +2), where π is the natural ﬁbration from SO(n +2)to Sn+1.

Similarly, if M n is a proper Dupin hypersurface of Sn+1, then the lift M ∗ of M to SO(n +2)is a weakly equifocal submanifold. If M is proper Dupin, but not isoparametric, then it follows immediately that M ∗ is an inhomogeneous weakly equifocal hypersurface of SO(n +2). It is much more difficult to find inhomogeneous equifocal hypersurfaces in SO(n +2). This will be done in the next section. Let M be an equifocal submanifold in the symmetric space N and v a parallel normal field on M. Then the parallel set Mv = ηv(M) is an immersed manifold since the end point map ηv : M → N has constant rank.

6.7 Proposition. Let M be a compact, equifocal submanifold in a compact, “refFsc symmetric space N, and v a parallel normal ﬁeld. If v is not focal, then Mv

29 is equifocal if and only if ηv : M → Mv is one-to-one. Moreover, if ηv is not one-to-one then (1) ηv : M → Mv is a finite cover, (2) the normal holonomy of Mv is nontrivial, but otherwise Mv satisfies all the conditions in the definition of an equifocal submanifold.

PROOF. We first prove that expx(ν(M)x) = expηv (x)(ν(Mv)ηv (x)) if ηv(x) is not a focal point of M with respect to x. Theorems 1.10 and 1.9 imply that the connected components of M˜ = π−1(φ−1(M)) are isoparametric. Therefore the corresponding statement is true for M˜ and M˜ v˜, see [Te3]. The normal spaces of M˜ and M˜ v˜ are the horizontal lifts of the normal spaces of M and Mv respectively which implies what we wanted to prove. It also follows that the normal bundle of Mv is abelian. By Lemma 6.3, ν(Mv) is flat if ν(M˜ v˜) is flat. We need to show that the normal bundle of Mv has trivial holonomy if ηv is one-to-one. Let w be a parallel normal vector field on M. Then dηv(w) will give rise to a globally defined parallel normal field on Mv since ηv is one-to-one. This shows that Mv is equifocal if ηv is one-to-one. If ηv is not one-to-one, let p and q be two different points in M such that ηv(p)=ηv(q). Let α(t) be a curve connecting p and q. Then β(t)=ηv(α(t)) is a closed curve in Mv and dηv(v) will give rise to a parallel normal field along β(t) that does not close up in β(0) = β(1). This shows that the holonomy of the normal bundle is nontrivial. It is obvious that the focal structure of Mv is parallel.

6.8 Corollary. Let M,N,v and ηv be as in Proposition 6.7. Then one of the “refHd following statements holds: (1) ηv : M → Mv is a diffeomorphism, (2) ηv : M → Mv is a finite cover, (3) ηv : M → Mv is a fibration and Mv is a focal submanifold of M.

6.9 Remark. As mentioned in the introduction, if N is a real projective “ref space and M a distance sphere in N, then M is equifocal. Its parallel manifolds are spheres, a projective hyperplane and a point. The projective hyperplane is not equifocal since it does not have trivial normal holonomy. As a parallel submanifold M doubly covers the projective hyperplane by the corresponding end point map ηv. Note that the lift of M to the Hilbert space V has two connected components.

6.10 Proposition. Let M be an equifocal submanifold of N, and v a parallel “refFscc normal ﬁeld on M. Then Mv is embedded. Moreover, if v1,v2 are two parallel ∩ ∅ normal ﬁelds on M and Mv1 Mv2 = then Mv1 = Mv2 .

PROOF. A connected component of the lift M˜ v˜ to V is either an isoparametric submanifold or a focal submanifold of the isoparametric submanifold M˜ and hence embedded. It follows that Mv must be embedded. If two parallel manifolds of M meet without coinciding, then the same thing is true for their lifts.

30 But we know that parallel manifolds of an isoparametric submanifolds cannot meet without coinciding. One consequence of Propositions 6.7 and 6.10 is that M and its parallel submanifolds give rise to an ‘orbit like foliation’ of M. There are three types of leaves: ‘principal’ when Mv is equifocal, ‘exceptional’ when the dimension of Mv is the same as that of M but the normal holonomy is nontrivial, and ‘singular’ when Mv is a focal submanifold. We will see later that we can exclude ‘exceptional’ leaves when the ambient space N is simply connected.

6.11 Proposition. If N is a simply connected, compact symmetric space, and “refHe M is a connected submanifold of N, then the lift M ∗ = π−1(M) is connected. If furthermore G is simply connected then M˜ = φ−1(M ∗) is connected. PROOF. Since N = G/K is simply connected, K is connected. So the exact sequence of homotopy of the fibration M ∗ → M with fiber K implies that M ∗ is connected. We now prove that the lift M˜ = φ−1(π−1(M)) of M to V is connected. We know fromthe proof of Proposition 5.4 that M˜ is diffeomorphic to the path space P (G, p∗ × M ∗), where M ∗ = π−1(M) and p∗ is a lift of p. ∗ ∗ ∗ Notice that P (G, p × M ) fibers over M with fiber Ωp∗ (G). Hence

∗ ∗ ∗ ... → π0(Ωp∗ (G)) → π0(P (G, p × M ) → π0(M ).

Since G is simply connected we have that π0(Ωp∗ (G)) = π1(G)=0.

6.12 Corollary. Let N be a simply connected, compact symmetric space, M “refHf a codimension r equifocal submanifold of N, and v a parallel normal ﬁeld. If v is not focal then ηv : M → Mv is a diﬀeomorphism and Mv is also equifocal.

PROOF. By Proposition 6.7, it suffices to prove that ηv is one to one. Suppose x1,x2 ∈ M such that ηv(x1)=ηv(x2)=x. We can assume that N = G/K and G is simply connected. Let v˜ be the horizontal lift of v to M˜ via π ◦φ. By Lemma 5.12 and 6.4, v˜ is a non-focal, parallel normal field on M˜ . Since M˜ is connected by Proposition 6.11 and is isoparametric in V , ηv˜ : M˜ → M˜ v˜ is a diffeomorphism. But by Lemma 5.11 and 6.2, ηv˜ maps the fiber Yi over xi of π ◦ φ diffeomorphically onto the fibers over x for i =1, 2. Because ηv˜ is a diffeomorphism, Y1 = Y2. In particular, this proves that x1 = x2.

6.13 Theorem. Let M be an equifocal submanifold of the symmetric space “refFss N. Then M is totally focal in N. PROOF. Let c(t) be a geodesic that meets M orthogonally at t =0and satisﬁes c(1) ∈ Mv for some focal normal ﬁeld v. We have to show that c(1) is a focal point of M in direction c(0). Let ˜c(t) be a lift to V . Then ˜c(0) lies in −1 −1 M˜ = φ (π (M)) and ˜c(1) ∈ M˜ v˜. It follows from Lemmas 5.12 and 6.4 that the components of M˜ v˜ are focal submanifolds of M˜ . It now follows from[Te3]

31 that ˜c(1) is a focal point of M˜ in direction ˜c(0). Hence c(1) is a focal point of M in direction c(0) by Lemmas 5.12 and 6.4.

It is easy to see that a k-ﬂat in a compact, rank k symmetric space N is a closed torus. But an r-ﬂat with r

6.14 Theorem. Let N be a connected, compact, symmetric space of semi- “refHc simple type equipped with the bi-invariant metric induced from the Killing form on G. Suppose H acts on N isometrically, and x ∈ N is a regular point of the · · H-action such that exp(ν(H x)x) is contained in some ﬂat. Then expx(ν(H x)x) is closed in N. This theoremimpliesthat if M is a homogenous, equifocal submanifold of N then exp(ν(M)x) is closed. The following theoremgeneralizes this fact to an arbitrary equifocal submanifold:

6.15 Theorem. Let N be as in Theorem 6.14, and M a closed equifocal sub- “refFtt ∈ manifold of N. Then expx(ν(M)x) is a closed torus for all x M. PROOF. It is evident that if we lift M to the Lie group G and prove the theoremfor M ∗, then it also follows for M in N. So we may assume that N = G. It is proved in [HPTT2] that if T is the Lie algebra of a torus in G and A is a linear subspace of T , then exp(A) is a closed torus if and only if exp(A⊥) is, where A⊥ is the orthogonal complement of A in T . The proof of this theoremis similarto the proof of Theorem6.14 in [HPTT2] with minor changes. We repeat it for sake of completeness. Set A = ν(M)x and A ¯ A = expx( ). Assume A is not closed. We set B = A. Then B is a torus, and the Lie algebra B of B is abelian. Let A1 denote the orthogonal complement of A in B, and A1 = exp(A1). We ﬁrst prove that B is transversal to the orbit M. To see this, notice that A is orthogonal to M whenever A meets M since the parallel manifolds of M give rise to an orbit like foliation whose leaves are met orthogonally by A. More precisely, let Ma denote the parallel submanifold of M through a ∈ A, then aA⊥T (Ma)a and Ma = M if a ∈ A ∩ M. Since B is the closure of A it follows that bA⊥TMb for every b ∈ B ∩ M which implies that B is transversal to M since TBb = bB contains bA. By transversality, B ∩ M is a compact submanifold of G. Next we show that B ∩ M = A1. In fact, one sees easily that T (B ∩ M)b = bA1 for every b ∈ B ∩ M. But A1 is the integral submanifold through e of the distribution (g)=gA1 deﬁned on G. Hence B ∩ M = A1. Since B and A1 are tori we have that A is a torus.

6.16 Corollary. Let N be as in Theorem 6.14, and M a closed equifocal hy- “refFtc persurface of N. Then every normal geodesic to M in N is a circle. 6.17 Proof of Theorem 1.8. “ref −1 −1 Let v ∈ ν(M)x, x˜ ∈ φ (π (x)), and v˜ the horizontal lift of v by π ◦ φ. ◦ Since π φ is a Riemannian submersion, expx(v)=π(φ(˜x +˜v)). We prove each itemof the theoremseperately below:

32 (a): By Corollary 5.11 (3) and Lemma 6.2 (3), π ◦ φ maps the focal leaf of Fv˜ diffeomorphically onto the focal leaf of Fv. Then (a) follows fromProposition 5.17 (2). (b) is proved in Theorem6.15. (c): The lift M˜ = φ−1(π−1(M)) is an isoparametric submanifold and has therefore an affine Coxeter group W acting on its affine normal spaces, see [Te3]. Since d(π ◦ φ) maps the affine normal spaces of M˜ isometrically onto the normal spaces of M, we have that W acts on ν(M)x for all x ∈ M. Then (c) (1) follows from standard results of isoparametric submanifolds in Hilbert space. To ˜ ∩ ˜ V · V prove (c) (2), we note that M (˜x + ν(M)x˜) = expx˜ (W 0), where exp is the exponential map for V .Soexp(W · 0) ⊂ M ∩ Tx. Conversely, if y ∈ Tx ∩ M then there exists a parallel normal field v on M such that y = exp(v(x)). But y = π(φ(˜x +˜v(˜x)), where v˜ is the horizontal lift of v. Then M˜ v˜ ∩ M˜ = ∅. But M˜ is connected and isoparametric. So M˜ = M˜ v˜. Hence v˜(x) ∈ W · 0 and y ∈ expV (W · 0). This proves (c) (2). (d): Using results for the isoparametric submanifold M˜ in V , (2) follows from(1). So we need only prove (1). Since exp(ν(M)x) is contained in some flat, expx is a local isometry from ν(M)x onto Tx. So to prove (d) (1) it suffices to prove that expx is one to one on Dx. Suppose not. Then there exist parallel V V normal fields v˜1 and v˜2 on M˜ such that exp (tv˜1(˜x)), exp (tv˜2(x)) ∈ Dx for ≤ ≤ all 0 t 1 and expx(v1(x)) = expx(v2(x)) = p, where x = π(φ(˜x)) and ◦ ∈ ∩ vi = d(π φ)(˜vi). Note that p Mv1 Mv2 . By Proposition 6.10, Mv1 = Mv2 . ˜ ◦ −1 ˜ But Mv˜i =(π φ) (Mvi ). Since M is connected and isoparametric, v˜1 =˜v2. Hence v1 = v2, which proves (d) (1). (e) is Theorem6.13. (f): Item(1) is proved in Corollary 6.12. It was proved in Corollary 6.8 that if exp(v(a)) is a focal point, then ηv is a fibration. By Corollary 5.11 (3) and ◦ −1 −1 ∈ Lemma 6.2 (3), π φ maps ηv˜ (˜y) diffeomorphically onto ηv (y) for y Mv. −1 ˜ −1 But ηv˜ (˜y) is the slice of the isoparametric submanifold M in V .Soηv˜ (˜y) is an isoparametric submanifold of y + ν(M˜ y˜)y˜. This proves (2). Item(3) can be proved exactly the same way as (c)(2). (g) follows fromProposition 6.10 and standard results for isoparametricsub- manifolds in V . (h): Note that since M˜ is isoparametric, the distance squared function fa −1 on M˜ v˜ is non-degenerate in the sense of Bott and is perfect if a ∈ (π ◦ φ) (p) (cf. [Te3]). Using the same argument as in the proof of Proposition 5.4, M˜ v˜ ∗ ∗ is diffeomorphic to P (G, p × M ) and fa corresponds to the energy functional ∗ −1 ∗ ∗ E, where p ∈ π (p). Next we prove that P (N,p × Mv) and P (G, p × M ) are homotopy equivalent. For this we can assume that p lies in M and hence ∗ ∗ ∗ ∗ that p lies in M . We have a fibration π∗ : P (G, p × M ) → P (N,p × M) with fiber the space of paths in the coset p∗K starting in p∗. Since the fiber is obviously contractible it follows that P (N,p × M) and P (G, p∗ × M ∗) are homotopy equivalent. Since E corresponds as in the proof of Proposition 5.4 to

33 the distance squared function of an isoparametric submanifold in V , the indices and Morse linking cycles at critical points of E are given explicitly (cf. [Te3]) and (1), (2) follow. (i): It is known that V = ∪{Dx˜ | x˜ ∈ M˜ } and has properties analogous to (1), (2) and (3) (cf. [Te3]). So (i) follows from(d).

As a consequence of Theorem1.8, we can describe the space of parallel submanifolds of an equifocal submanifold in the simply connected case.

6.18 Corollary. Let M be an equifocal submanifold of a simply connected “refFs semi-simple symmetric space N. Then the space of parallel submanifolds of M with the quotient topology is a simplex where the boundary points correspond to the focal manifolds. Note that Theorem1.6 is a special case of Theorem1.8 when M is an equifocal hypersurface. So the only theoremthat remainsto prove is 1.7.

6.19 Proof of Theorem 1.7. “ref By Theorem1.9 (3) and 1.10 (3), the connected components of M ∗ are weakly equifocal in G, and similarly the connected components of M˜ are weakly isoparametric in V . The proof that an inﬁnite dimensional isoparametric submanifold is taut (cf. [Te3]) shows also that an inﬁnite dimensional weakly isoparametric submanifold in V is taut. So (b) and (c) follow. (a) follows fromProposition 5.17 (1) and Lemma 6.2 (3).

7. Inhomogeneous examples in Hilbert spaces

The main result of this section is to show that there exist inhomogeneous equifocal hypersurfaces in SO(n) and H0([0, 1], SO(n)) for certain numbers n. By an inhomogeneous submanifold we mean that it is not an orbit of a subgroup of the isometry group of the ambient space. Let Sn = SO(n+1)/SO(n), π : SO(n+1) → Sn the natural fibration, and φ : H0([0, 1], SO(n + 1)) → SO(n +1) the parallel transport map as before. Let M ⊂ Sn be a submanifold, M ∗ = π−1(M), and M˜ = φ−1(M ∗). First, we will derive explicit formulas relating the shape operators of M, M ∗ and M˜ . In order to do this, we recall some facts about Riemannian submersions. Let π : E → B be a Riemannian submersion, M a submanifold of B, and M ∗ = π−1(M). Let X and Y be vector fields on E, and define

∇ ∇ NX Y =( Xh Yh)v +( Xh Yv)h, where the indices h and v refer to horizontal and vertical components respectively. Then N is tensorial in X and Y . In fact, N is one of the O’Neill tensors for the

34 Riemannian submersion π. If both X and Y are horizontal, then

NX Y = −NY X =(∇X Y )v.

7.1 Proposition. Let π : E → B be a Riemannian submersion, M a sub- “refFeg ∗ −1 ∗ ∗ manifold of B, v ∈ ν(M)x and u ∈ TMx. Let M = π (M), and v ,u the horizontal lifts of v, u at y ∈ π−1(x) repectively. Then

∗ ∗ ∗ ∗ ∗ Av∗ u =(Avu) + Nv u

∗ ∗ ∗ where A and A are the shape operators of M and M respectively and (Avu) is the horizontal lift of Avu. ∗ ∗ PROOF. Since the horizontal component of ∇ u∗ v coincides with the hor- ∇ ∗ ∗ ∗ izontal lift of uv, the horizontal component of Av∗ u is (Avu) . The vertical ∗ ∗ ∗ component of ∇ u∗ v is tangent to M and hence is equal to the vertical compo- − ∗ ∗ ∗ ∗ nent of Av∗ u . Since both u and v are horizontal, the vertical component of ∗ ∗ ∗ ∗ ∗ ∗ ∇ ∗ ∗ − ∗ u v is equal to Nu v = Nv u . This proves that the vertical part of Av∗ u ∗ is equal to Nv∗ u .

→ 7.2 Proposition. Let G/K be a symmetric space, and π : G G/K the “refAa corresponding Riemannian submersion. If v∗ and u∗ are horizontal vectors in TeG, then ∗ 1 ∗ ∗ N ∗ u = [v ,u ]. v 2

PROOF. Since left invariant vector ﬁelds deﬁned by u∗ and v∗ are horizontal ∗ and Nv∗ u is a tensor,

∗ ∗ 1 ∗ ∗ N ∗ u =(∇ ∗ u ) = [v ,u ] . v v v 2 v

But G/K is a symmetric space implies that [v∗,u∗] is vertical. So we are done.

→ 7.3 Proposition. Let G/K be a compact, symmetric space, π : G G/K “refAb the natural ﬁbration, M a submanifold of G/K, and M ∗ = π−1(M). Assume ∗ ∗ that p = π(e) ∈ M. Let X ∈ TMp,ξ ∈ ν(M)p, X and ξ the horizontal lift of ∗ h ∗ X and ξ at e respectively, and Xv a vertical vector at e. Let Aξ and Aξ∗ be the shape operators of M and M ∗ respectively. Then ∗ ∗ ∗ 1 ∗ ∗ (1) Aξ∗ Xh =(AξX) + 2 [ξ ,Xh], ∗ ∗ − 1 ∗ ∗ (2) Aξ∗ (Xv )= 2 [ξ ,Xv ], which is horizontal.

35 PROOF. (1) follows from Lemma 4.4 and Proposition 7.2. To prove (2), ∇∗ ∗ ∗ we first notice that X∗ ξ depends only on ξ along a vertical curve with tangent ∗ v ∈ vector Xv at e. By Lemma 6.1 (2), the horizontal lift of ξ along k K is right ∗ ∗ invariant. So we may assume that both Xv and ξ are right invariant vector fields. ∇∗ ∗ 1 ∗ ∗ ∇∗ ∗ Hence X∗ ξ = 2 [ξ ,Xv ]. Since the fibers of π are totally geodesic, X∗ ξ is v v ∗ horizontal. Now let G = K + P be the Cartan decomposition, and A = ν(M )e. Because A is abelian and the metric on G is Ad-invariant, we have [A, K] ⊥A. A K ⊂ ∗ This implies that [ , ] TMe and proves (2).

→ n 7.4 Theorem. Let π : SO(n +1) S be the natural ﬁbration associated “refAg to Sn, φ : H0([0, 1], SO(n + 1)) → SO(n +1)the parallel transport map, M a submanifold of Sn, M ∗ = π−1(M), and M˜ = φ−1(M ∗). Let x˜ ∈ M˜ , x∗ = φ(˜x), and x = π(x∗). Let ξ be a unit normal vector of M at x, and ξ∗, ˜ξ the horizontal lift of ξ. Let Eλ(x, ξ) denote the eigenspace of Aξ for the eigenvalue λ, and ∗ ∗ ∗ E (x ,ξ ), E˜λ(˜x, ˜ξ) deﬁnd similarly. Then we have λ ∗ ∗ (1) dπx∗ maps E csc θ+cot θ (x ,ξ ) isomorphically onto Ecot θ(x, ξ), and the inverse 2 is given by Y → Y ∗, ∗ ∗ ∗ − ∗−1 ∗ ∗ ∈ ∗ ∗ ∗ Y = x (Xh + (csc θ cot θ)[x ξ ,Xh]) E csc θ+cot θ (x ,ξ ), 2 ∈ ∗ ∈P ∗ ∗ where Y Ecot θ(x, ξ) and Xh such that x Xh is the horizontal lift of Y , ˜ ˜ ∗ ∗ ∗ (2) dφx˜ maps E 1 (˜x, ξ) isomorphically to E csc θ+cot θ (x ,ξ ), and it is given explic- θ 2 itly as follows:

˜ θ −1 − ∗ − − ∗ ∈ ˜ ˜ Y = g (cos(θ(t 1))Xh sin(θ(t 1))Xv )g E 1 (˜x, ξ) sin θ θ −→dφ ∗ ∗ ∗ − ∗ ∈ ∗ ∗ ∗ Y = x (Xh + (csc θ cot θ)Xv ) E (x ,ξ ),

∗ ∗−1 ∗ ∗ ∈ 1 where Xv =[x ξ ,Xh] and g H ([0, 1],G) satisﬁes the condition that g(1) = e and g ∗ x˜ = 0ˆ.

Before proving this theorem, we need the following Lemma: cot θ+csc θ 7.5 Lemma. With the same notation as in Theorem 7.4, if λ = 2 = “refAf 1 θ ∗ ∗ ∈ ∗ ∗ 2 cot 2 is an eigenvalue of Aξ∗ at x M with multiplicity m, then exp(θξ ) is a focal point of M ∗ with respect to x∗ of multiplicity m. ∗ ∈ ∗ ∗ ∗ PROOF. We may assume that g = e. Let Y Eλ, y (t) a curve in M such that (y∗)(0) = Y ∗, and v∗(t) be the parallel normal ﬁeld along y∗(t) with v∗(0) = θξ∗. By Lemma 3.4, we have

∗ ∗ ∗ ∗ ∗ ∗ ∗ − Py (1, 0)dηθξ∗ ((v ) (0)) = (D1(θξ ) D2(θξ )Aθξ∗ )(Y ),

36 ∗ where Py∗ (1, 0) is the parallel translation from 1 to 0 along y , and η is the end n point map. Using the deﬁnition of the operators Di and the fact that S has only one root, we get

θ 2 θ D (θξ∗)(Y ∗) = cos Y ∗,D(θξ∗)(Y ∗)= sin Y ∗. 1 2 2 θ 2 ∗ ∗ This proves that dηθξ∗ ((v ) (0)) = 0.

7.6Proof of Theorem 7.4. By left translation, it suﬃces to prove (1) for ∗ ∗ ∗ ∗ “ref x = e. Suppose Aξ(X) = cot θX. Set Xv =[ξ ,Xh]. Then by Proposition 7.3 we have ∗ ∗ ∗ 1 ∗ ∗ ∗ 1 ∗ A ∗ (X ) = cot θX + [ξ ,X ] = cot θX + X , ξ h h 2 h h 2 v ∗ ∗ 1 ∗ ∗ 1 ∗ A ∗ (X )=− [ξ ,X ]= X , ξ v 2 v 2 h ∗ 2 ∗ − ∗ where the last equality follows fromthe fact that ad(ξ ) (Xh)= Xh. A direct ∗ − ∗ ∗ computation shows that Xh + (csc θ cot θ)Xv is in E csc θ+cot θ . 2 ∗ ∗ csc θ+cot θ ∗ ∗ ∗ ∗ Conversely, suppose A (Y )= 2 Y . Let Y = Yh + Yv be the decomposition into horizontal and vertical components, and Y = dπ(Y ∗). Then ∗ Yh is the horizontal lift of Y at e. Using Proposition 7.3, we get

∗ ∗ ∗ ∗ ∗ ∗ Aξ∗ (Y )=Aξ∗ (Yh )+Aξ∗ (Yv ) 1 1 =(A (Y ))∗ + [ξ∗,Y∗] − [ξ∗,Y∗] ξ 2 h 2 v csc θ + cot θ = (Y ∗ + Y ∗). 2 h v Comparing the vertical components of both sides of the above equation, we get

∗ − ∗ ∗ Yv = (csc θ cot θ)[ξ ,Yh ].

Since G/K is of rank one,

∗ ∗ − ∗ ∗ ∗ − − ∗ [ξ ,Yv ] = (csc θ cot θ)[ξ , [ξ ,Yh ]] = (csc θ cot θ)Yh .

∗ ∗ Comparing the horizontal component then gives (Aξ(Y )) = (cot θY ) . This ∗ proves that dπ(Y )=Y ∈ Ecot θ. To prove (2), we ﬁrst assume that x˜ = 0ˆ. Recall that the parallel transport map φ maps focal points of M˜ in H0([0, 1], G) to focal points of M ∗ in G, and maps normal geodesics of M˜ to normal geodesics of M ∗. Note also that given ∈ ˜ 1 ˜ v ν(M)u, u + λ v is a multiplicity m focal point of M with respect to u if and only if λ is a principal curvature of M˜ with multiplicity m. So as a consequence

37 ˜ ∗ of Lemma 7.5, we have dφ maps E 1 isomorphically onto E csc θ+cot θ . In fact, the θ 2 ˜ → ∗ inverse of dφ : E 1 E csc θ+cot θ can be given explicitly as follows: By Lemma 5.7, θ 2 we have ∗ 1 ∗ ∗ ∗ ∗ A˜ (Z˜ )=[Z,ξ˜ ]+ [x (0),ξ ] − A ∗ (x (0)), (∗) ˜ξ 2 ξ ∗ −˜ ˜ ∈ ˜ ∗ ∈ ∗ where x (0) = Z(1). Now suppose Z E 1 . Then x (0) E csc θ+cot θ . By (1), θ 2 ∈ ∗ ∗ ∗ − there is X Ecot θ such that x (0) = Xh + s(θ)Xv , where s(θ) = csc θ cot θ. So equation (∗) becomes

1 ∗ 1 ∗ ∗ ∗ ∗ ∗ Z˜ =[Z,ξ˜ ]+ [X + s(θ)X ,ξ ] − A ∗ (x (0)). θ 2 h v ξ ∗ ∗ ∗ ∗ ∗ − ∗ ∗ ∈ ∗ Using [ξ ,Xh]=Xv , [ξ ,Xv ]= Xh, and x (0) E csc θ+cot θ , we get 2

1 Z˜ = −[ξ∗, Z˜ + X∗ − cot θX∗]. θ h v

Set ˜ ˜ ∗ − ∗ Y = Z + Xh cot θXv . Then ˜ − ∗ ˜ ˜ − ∗ Y = θ[ξ , Y ], Y (1) = csc θXv . This initial value problemcan be solved explicitly, and the solution is

˜ − ∗ − ∗ Y (t) = (cos(θt) cot θ sin(θt)) Xh (sin(θt) + cot θ cos(θt))Xv , 1 = − {sin((t − 1)θ)X∗ + cos((t − 1)θ)X∗}. sin θ h v

So θ X˜ = −Z˜ = (cos(θ(t − 1))X∗ − sin(θ(t − 1))X∗ sin θ h v ∗ is in E˜ 1 . To summarize, we have shown that if π(e) ∈ M, x = e, and x˜ = 0ˆ, θ then dφ ˜ −→0ˆ ∗ −→dπe E 1 E csc θ+cot θ Ecot θ θ 2 are isomorphisms and they are given explicitly as

θ dφ (cos(θ(t − 1))X∗ − sin(θ(t − 1))X∗) −→ X∗ + (csc θ − cot θ)X∗, sin θ h v h v −→dπ ∗ X = dπ(Xh).

38 Next we compute the formuals relating the curvature distributions at an arbitrary point x ∈ M. Let x∗ ∈ π−1(x) and x˜ ∈ φ−1(x∗). Choose g ∈ H1([0, 1],G) such that g(1) = e and g ∗ x˜ = 0ˆ. Recall that

φ(g ∗ x˜)=g(0)φ(˜x)g(1)−1 = g(0)x∗ = φ(0)ˆ = e.

∗ −1 0 So x = g(0) . Because Fg(y)=g ∗ y is an isometry of H ([0, 1], G), we can translate the computation at 0ˆ for Fg(M˜ ) to x˜ by Fg to obtain the formula stated in (2).

In the remainder of this section, we will prove that there are inhomogeneous isoparametric hypersurfaces in SO(n) and in the Hilbert space H0([0, 1]; SO(n)) for certain n. These examples are based on the isoparametric hypersurfaces of Clifford type in spheres that were found by Ferus, Karcher, and Münzner [FKM]. We first descibe the Clifford examples briefly following [PTh]. Let

C = {E1,...,Em−1} be a systemof skew - × --matrices satisfying

EiEj + EjEi = −2δijId.

We call C a Clifford system. Such Clifford systems are in one-to-one corre- m−1 spondence with orthogonal representations of the Clifford algebra Cm−1 of R endowed with a negative definite metric. We say that u, v ∈ Rl are Clifford orthogonal if

u, v = E1u, v = ...= Em−1u, v =0.

The pairs (u, v) of Cliﬀord orthogonal vectors satisfying

1 u, u = v, v = 2

2l−1 forma submanifold V2(C) in S . We call V2(C) the Cliﬀord-Stiefel manifold of l 2l−1 C-orthonormal 2-frames in R . The tubes around V2(C) in S turn out to be isoparametric hypersurfaces, which have four distinct principal curvatures λ1 > ···>λ4 with multiplicities m1,m2,m1,m2, where m1 = m and m2 = - − m − 1. Using the classiﬁcation of homogeneous isoparametric hypersurfaces, one sees that these examples are inhomogeneous if m =1 , 2,or4 and (m, -) =(9 , 16). (See [FKM] for a detailed discussion of these examples.) + Using the same notation as in [FKM], we set M = V2(C), which is one of the focal manifolds in the isoparametric family. It is proved in section 5 in

39 [FKM] that the set L of points x in the focal submanifold M + such that there + are pairwise orthogonal unit normal vectors ξ0,...,ξ3 in ν(M )x such that

3 ≥ dim ker Aξi 3 i=0

+ is a non-empty proper subset of M if the multiplicities satisfy 9 ≤ 3m1 4 except by finitely many low dimensional examples. It now follows that M + and its family of isoparametric hypersurfaces are inhomogeneous without using the classification of the homogeneous ones. Applying Theorems 1.9(3) and 1.10(3) to these Clifford examples in S2-−1,we obtain many isoparametric hypersurfaces in H0([0, 1], SO(2-)) and in SO(2-). In the following, we prove that these examples are not homogenous. Because classifications of equifocal homogeneous hypersurfaces in compact Lie groups and in Hilbert spaces are not known, we will use the result of Ferus-Karcher-Münzner about the set L to prove the inhomogeneity. We state this more precisely in the following theorem.

7.7 Theorem. Let M be an isoparametric hypersurface of Clifford type in “refih S2-−1 satisfying 6 ≤ 2m<-+8and m =4 . Then (i) M ∗ = π−1(M) is an inhomogeneous equifocal hypersurface in SO(2-), where π : SO(2-) → S2-−1 is the Riemannian submersion obtained by the identi- fication S2-−1 = SO(2-)/SO(2- − 1), (ii) M˜ = φ−1(M ∗) is an inhomogeneous isoparametric hypersurface in V , where V = H0([0, 1]; SO(2-)) and φ : V → SO(2-) is the parallel transport map.

We will need the following simple Lemma:

7.8 Lemma. Let G = K + P be a Cartan decomposition corresponding to “refAj Sn−1 = SO(n)/SO(n − 1). Suppose a ∈Psuch that ad(a)2 |P ∩A⊥ = − id, and x, y, z ∈Pare orthogonal to a.Ifr =[a, x]=[y, z] then r =0and x =0.

PROOF. Let GC (λ) denote the eigenspace of ad(a) on GC = G⊗C cor- ⊥ responding to the eigenvalue λ. Then a ∩P ⊂ GC (i)+GC (−i). It follows from [GC (λ1), GC (λ2)] ⊂GC (λ1 + λ2) that [a, x] ∈GC (i)+GC (−i) and [y, z] ∈GC (2i)+GC (−2i)+GC (0). Hence r =[a, x]=[y, z]=0. Since ad(a)2 = − id on A⊥ ∩P, x =0.

40 7.9 Proof of Theorem 7.7. “ref Let L denote the set of points x in the focal submanifold M + such that there + are orthonormal vectors ξ1,...,ξ4 in ν(M )x such that

4 dim E0(x, ξi) ≥ 3. i=1

It is proved in [FKM] that L is a proper subset of M +. Let O denote the set of points x˜ in M˜ + =(π ◦ φ)−1(M +) such that there are orthonormal vectors + ˜ξ1,...,˜ξ4 in ν(M˜ )x˜ such that

4 dim (E˜ 2 (˜x, ˜ξi)+E˜− 2 (˜x, ˜ξi) ≥ 3. π π 1

∗ ∗ ∗ ∗ Let x = φ(˜x), x = π(x ), ξ = dφ(˜ξi) and ξi = dπ(ξ ). Since both π and φ are i∗ ∗ ∗ i { }⊂ + ∗ { }⊂ + Riemannian submersion, ξ1 ,...,ξ4 ν(M )x and ξ1,...,ξ4 ν(M )x are orthonormal. Choose g ∈ H1([0, 1],G) such that g(1) = e and g ∗ x˜ = 0ˆ. We claimthat

4 E˜ 2 (˜x, ˜ξi)+E˜− 2 (˜x, ˜ξi) π π 1 4 πt = sin( )g−1X∗g | x∗X∗ is the horizontal lift of X ∈ E (x, ξ ) . 2 h h 0 i 1

In particular, this shows that O = φ−1(π−1(L)). By deﬁnition of O, M˜ + being homogeneous would imply O is either equal to M˜ + or an empty set. Since L is a proper non-empty subset of M +, O is a proper subset of M˜ +. Hence M˜ + is inhomogeneous. To prove our claim, we ﬁrst note that by Theorem 7.4 any vector in E˜ 2 (˜x, ˜ξ) is of the form π πt πt g sin X∗ + cos [ξ∗,X∗] g−1 2 h 2 h for some X ∈ E0(x, ξ), and any vector in E˜ −2 (˜x, ˜ξ) is of the form π πt πt g sin Y ∗ − cos [ξ∗,Y∗] g−1 2 h 2 h for some Y ∈ E0(x, ξ). Therefore a typical element in E˜ 2 (˜x, ˜ξ)+E˜− 2 (˜x, ˜ξ) is of π π the form πt πt g sin (X + Y )∗ + cos [ξ∗, (X − Y )∗ ] g−1. 2 h 2 h 41 Now suppose Z ∈∩(E˜ 2 (˜x, ˜ξi)+E˜− 2 (˜x, ˜ξi)). Then there exist X(i),Y(i) ∈ π π E0(x, ξi) for i =1,...,4 such that πt πt − Z = g sin( )X(i)∗ + cos( )[x∗ 1ξ∗,Y(i)∗ ] g−1, 2 h 2 i h ∗ ∗ ∗ ∗ ∗ where x X(i)h and x Y (i)h are the horizontal lifts of X(i) and Y (i) at x re- ∗ ∗ ∗−1 ∗−1 ∗ ∗−1 spectively. Since Z is continuous and x = g(0), Z(0) = x [x ξi ,Y(i)h]x . This implies that ∗−1 ∗ ∗ ∗−1 ∗ ∗ [x ξi ,Y(i)h]=[x ξj ,Y(j)h] for any 1 ≤ i, j ≤ 4. Next we claimthat Z(0) = 0. To see this, we may assume ∗ ∗ ∗ ∗ ∗ ∗ ∗ x = e. Since ξ1 ,...,ξ4 are in ν(M )e, they are perpendicular to E± 1 (x ,ξi ). ∗ ∗ ∗ ∗ 2 Now let a = ξ1 , x = Y (1) , y = ξ2 and z = Y (2) in Lemma 7.8. It then follows ∗ h∗ ∗ h that r = Z(0)=[ξ1 ,Y(1)h]=0and Y (1)h =0.So πt πt − πt Z = g sin( )X(i)∗ + cos( )[x∗ 1ξ∗,Y(i)∗ ] g−1 = g sin( )X(i)∗ g−1. 2 h 2 i h 2 h Moreover, ∗ ∗ Z(1) = X(1)h = ...= X(4)h. ∗ ∗ ∈ 4 So x Z(1) = Xh, the horizontal lif of some X 1 E0(x, ξi), and πt Z = sin gX∗g−1. 2 h This proves our claimand completesthe proof of (2). ∗ Now suppose M + = π−1(M) is the orbit of some subgroup H of G × G through x∗ = ea, where G = SO(2-). It is proved in [Te5] that M˜ + is then the orbit of the isometric action of

P (G, H)={g ∈ H1([0, 1],G) | (g(0),g(1)) ∈ H} through the constant path aˆ, a contradiction. This proves (1).

8. Open Problems

1. Suppose M is an equifocal submanifold of a simply connected, compact symmetric space N of codimension r ≥ 2 such that the action of the associated aﬃne Weyl group on ν(M)p is irreducible. Is M homogeneous, i.e., is M an orbit of some hyperpolar action on N? This is true for irreducible equifocal (i.e., isoparametric) hypersurfaces in Euclidean spaces if the codimension is at least three, see [Th2].

42 2. By Theorem1.6, the normalgeodesic γx of an equifocal hypersurface in a compact, semi-simple symmetric space N is closed and there are 2g focal points in γx. Is there a generalization of Münzner’s theoremfor isoparametric hypersurfaces in spheres saying that g =1, 2, 3, 4 or 6? Wu proved in [Wu2] that if N is a complex or quaternionic projective space then g =1, 2 or 3. 3. Can H∗(M,Z2) be computed explicitly in terms of the associated affine Weyl group and multiplicities? 4. If M is an irreducible, codimension r ≥ 2 isoparametric submanifold of an infinite dimensional Hilbert space, is M homogeneous? 5. Let y be defined as in Theorem1.8. Is there a finite group acting on the normal torus Tx of an equifocal submanifold in a simply connected symmetric space that is simply transitive on the set of chambers {y | y ∈ Tx ∩ M}? 6. Is there a similar theory for equifocal submanifolds in simply connected, non-compact, symmetric spaces? 7. Lie sphere geometry of Sn (see [Pi], [CC]) can be naturally extended to compact symmetric spaces. To be more precise, let N be a simply connected, compact symmetric space, and the unit tangent bundle T 1N be equipped with the natural contact structure. Given an immersed Legendre 1 1 submanifold f : X → T N and t ∈ R, let ft : X → T N denote the map ft(u)=d exptu(tu). We call λ a multiplicity m focal radius of X along u0

if fλ is singular at u0 and the dimension of the kernel of d(ft)u0 is m.A connected k-dimensional submanifold S of X is called a focal leaf of X if there exists a smooth function λ : S → R such that λ(u) is a multiplicity k focal radius of X along u and ker(d(fλ(u))u)=TSu for all u ∈ S. A Legen- dre submanifold X of T 1N is called Dupin if every focal leaf projects down to an intersection of geodesic spheres in N. A Dupin Legendre submanifold is called proper if the focal radii have constant multiplicities. Note that if M is an immersed submanifold of N then the unit normal bundle ν1(M) is an immersed Legendre submanifold of T 1N. Moreover, a hypersurface M is weakly equifocal if and only if ν1(M) is a proper, Dupin Legendre submanifold of T 1N. It follows fromresults of this paper that Dupin Legendre submanifolds of T 1N share many of the same properties as Dupin Legendre submanifolds (these are called Dupin Lie geometric hypersurfaces in [Pi]) of T 1Sn. So the following questions arise naturally: (a) What is the group of diffeomorphisms g : N → N such that the induced map dg : T 1N → T 1N maps Dupin Legendre submanifolds to Dupin Legendre submanifolds? When N = Sn, this group is the group of conformal diffeomorphisms (Möbius transforamtions). (b) What is the group of contact transformations of T 1N that map Dupin Legendre submanifolds to Dupin Legendre submanifolds? Or equivalently, what is the group of contact transformations of T 1N that maps Legendre spheres to Legendre spheres? Here a Legendre sphere in T 1N is defined to be either the unit normal bundle of a geodesic hypersphere or a fiber of the projection π : T 1N → N. When N = Sn, this

43 group is the group of Lie sphere transformations, which is isomorphic to O(n +1, 2)/Z2 and is generated by conformal transformations and the parallel translations ft (cf. [CC], [Pi]). (c) A compact submanifold M in a compact symmetric space N is called taut if for generic p ∈ N the energy functional E : P (N,p × M) → R is a perfect Morse function. Is a taut submanifold Dupin? Is tautness invariant under the transformations in questions (a) and (b)?

44 References

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Gudlaugur Thorbergsson Chuu-lian Terng Mathematisches Institut Department of Mathematics der Universität zu Köln Northeastern University Weyertal 86–90 Boston, MA 02115 D-50931 Köln, Germany email: [email protected] email:[email protected]