Compact Lie groups are ‘perfect’ entities in modern mathematics because: 

(a)

They are differentiable (and even analytical) manifolds of finite dimension and therefore can be analysed with the help of the most powerful tool of mathematics: ordinary and partial differential operators (equations);




(b)

Denoting by £(G) or L(G) = T e (G) the tangent space to G in unity e, we find that L(G) can be identified with the set of left-invariant vector fields on G (or with the set of appropriate differential operators of order one). Thus L(G) naturally becomes a Lie algebra: on L(G) there exists a structure [·, ·] which satisfies the axioms of Lie algebra (see below);




(c)

This rich algebraic structure, in turn, makes it possible to apply the powerful algebraic tools (theory of Lie algebras);




(d)

An abstract (finite dimensional) Lie algebra g defines (up to an isomorphism) a simply connected Lie group G such that £(G) = g. Therefore investigations of the Lie group G can be to large extent replaced by investigations of its Lie algebra g;




(e)

Compactness of a Lie group G makes it possible to use the theory of representations of compact groups developed by Weyl and Peter. This theory is extraordinarily beautiful and simple: it reduces to large extend to the spectral theory of compact operators;




(f)

And finally every complex, compact (connected) commutative Lie group is a torus. Moreover, as it was shown by E. Cartan, every compact Lie group contains a maximal torus T (any other such a torus is conjugated T 1 = gTg −1 ); and any point g 0 of the group G belongs to some maximal torus: G = U g gTg −1. This fundamental Cartan theorem makes it possible to apply to T (and thus to the whole of G) the theory of representations of abelian groups: every representation of the torus T decomposes into the finite (and orthogonal) sum of unit representations (over ℂ), the so called weights. Thus the knowledge of these weights is the fundamental element of the theory of representations of compact Lie groups.

Representations of Compact Lie Groups

Isoparametric foliations and their buildings

Manifolds all of whose geodesics are closed

This chapter discusses isoparametric hypersurfaces and their generalizations The chapter presents a survey of isoparametric hypersurfaces, discusses Dupin hypersurfaces, and describes isoparametric submanifolds in ambient spaces that are finite dimensional Euclidean spaces or infinite dimensional Hilbert spaces The taut submanifolds in Riemannian manifolds are described The classification of isoparametric hypersurfaces in spheres with three principal curvatures is related to the various characterizations of the standard embeddings of the projective planes The chapter discusses isoparametric hypersurfaces in spheres with four different principal curvatures, all of which are assumed to have the same multiplicity The Clifford examples of Ferus, Karcher, and Munzner together with the homogeneous hypersurfaces are all known examples of isoparametric hypersurfaces in spheres A difference between isoparametric and Dupin hypersurfaces is that although the parallel hypersurfaces of the Dupin ones are also Dupin, they do not foliate the ambient space as the isoparametric ones do The hypersurface of a sphere is isoparametric if and only if it is equifocal

Chapter 10 - A Survey on Isoparametric Hypersurfaces and Their Generalizations

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 topics 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 from the 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 reduces 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

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Submanifold Geometry in Symmetric Spaces

Basics of Submanifold Theory in Space Forms The fundamental equations for submanifolds of space forms Models of space forms Principal curvatures Totally geodesic submanifolds of space forms Reduction of the codimension Totally umbilical submanifolds of space forms Reducibility of submanifolds Submanifold Geometry of Orbits Isometric actions of Lie groups Existence of slices and principal orbits for isometric actions Polar actions and s-representations Equivariant maps Homogeneous submanifolds of Euclidean spaces Homogeneous submanifolds of hyperbolic spaces Second fundamental form of orbits Symmetric submanifolds Isoparametric hypersurfaces in space forms Algebraically constant second fundamental form The Normal Holonomy Theorem Normal holonomy The normal holonomy theorem Proof of the normal holonomy theorem Some geometric applications of the normal holonomy theorem Further remarks Isoparametric Submanifolds and Their Focal Manifolds Submersions and isoparametric maps Isoparametric submanifolds and Coxeter groups Geometric properties of submanifolds with constant principal curvatures Homogeneous isoparametric submanifolds Isoparametric rank Rank Rigidity of Submanifolds and Normal Holonomy of Orbits Submanifolds with curvature normals of constant length and rank of homogeneous submanifolds Normal holonomy of orbits Homogeneous Structures on Submanifolds Homogeneous structures and homogeneity Examples of homogeneous structures Isoparametric submanifolds of higher rank Normal Holonomy of Complex Submanifolds Polar-like properties of the foliation by holonomy tubes Shape operators with some constant eigenvalues in parallel manifolds The canonical foliation of a full holonomy tube Applications to complex submanifolds of Cn with nontransitive normal holonomy Applications to complex submanifolds of CPn with nontransitive normal holonomy The Berger-Simons Holonomy Theorem Holonomy systems The Simons holonomy theorem The Berger holonomy theorem The Skew-Torsion Holonomy Theorem Fixed point sets of isometries and homogeneous submanifolds Naturally reductive spaces Totally skew one-forms with values in a Lie algebra The derived 2-form with values in a Lie algebra The skew-torsion holonomy theorem Applications to naturally reductive spaces Submanifolds of Riemannian Manifolds Submanifolds and the fundamental equations Focal points and Jacobi fields Totally geodesic submanifolds Totally umbilical submanifolds and extrinsic spheres Symmetric submanifolds Submanifolds of Symmetric Spaces Totally geodesic submanifolds Totally umbilical submanifolds and extrinsic spheres Symmetric submanifolds Submanifolds with parallel second fundamental form Polar Actions on Symmetric Spaces of Compact Type Polar actions - rank one Polar actions - higher rank Hyperpolar actions - higher rank Cohomogeneity one actions - higher rank Hypersurfaces with constant principal curvatures Polar Actions on Symmetric Spaces of Noncompact Type Dynkin diagrams of symmetric spaces of noncompact type Parabolic subalgebras Polar actions without singular orbits Hyperpolar actions without singular orbits Polar actions on hyperbolic spaces Cohomogeneity one actions - higher rank Hypersurfaces with constant principal curvatures Appendix: Basic Material Exercises appear at the end of each chapter.

Submanifolds and Holonomy

We prove, in a purely geometric way, that there are no connected irreducible proper subgroups of SO(N,1). Moreover, a weakly irreducible subgroup of SO(N,1) must either act transitively irreducible subgroup of SO(N,1) must either act transitively on the hyperbolic space or on a horosphere. This has obvious consequences for Lorentzian holonomy and in particular explains clasification results of Marcel Berger's list (e.g. the fact that an irreducible Lorentzian locally symmetric space has constant curvatures). We also prove that a minimal homogeneous submanifold of hyperbolic space must be totally-geodesic.

The geometry of homogeneous submanifolds of hyperbolic space

We define and study isoparametric submanifolds of general ambient spaces and of arbitrary codimension. In particular we study their behaviour with respect to Rie- mannian submersions and their lift into a Hilbert space. These results are used to prove a Chevalley type restriction theorem which relates by restriction eigenfunctions of the Lapla- cian on a compact Riemannian manifold which contains an isoparametric submanifold with flat sections to eigenfunctions of the Laplacian of a section. A simple example of such an isoparametric foliation is given by the conjugacy classes of a compact Lie group and in that case the restriction theorem is a (well known) fundamental result in representation theory. As an application of the restriction theorem we show that isoparametric submanifolds with flat sections in compact symmetric spaces are level sets of eigenfunctions of the Laplacian and are hence related to representation theory. In addition we also get the following results. Isoparametric submanifolds in Hilbert space have globally flat normal bundle, and a general result about Riemannian submersions which says that focal distances do not change if a submanifold of the base is lifted to the total space.

Isoparametric submanifolds and a Chevalley-type restriction theorem

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Isoparametric submanifolds and their homogeneous structures

A submanifold has by definition constant principal curvatures if the eigenvalues of the shape operators Aξ are constant for any parallel normal field ξ along any curve. It is shown that a submanifold of Euclidean space has constant principal curvatures if and only if it is an isoparametric or a focal manifold of an isoparametric submanifold. Furthermore a "homogeneous slice theorem" is proved which says that the fibres of the projection of an isoparametric submanifold onto a full focal manifold are homogeneous isoparametric. To this end it is shown that any two points of an isoparametric submanifold can be connected by a piecewise differentiable curve whose pieces are tangent to one of the simultaneous eigenspaces Ei, i ∈ I, of the shape operators provided that the corresponding reflections generate the Weyl group of the isoparametric submanifold.

Carlos Olmos

Papers

Submanifolds and Holonomy

The geometry of homogeneous submanifolds of hyperbolic space

Isoparametric submanifolds and a Chevalley-type restriction theorem

Isoparametric submanifolds and their homogeneous structures

Submanifolds with constant principal curvatures and normal holonomy groups