Carlos Olmos

The geometry of homogeneous submanifolds of hyperbolic space

Abstract. 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 Bergers 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 normal holonomy group

We prove that the restricted normal holonomy group of a submanifold of a space of constant curvature is compact and that the nontrivial part of its representation on the normal space is the isotropy representation of a semisimple Riemannian symmetric space.

The skew-torsion holonomy theorem and naturally reductive spaces

Abstract We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul-Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split off, locally, a sphere or a compact Lie group with a bi-invariant metric. From this it follows easily how to obtain the full isometry group of a naturally reductive space. This generalizes known classification results of Onishchick, for normal homogeneous spaces with simple group of isometries, and Shankar, for homogeneous spaces of positive curvature. This also answers a question posed by J. Wolf and Wang–Ziller. Namely, to explain why the presentation group of an isotropy irreducible space, strongly or not, cannot be enlarged (except for spheres, or for compact simple Lie groups with a bi-invariant metric).

Orbits of rank one and parallel mean curvature

Let Mn ( n > 2 ) be a (extrinsic) homogeneous irreducible full submanifold of Euclidean space with rank(M) = k > 1 (i.e., it admits k > 1 locally defined, linearly independent parallel normal vector fields). We prove that M must be contained in a sphere. This result toghether with previous work of the author about homogeneous submanifolds of higher rank imply, in particular, the following theorem: A homogeneous irreducible submanifold of Euclidean space with parallel mean curvature vector is either minimal, or minimal in a sphere, or an orbit of the isotropy representation of a simple symmetric space.

Copolarity of isometric actions

We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity k representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Sainelson.

Variationally complete representations are polar

A recent result of C. Gorodski and G. Thorbergsson, involving classification, asserts that a variationally complete representation is polar. The aim of this paper is to give a conceptual and very short proof of this fact, which is the converse of a result of Conlon. The concept of a variationally complete action was introduced by R. Bott [B] in 1956. Two years later, Bott and Samelson [BS] proved that s-representations (i.e. isotropy representation of semisimple symmetric spaces) are variationally complete. This class of representations contains examples of what L. Conlon [C] called polar representations, or more generally hyperpolar actions. He proved that a hyperpolar action of a compact Lie group on a complete Riemannian manifold is variationally complete. Polar representations were classified by J. Dadok [D] who proved that they are orbit equivalent to s-representations (see also [EH]). Recently, C. Gorodski and G. Thorbergsson classified variationally complete representations of compact Lie groups [GT]. From this classification they obtained that a variationally complete representation is also orbit equivalent to an s-representation (from this they obtained, with different methods, Dadok’s list). So, they obtained the following equivalent theorem, some of whose history can be found in [TT, p. 196]. Theorem ([GT]). A variationally complete orthogonal representation of a compact Lie group is polar. The object of this short note is to give a direct and geometric proof of the above theorem. Recall that a compact connected Lie subgroup G of SO(n) acts polarly on R if there exists an affine subspace which meets orthogonally all G-orbits. This is equivalent to the fact that the tangent space Tv(G.v) of a principal orbit G.v contains the tangent spaces of all orbits through points in the normal space νv(G.v). The G-action is called variationally complete if any G-transversal Jacobi field (i.e. produced by variations of transversal geodesics) that is tangent to orbits at two points is the restriction of some Killing field on R induced by the action. Recall that a geodesic γ(t) in R is G-transversal if it is orthogonal to the G-orbit through Received by the editors November 9, 2000 and, in revised form, December 6, 2000. 1991 Mathematics Subject Classification. Primary 53C40; Secondary 53C35.

Curvature invariants, Killing vector fields, connections and cohomogeneity

A direct, bundle-theoretic method for defining and extending local isometries out of curvature data is developed. As a by-product, conceptual direct proofs of a classical result of Singer and a recent result of the authors are derived.

Compact homogeneous Riemannian manifolds with low coindex of symmetry

We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds whose co-index of symmetry is less or equal than three. We will also construct many examples which arise from the theory of polars and centrioles in Riemannian symmetric spaces of compact type.

Level sets of scalar Weyl invariants and cohomogeneity

We prove that the cohomogeneity of a Riemannian manifold coincides locally with the codimension of the foliation by regular level sets of the scalar Weyl invariants.

A geometric proof of the Karpelevich–Mostow theorem

In this paper we give a geometric proof of the Karpelevichs theorem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non compact type, has a totally geodesic orbit. In fact, this is equivalent to a well-known result of Mostow about existence of compatible Cartan decompositions.