Finiteness results for equifocal hypersurfaces in compact symmetric spaces

We prove that there are only finitely many diffeomorphism types of curvature-adapted equifocal hypersurfaces in a simply connected compact symmetric space. Moreover, if the symmetric space is of rank one, the result can be strengthened by dropping the condition curvature-adapted.     

Skew-product decompositions of Brownian motions on manifolds: A probabilistic aspect of the Lichnerowicz-Szabo theorem

We show that the only compact simply connected manifolds for which the radial part of Brownian motion enjoys the Markov property are compact two points homogeneous spaces, i.e. rank one symmetric spaces.     

On the geometry of spaces of oriented geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kähler structure on L(M).     

On harmonic and asymptotically harmonic homogeneous spaces

We classify noncompact homogeneous spaces which are Einstein and asymptotically harmonic. This completes the classification of Riemannian harmonic spaces in the homogeneous case: Any simply connected homogeneous harmonic space is flat, or rank-one symmetric, or a nonsymmetric Damek–Ricci space. Independently, Y. Nikolayevsky has obtained the latter classification under the additional assumption of nonpositive sectional curvatures [N2]. Supported in part by DFG priority program “Global Differential Geometry” (SPP 1154). Received: September 2004; Revision: June 2005; Accepted: September 2005     

Cohomogeneity one manifolds with a small family of invariant metrics

In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible symmetric space. As a corollary such manifolds admit an invariant metric with non-negative sectional curvature.     

On the volume of the intersection of two geodesic balls

The first author and D. Kunszenti-Kovács (2010) [1] proved that if the volume of the intersection of three geodesic balls of a complete connected Riemannian manifold depends only on the center-center distances and the radii of the balls, then the manifold is one of the simply connected spaces of constant curvature. In this paper, we study the geometrical consequences of the analogous condition for pairs of geodesic balls. We show that in a complete, connected and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. It is also shown that if in a Riemannian manifold the volume of the intersection of two small geodesic balls of equal radii depends only on the distance between the centers and the common value of the radii, then the space is Einstein, and if we assume in addition that the space is symmetric, then it must be Osserman and hence two-point homogeneous.     

Einstein homogeneous bisymmetric fibrations

We consider a homogeneous fibration G/L → G/K, with symmetric fiber and base, where G is a compact connected semisimple Lie group and L has maximal rank in G. We suppose the base space G/K is isotropy irreducible and the fiber K/L is simply connected. We investigate the existence of G-invariant Einstein metrics on G/L such that the natural projection onto G/K is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber K/L is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the G-invariant Einstein metrics with totally geodesic fibers for the first type. For the second type, we classify all these metrics when G is an exceptional Lie group. If G is a classical Lie group we classify all such metrics which are the orthogonal sum of the normal metrics on the fiber and on the base or such that the restriction to the fiber is also Einstein.     

A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces

A submanifold in a symmetric space is called equifocal if it has a globally flat abelian normal bundle and its focal data is invariant under normal parallel transportation. This is a generalization of the notion of isoparametric submanifolds in Euclidean spaces. To each equifocal submanifold, we can associate a Coxeter group, which is determined by the focal data at one point. In this paper we prove that an equifocal submanifold in a simply connected compact symmetric space is a non-trivial product of two such submanifolds if and only if its associated Coxeter group is decomposable. As a consequence, we get a similar splitting result for hyperpolar group actions on compact symmetric spaces. These results are an application of a splitting theorem for isoparametric submanifolds in Hilbert spaces by Heintze and Liu.

     

A Characterization of Homogeneous Spaces with Positive Hyperbolic Rank

Here we present a necessary and sufficient condition for a negatively curved homogeneous space to be a rank one symmetric space. In particular, we show rigidity if such a space has positive hyperbolic rank greater than equal to that of its 'Abelian direction'. The notion of hyperbolic-rank extends the notion of rank in negatively curved spaces. The central theorem is an analogue of a result by U. Hamenstädt for compact negatively curved manifolds. We also provide an example of a nonsymmetric hyperbolic rank two homogeneous space, demonstrating the sharpness of the theorem.     

Enlarging totally geodesic submanifolds of symmetric spaces to minimal submanifolds of one dimension higher

We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces.

     

A homological criterion for reducibility of analytic spaces,with application to characterizing the theta divisor of a product of two general principally polarized abelian varieties

A closed subset of pure codimension one in an analytic space, consisting entirely of local normal crossings double points, is called an ordinary rank two double locus. We give a topologically computable upper bound on the number of connected components of an ordinary rank two double locus in a given space. This leads to criteria for global reducibility of spaces. The first is that a simply connected space with a non empty ordinary rank two double locus is always reducible. A finer criterion implies that a principally polarized abelian variety A is isomorphic to a product of two positive dimensional principally polarized abelian varieties, each with smooth theta divisor, if and only if the theta divisor of A contains a non empty ordinary rank two double locus. Analogous reducibility results apply to certain complete intersection varieties, and to divisors on such varieties.     

Symmetry in the geometry of metric contact pairs

We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold with decomposable ? is a Calabi‐Eckmann manifold or the Riemannian product of a sphere and . We show that a complete, simply connected, normal metric contact pair manifold with decomposable ?, such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, is the product of globally ?‐symmetric spaces or the product of a globally ?‐symmetric space and . Moreover in the first case the manifold fibers over a locally symmetric space endowed with a symplectic pair.     

A note on the uniqueness of the canonical connection of a naturally reductive space

We extend the result in Olmos and Reggiani (J. Reine Angew. Math. 664:29–53, 2012) to the non-compact case. Namely, we prove that the canonical connection on a simply connected and irreducible naturally reductive space is unique, provided the space is not a sphere, a compact Lie group with a bi-invariant metric or its symmetric dual. In particular, the canonical connection is unique for the hyperbolic space when the dimension is different from three. We also prove that the canonical connection on the sphere is unique for the symmetric presentation. Finally, we compute the full isometry group (connected component) of a compact and locally irreducible naturally reductive space.     

On the Cohomology of Hermitian Symmetric Spaces

A formula for the value of a power of a two-dimensional cohomology class of a simply connected compact Hermitian symmetric space M at an arbitrary homology class of this space is derived.     

On secant varieties of compact Hermitian symmetric spaces

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three—with one exception, the secant variety of the 21-dimensional spinor variety in P⁶³ where we show that the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.     

On the simple connectedness of hyperplane complements in dual polar spaces, II

Suppose Δ is a dual polar space of rank n and H is a hyperplane of Δ. Cardinali, De Bruyn and Pasini have already shown that if n≥4 and the line size is greater than or equal to 4 then the hyperplane complement Δ−H is simply connected. This paper is a follow-up, where we investigate the remaining cases. We prove that the hyperplane complements are simply connected in all cases except for three specific types of hyperplane occurring in the smallest case, when the rank and the line size are both 3.     

Point leaf maximal singular Riemannian foliations in positive curvature

We generalize the notion of fixed point homogeneous isometric group actions to the context of singular Riemannian foliations. We find that in some cases, positively curved manifolds admitting these so-called point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces. In all cases, manifolds admitting such foliations are cohomology CROSSes or finite quotients of them. Among non-simply connected manifolds, we find examples of such foliations which are non-homogeneous.     

The higher rank rigidity theorem for manifolds with no focal points

We say that a Riemannian manifold M has rank M ≥ k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns–Spatzier, later generalized by Eberlein–Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank k ≥ 2 (the “higher rank” assumption) whose isometry group Γ satisfies the condition that the Γ-recurrent vectors are dense in SM is a symmetric space of noncompact type. This includes, for example, higher rank M which admit a finite volume quotient. We adapt the method of Ballmann and Eberlein–Heber to prove a generalization of this theorem where the manifold M is assumed only to have no focal points. We then use this theorem to generalize to no focal points a result of Ballmann–Eberlein stating that for compact manifolds of nonpositive curvature, rank is an invariant of the fundamental group.