Proofs of Conjectures about Singular Riemannian Foliations

A singular foliation on a complete Riemannian manifold M is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that if the distribution of normal spaces to the regular leaves is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section), which meets every leaf orthogonally. In addition the set of regular points is open and dense in each section. This result generalizes a result of Boualem and solves a problem inspired by a remark of Palais and Terng and a work of Szenthe about polar actions. We also study the singular holonomy of a singular Riemannian foliation with sections (s.r.f.s. for short) and in particular the tranverse orbit of the closure of each leaf. Furthermore we prove that the closures of the leaves of a s.r.f.s on M form a partition of M which is a singular Riemannian foliation. This result proves partially a conjecture of Molino.     

Polar actions on Hermitian and quaternion-Kähler symmetric spaces

We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classification of polar actions on Hermitian symmetric spaces. In the second part we prove that polar actions on Wolf spaces are quaternion-coisotropic and that isometric actions on these spaces admit an orbit of special type, analogous to the existence of a complex orbit for an isometric action on a compact homogeneous simply connected Kähler manifold.     

Polar and coisotropic actions on Kähler manifolds

The main result of the paper is that a polar action on a compact irreducible homogeneous Kähler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.

     

Singular riemannian foliations with sections,transnormal maps and basic forms

A singular riemannian foliation on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold Σ that meets every leaf of orthogonally and whose dimension is the codimension of the regular leaves of . We prove that the algebra of basic forms of M relative to is isomorphic to the algebra of those differential forms on Σ that are invariant under the generalized Weyl pseudogroup of Σ. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos. Marcos M. Alexandrino and Claudio Gorodski have been partially supported by FAPESP and CNPq.     

Polar actions on Damek-Ricci spaces

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular, we show that non-trivial polar actions exist on all Damek-Ricci spaces.     

Groups acting transitively on compact CR manifolds of hypersurface type

Let be a compact homogeneous manifold with acting effectively and with a -invariant CR structure of hypersurface type; then any maximal compact subgroup acts transitively on .

     

Tangential flatness and global rigidity of higher rank lattice actions

We establish the continuous tangential flatness for orientable
weakly Cartan actions of higher rank lattices. As a corollary, we obtain the global rigidity of Anosov Cartan actions.

     

Cohomogeneity one actions on noncompact symmetric spaces of rank one

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces , . For the quaternionic hyperbolic spaces , , we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Élie Cartan.

     

Free actions of -groups on products of lens spaces

Let be an odd prime number. We prove that if acts freely on a product of equidimensional lens spaces, then . This settles a special case of a conjecture due to C. Allday. We also find further restrictions on non-abelian -groups acting freely on a product of lens spaces. For actions inducing a trivial action on homology, we reach the following characterization: A -group can act freely on a product of lens spaces with a trivial action on homology if and only if and has the -extension property. The main technique is to study group extensions associated to free actions.

     

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.     

Polar actions on compact rank one symmetric spaces are taut

We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z ₂-coefficients.The second author was supported in part by FAPESP and CNPq.     

Some remarks on embedded hypersurfaces in hyperbolic space of constant curvature and spherical boundary

We consider embedded hypersurfacesM in hyperbolic space with compact boundaryC and somer ^th mean curvature functionH _r a positive constant. We investigate when symmetries ofC are symmetries ofM. We prove that if 0H _r1 andC is a sphere thenM is a part of an equidistant sphere. Forr=1 (H ₁ is the mean curvature) we obtain results whenC is convex.     

Torus actions on symplectic orbi-spaces

Which -dimensional orbi-spaces have effective symplectic - torus actions? As shown by Lerman and Tolman (1997) and Watson (1997), this question reduces to that of characterizing the finite subgroups of centralizers of tori in the real symplectic group . We resolve this question, and generalize our method to a calculation of the centralizers of all tori in .

     

Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry

In the study of the collapsed manifolds with bounded sectional curvature,the following two results provide basic tools:a(singular)fibration theorem by K.Fukaya[J.Differential Gcom.,1987.25(1):139156]and J.Cheeger,K.Fukaya,and M.Gromov[J.Airier.Math.Soc.,1992.5(2):327372],and the stability for isometric compact Lie group actions on manifolds by R.S.Palais[Bull.Amer.Math.Soc.,1961,67(4):362364]and K.Grove and H.Karcher[Math.Z..1973,132:1120].The main results in this paper(partially)generalize the two results to manifolds with local bounded Ricci covering geometry.     

Some examples of free actions on products of spheres

If G₁ and G₂ are finite groups with periodic Tate cohomology, then G₁×G₂ acts freely and smoothly on some product Sⁿ×Sⁿ.     

Log pairs on hypersurfaces of degreeN in {N{\text{ in }}\mathbb{P}^N

The objective of this paper is to study the birational structure of smooth hypersurfaces of degreeN in by examining properties of moving log pairs on them. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp 131–138, July, 2000.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.     

Classification of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III

We classify a certain class of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III. The structural analysis of type III factors and the canonical extension of endomorphisms introduced by Izumi are our main technical tools.     

Topological,affine and isometric actions on flat riemannian manifolds II

Explicit examples of finite subgroups of the group of homotopy classes of self-homotopy equivalences of some flat Riemannian manifolds which cannot be lifted to effective actions are given. It is also shown that no finite subgroups of the kernel of π₀(Homeo(M))→Out π₁(M) can be lifted back to Homeo(M), for a large class of flat manifolds M. Some results of an earlier paper by the authors are refined and related to recent work of R. Schoen and S.T. Yau.     

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

On quadratic hypersurfaces in $\mathbb {H}^2$, we find the explicit forms of tangential Cauchy‐Fueter operators and associated tangential Laplacians □_b. Then by using the Fourier transformation on the associated nilpotent Lie groups of step two, we construct the relative fundamental solutions to the tangential Laplacians and Szegö kernels on the nondegenerate quadratic hypersurfaces. It is different from the complex case that the quaternionic tangential structures on the nondegenerate quadratic hypersurfaces in $\mathbb {H}^2$ cannot be reduced to one standard model and the non‐homogeneous tangential Cauchy‐Fueter equations are solvable even in many convex cases.