Homogeneous and inhomogeneous manifolds

All metaLindelöf, and most countably paracompact, homogeneous manifolds are Hausdorff. Metacompact manifolds are never rigid. Every countable group can be realized as the group of autohomeomorphisms of a Lindelöf manifold. There is a rigid foliation of the plane.

     

On the Singular Set of Free Interface in an Optimal Partition Problem

We study the singular set of free interface in an optimal partition problem for the Dirichlet eigenvalues. We prove that its upper (n − 2) -dimensional Minkowski content, and consequently its (n − 2) -dimensional Hausdorff measure, are locally finite. We also show that the singular set is countably (n − 2) -rectifiable; namely, it can be covered by countably many C¹ -manifolds of dimension (n − 2) , up to a set of (n − 2) -dimensional Hausdorff measure zero. Our results hold for optimal partitions on Riemannian manifolds and harmonic maps into homogeneous trees as well. © 2019 Wiley Periodicals, Inc.     

Invariants for proper metric spaces and open Riemannian manifolds

We introduce uniform structures of proper metric spaces and open Riemannian manifolds, characterize their (arc) components, present new invariants like e.g. Lipschitz and Gromov–Hausdorff cohomology, specialize to uniform triangulations of manifolds and prove that the presence of a spectral gap above zero is a bounded homotopy invariant.     

Lipschitz-volume rigidity on limit spaces with Ricci curvature bounded from below

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov–Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.     

Bilipschitz homogeneous Jordan curves

We characterize bilipschitz homogeneous Jordan curves by utilizing quasihomogeneous parameterizations. We verify that rectifiable bilipschitz homogeneous Jordan curves satisfy a chordarc condition. We exhibit numerous examples including a bilipschitz homogeneous quasicircle which has lower Hausdorff density zero. We examine homeomorphisms between Jordan curves.

     

Odd symplectic flag manifolds

We define the odd symplectic Grassmannians and flag manifolds, which are smooth projective varieties equipped with an action of the odd symplectic group, analogous to the usual symplectic Grassmannians and flag manifolds. Contrary to the latter, which are the flag manifolds of the symplectic group, the varieties we introduce are not homogeneous. We argue nevertheless that in many respects the odd symplectic Grassmannians and flag manifolds behave like homogeneous varieties; in support of this claim, we compute the automorphism group of the odd symplectic Grassmannians and we prove a Borel-Weil-type theorem for the odd symplectic group.     

On twisted contact groupoids and on integration of twisted Jacobi manifolds

We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic groupoids with twisted contact ones. Some examples for each structure are presented.     

Closed geodesics in homology classes on hyperbolic manifolds with cusps

On a class of hyperbolic manifolds with infinite volume, we give an asymptotic estimate for the number of closed geodesics in a given homology class. We show that, in certain cases, the existence of parabolic transformations in the fundamental group Γ of these manifolds has an effect on this estimate. This happens when the Hausdorff dimension of the limit set of Γ is less than 3/2. The geometrical meaning of this critical value remains to be understood.     

A Blow-up Theorem for regular hypersurfaces on nilpotent groups

 We obtain an intrinsic Blow-up Theorem for regular hypersurfaces on graded nilpotent groups. This procedure allows us to represent explicitly the Riemannian surface measure in terms of the spherical Hausdorff measure with respect to an intrinsic distance of the group, namely homogeneous distance. We apply this result to get a version of the Riemannian coarea forumula on sub-Riemannian groups, that can be expressed in terms of arbitrary homogeneous distances. We introduce the natural class of horizontal isometries in sub-Riemannian groups, giving examples of rotational invariant homogeneous distances and rotational groups, where the coarea formula takes a simpler form. By means of the same Blow-up Theorem we obtain an optimal estimate for the Hausdorff dimension of the characteristic set relative to C ^1,1 hypersurfaces in 2-step groups and we prove that it has finite Q–2 Hausdorff measure, where Q is the homogeneous dimension of the group. Received: 6 February 2002 Mathematics Subject Classification (2000): 28A75 (22E25)     

Three dimensional homogeneous Finsler manifolds

In this paper we study three dimensional homogeneous Finsler manifolds. We first obtain a complete list of the three‐dimensional homogeneous manifolds which admit invariant Finsler metrics. Then we consider invariant Randers metrics and present the classification of three dimensional homogeneous Randers spaces under isometrics.     

Detecting Hilbert manifolds among isometrically homogeneous metric spaces

We detect Hilbert manifolds among isometrically homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H, where G is a metrizable topological group and H is a closed balanced subgroup of G.     

Three- and Four-Dimensional Einstein-like Manifolds and Homogeneity

The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three-dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podestá and A. Spiro, and illustrating a striking contrast with the three-dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).     

A version of Turán’s problem for positive definite functions of several variables

We present a systematic approach to solving the problem of affine homogeneity of real hypersurfaces in the three-dimensional complex space. This question is an important part of the general problem of holomorphic classification of homogeneous real hypersurfaces in three-dimensional complex spaces. In contrast to the two-dimensional case, the whole problem (just as its affine part) has not yet been fully studied, although there exist a large number of examples of homogeneous manifolds. We study only the class of tubular type surfaces, which is defined by conditions imposed on the 2-jet of their canonical equations and generalizes the class of tube manifolds. We discuss the procedure of describing all matrix Lie algebras corresponding to the homogeneous manifolds under consideration. In the class that we study, we distinguish four cases depending on the third-order Taylor coefficients of the canonical equations; in three of these cases, the Lie algebras and the corresponding affine homogeneous surfaces are completely described. The key point of the proposed approach is the solution of a large system of quadratic equations that corresponds to each of the homogeneous surfaces.     

两个非紧致齐性复解析流形

本文给出两个非紧致的齐性复解析流形．用它的齐性子流形构造出两个例外对称典型域的扩充空间，并由复流形上的运动群在超圆上的限制得到了两个例外对称典型域的仿射自同胚群，它们是闭的辛子群．     

Foliations, topology and geometry of 3-manifolds: R-covered foliations and transverse pseudo-Anosov flows

We analyse the topological and geometrical behavior of foliations on 3-manifolds. We consider the transverse structure of an R-covered foliation in a 3-manifold, where R-covered means that in the universal cover the leaf space of the foliation is Hausdorff. If the manifold is aspherical we prove that either there is an incompressible torus in the manifold; or there is a transverse pseudo-Anosov flow. It follows that manifolds with R-covered foliations satisfy the weak hyperbolization conjecture.     

Wedge removability of metrically thin sets and application to the CR-meromorphic extension

We give a wedge removability theorem for metrically thin sets of two codimensional Hausdorff null measure. Following [22], this removability theorem combined with the wedge removability theorem of [21] for closed subsets of two codimensional manifolds, gives a CR-meromorphic extension theorem in the greater codimensional case. Received: 28 August 1999; in final form: 10 April 2000 / Published online: 17 May 2001     

Properly discontinuous actions on Hilbert manifolds

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature.