Geometriae Dedicata最新文献

For a stratified group G, we construct a class of Lie groups endowed with a left-invariant distribution locally diffeomorphic to the flat distribution of G. Vice versa, we show that all Lie groups with a left-invariant distribution that is locally diffeomorphic to the flat distribution of G belong to the class we constructed, if the Lie algebra of G has finite Tanaka prolongation.

A closed Riemann surface S of genus (gge 2) is called cyclic p-gonal if it has an automorphism (rho ) of order p, where p is a prime, such that (S/langle rho rangle ) has genus 0. For (p=2), the surface is called hyperelliptic and (rho ) is an involution with (2g+2) fixed points. Classicaly, cyclic p-gonal surfaces can be characterized using Fuchsian groups. In this paper we establish a geometric characterization of cyclic p-gonal surfaces. Specifically, this is determined by collections of simple geodesic arcs on the surfaces and graphs associated to these arcs. In previous work, the author has given a geometric characterization of hyperelliptic surfaces in terms of simple, closed geodesics and graphs associated to these. The present work may be seen as an extension. Involutions, however, have properties that do not generalize to arbitrary automorphisms of order p. Hence, the number of vertices needed in the graphs used here is larger than that of the hyperelliptic case.

Three hundred years ago, Prince Rupert of Rhine showed that a unit cube has the property that one copy of it can be passed through a suitable hole in another copy. Under this situation, we say that a unit cube has the Rupert property. In the past years, there are many research studying about the Rupert property of many solids in (mathbb {R}^3). For higher dimensions, the n-dimensional cube and the regular n-simplex were studied to have the Rupert property. In this work, we focus on the Rupert property of some polyhedrons in n dimensions. In particular, we show that some particular n-dimensional simplices, generalized n-dimensional octahedrons and some related solids in (mathbb {R}^n) have the Rupert property using arbitrarily small rotations and translations.

We show that if a cusped Borel Anosov representation from a lattice (Gamma subset textsf{PGL}_2({{,mathrm{mathbb {R}},}})) to (textsf{PGL}_d({{,mathrm{mathbb {R}},}})) contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov.

We show that a length-minimizing disk inherits the upper curvature bound of the target. As a consequence we prove that harmonic discs and ruled discs inherit the upper curvature bound from the ambient space.

Hyperbolic fillings of metric spaces are a well-known tool for proving results on extending quasi-Moebius maps between boundaries of Gromov hyperbolic spaces to quasi-isometries between the spaces. For a hyperbolic filling Y of the boundary of a Gromov hyperbolic space X, one has a quasi-Moebius identification between the boundaries (partial Y) and (partial X). For CAT(-1) spaces, and more generally boundary continuous Gromov hyperbolic spaces, one can refine the quasi-Moebius structure on the boundary to a Moebius structure. It is then natural to ask whether there exists a functorial hyperbolic filling of the boundary by a boundary continuous Gromov hyperbolic space with an identification between boundaries which is not just quasi-Moebius, but in fact Moebius. The filling should be functorial in the sense that a Moebius homeomorphism between boundaries should induce an isometry between there fillings. We give a positive answer to this question for a large class of boundaries satisfying one crucial hypothesis, the antipodal property. This gives a class of compact spaces called quasi-metric antipodal spaces. For any such space Z, we give a functorial construction of a boundary continuous Gromov hyperbolic space (mathcal {M}(Z)) together with a Moebius identification of its boundary with Z. The space (mathcal {M}(Z)) is maximal amongst all fillings of Z. These spaces (mathcal {M}(Z)) give in fact all examples of a natural class of spaces called maximal Gromov hyperbolic spaces. We prove an equivalence of categories between quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces. This is part of a more general equivalence we prove between the larger categories of certain spaces called antipodal spaces and maximal Gromov product spaces. We prove that the injective hull of a Gromov product space X is isometric to the maximal Gromov product space (mathcal {M}(Z)), where Z is the boundary of X. We also show that a Gromov product space is injective if and only if it is maximal.

Abstract

We determine homogeneous semi-symmetric neutral manifolds of dimension 4. We also describe all the possible semi-symmetric curvature tensors on four-dimensional neutral vector spaces.

We view space-filling circle packings as subsets of the boundary of hyperbolic space subject to symmetry conditions based on a discrete group of isometries. This allows for the application of counting methods which admit rigorous upper and lower bounds on the Hausdorff dimension of the residual set of a generalized Apollonian circle packing. This dimension (which also coincides with a critical exponent) is strictly greater than that of the Apollonian packing.

Let M be an oriented three-dimensional Riemannian manifold of constant sectional curvature (k=0,1,-1) and let (SOleft( Mright) ) be its direct orthonormal frame bundle (direct refers to positive orientation), which may be thought of as the set of all positions of a small body in M. Given ( lambda in {mathbb {R}}), there is a three-dimensional distribution (mathcal { D}^{lambda }) on (SOleft( Mright) ) accounting for infinitesimal rototranslations of constant pitch (lambda ). When (lambda ne k^{2}), there is a canonical sub-Riemannian structure on ({mathcal {D}}^{lambda }). We present a geometric characterization of its geodesics, using a previous Lie theoretical description. For (k=0,-1), we compute the sub-Riemannian length spectrum of (left( SOleft( Mright) ,{mathcal {D}} ^{lambda }right) ) in terms of the complex length spectrum of M (given by the lengths and the holonomies of the periodic geodesics) when M has positive injectivity radius. In particular, for two complex length isospectral closed hyperbolic 3-manifolds (even if they are not isometric), the associated sub-Riemannian metrics on their direct orthonormal bundles are length isospectral.

Abstract

Let M be a smooth manifold and let (chi in Omega ^3(M)) be closed differential form with integral periods. We show the Lie 2-algebra (mathbb {L}(C_chi )) of sections of the (chi ) -twisted Courant algebroid (C_chi ) on M is quasi-isomorphic to the Lie 2-algebra of connection-preserving multiplicative vector fields on an (S^1) -bundle gerbe with connection (over M) whose 3-curvature is (chi ) .