scholarly journals Riemannian manifolds with local symmetry

2014 ◽  
Vol 06 (02) ◽  
pp. 211-236 ◽  
Author(s):  
Wouter van Limbeek
Keyword(s):  
Riemannian Manifolds ◽  
Local Symmetry ◽  
Universal Cover ◽  
Connected Components ◽  
Curved Manifolds ◽  
Locally Homogeneous ◽  
Simply Connected ◽  
Positively Curved ◽  
Aspherical Manifolds

We give a classification of many closed Riemannian manifolds M whose universal cover [Formula: see text] possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that [Formula: see text] has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for non-positively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.

scholarly journals The even Clifford structure of the fourth Severi variety

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Maurizio Parton ◽  
Paolo Piccinni
Keyword(s):  
Vector Bundle ◽  
Riemannian Manifolds ◽  
Severi Variety ◽  
Simply Connected

AbstractTheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl

scholarly journals Open manifolds with non-homeomorphic positively curved souls

2019 ◽  
Vol 169 (2) ◽  
pp. 357-376 ◽  
Author(s):  
DAVID GONZÁLEZ-ÁLVARO ◽  
MARCUS ZIBROWIUS
Keyword(s):  
Sectional Curvature ◽  
Existence Results ◽  
Positive Answer ◽  
Open Manifold ◽  
Positive Sectional Curvature ◽  
Open Manifolds ◽  
Curved Manifolds ◽  
Eschenburg Spaces ◽  
Simply Connected ◽  
Positively Curved

AbstractWe extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class of spaces that includes all Eschenburg spaces.

scholarly journals MANY FINITE-DIMENSIONAL LIFTING BUNDLE GERBES ARE TORSION

2021 ◽  
pp. 1-16
Author(s):  
DAVID MICHAEL ROBERTS
Keyword(s):  
Analogous Result ◽  
Connected Components ◽  
Fibre Bundles ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Crossed Modules ◽  
Higher Gauge Theory ◽  
Bundle Gerbes ◽  
Simply Connected

Abstract Many bundle gerbes are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson [‘A note on bundle gerbes and infinite-dimensionality’, J. Aust. Math. Soc.90(1) (2011), 81–92] proved that gerbes on simply-connected manifolds, built from finite-dimensional fibre bundles with connected fibres, always have a torsion $DD$ -class. I prove an analogous result for a wide class of gerbes built from principal bundles, relaxing the requirements on the fundamental group of the base and the connected components of the fibre, allowing both to be nontrivial. This has consequences for possible models for basic gerbes, the classification of crossed modules of finite-dimensional Lie groups, the coefficient Lie-2-algebras for higher gauge theory on principal 2-bundles and finite-dimensional twists of topological K-theory.

POSITIVELY CURVED MANIFOLDS WITH LARGE CONJUGATE RADIUS

2013 ◽  
Vol 05 (03) ◽  
pp. 333-344 ◽  
Author(s):  
BENJAMIN SCHMIDT
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Spaces ◽  
Injectivity Radius ◽  
Curved Manifolds ◽  
Rank One ◽  
Sectional Curvatures ◽  
Simply Connected ◽  
Positively Curved

Let M denote a complete simply connected Riemannian manifold with all sectional curvatures ≥1. The purpose of this paper is to prove that when M has conjugate radius at least π/2, its injectivity radius and conjugate radius coincide. Metric characterizations of compact rank one symmetric spaces are given as applications.

On Locally Nonhomogeneous Pseudo–Riemannian Manifolds with Locally Homogeneous Levi–Civita Connections

2003 ◽  
Vol 14 (05) ◽  
pp. 559-572 ◽  
Author(s):  
Oldřich Kowalski ◽  
Zdeněk Vlášek ◽  
Barbara Opozda
Keyword(s):  
Riemannian Manifolds ◽  
Vector Fields ◽  
Killing Vector ◽  
Higher Dimensions ◽  
Killing Vector Fields ◽  
The Difference ◽  
Locally Homogeneous ◽  
Full Classification

In this paper we make the first steps to a classification of (pseudo-) Riemannian manifolds which are not locally homogeneous but their Levi–Civita connections are homogeneous. The full classification is given for dimension n = 2; in higher dimensions we prove some substantial partial results. In more generality, we are also interested in the difference between the dimension of the algebra of affine Killing vector fields and that of the algebra of metric Killing vector fields (without any homogeneity properties).

scholarly journals Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four

2006 ◽  
Vol 58 (2) ◽  
pp. 282-311 ◽  
Author(s):  
M. E. Fels ◽  
A. G. Renner
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Isotropy Subgroup ◽  
Algebraic Classification ◽  
Ricci Flat ◽  
Einstein Spaces ◽  
Simply Connected

AbstractA method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2, 2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to ℝ4. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.

scholarly journals Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

2013 ◽  
Vol 65 (4) ◽  
pp. 757-767 ◽  
Author(s):  
Philippe Delanoë ◽  
François Rouvière
Keyword(s):  
Symmetric Space ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Direct Proof ◽  
Locally Symmetric Spaces ◽  
Locally Symmetric Space ◽  
Hopf Fibrations ◽  
Rank One ◽  
Simply Connected ◽  
Positively Curved

AbstractThe squared distance curvature is a kind of two-point curvature the sign of which turned out to be crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, and an indirect one (via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

scholarly journals A BOUNDARY VERSION OF CARTAN–HADAMARD AND APPLICATIONS TO RIGIDITY

2009 ◽  
Vol 01 (04) ◽  
pp. 431-459 ◽  
Author(s):  
JEAN-FRANÇOIS LAFONT
Keyword(s):  
Riemannian Manifolds ◽  
Universal Cover ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Negatively Curved ◽  
Universal Covers ◽  
Sample Application ◽  
Positively Curved ◽  
Boundary Version ◽  
Hadamard Theorem

The classical Cartan–Hadamard theorem asserts that a closed Riemannian manifold Mn with non-positive sectional curvature has universal cover [Formula: see text] diffeomorphic to ℝn, and a by-product of the proof is that [Formula: see text] is homeomorphic to Sn-1. We prove analogues of these two results in the case where Mn has a non-empty totally geodesic boundary. More precisely, if [Formula: see text], [Formula: see text] are two negatively curved Riemannian manifolds with non-empty totally geodesic boundary, of dimension n ≠ 5, we show that [Formula: see text] is homeomorphic to [Formula: see text]. We show that if [Formula: see text] and [Formula: see text] are a pair of non-positively curved Riemannian manifolds with totally geodesic boundary (possibly empty), then the universal covers [Formula: see text] and [Formula: see text] are diffeomorphic if and only if the universal covers have the same number of boundary components. We also show that the number of boundary components of the universal cover is either 0, 2 or ∞. As a sample application, we show that simple, thick, negatively curved P-manifolds of dimension ≥ 6 are topologically rigid. We include some straightforward consequences of topological rigidity (diagram rigidity, weak co-Hopf property, and the Nielson problem).

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