Some Borsuk-Ulam-type theorems for maps from Riemannian manifolds into manifolds

1989 ◽  
Vol 111 (1-2) ◽  
pp. 61-67
Author(s):  
Duan Hai-bao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Topological Properties ◽  
Locally Symmetric Spaces ◽  
Cut Points ◽  
Continuous Map ◽  
Ulam Theorem

SynopsisSuppose f: M →N is a continuous map from a Riemannian manifold (M, d) into a manifold N. The main result of this paper is to give some conditions under which f identifies a pair of cut points. This result leads to generalisations of the classical Borsuk-Ulam theorem. As a consequence some topological properties of locally symmetric spaces are discovered.

Holonomy of Sub-Riemannian Manifolds

1997 ◽  
Vol 08 (03) ◽  
pp. 317-344 ◽  
Author(s):  
Elisha Falbel ◽  
Claudio Gorodski ◽  
Michel Rumin
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Smooth Manifold ◽  
Symmetric Spaces ◽  
Type Theorem ◽  
Central Symmetry ◽  
Contact Type ◽  
Riemannian Symmetric Spaces

A sub-Riemannian manifold is a smooth manifold which carries a distribution equipped with a metric. We study the holonomy and the horizontal holonomy (i.e. holonomy spanned by loops everywhere tangent to the distribution) of sub-Riemannian manifolds of contact type relative to an adapted connection. In particular, we obtain an Ambrose–Singer type theorem for the horizontal holonomy and we classify the holonomy irreducible sub-Riemannian symmetric spaces (i.e. homogeneous sub-Riemannian manifolds admitting an involutive isometry whose restriction to the distribution is a central symmetry).

Recurrent conformal 2-forms on pseudo-Riemannian manifolds

2014 ◽  
Vol 11 (06) ◽  
pp. 1450056 ◽  
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Higher Dimensions ◽  
Topological Properties ◽  
Lorentzian Manifolds ◽  
Differential Structure ◽  
Arbitrary Signature

In this paper, we introduce the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature. Some theorems already proved for the same differential structure on a Riemannian manifold are proven to hold in this more general contest. Moreover other interesting results are pointed out; it is proven that if the associated covector is closed, then the Ricci tensor is Riemann compatible or equivalently, Weyl compatible: these notions were recently introduced and investigated by one of the present authors. Further some new results about the vanishing of some Weyl scalars on a pseudo-Riemannian manifold are given: it turns out that they are consequence of the generalized Derdziński–Shen theorem. Topological properties involving the vanishing of Pontryagin forms and recurrent conformal 2-forms are then stated. Finally, we study the properties of recurrent conformal 2-forms on Lorentzian manifolds (space-times). Previous theorems stated on a pseudo-Riemannian manifold of arbitrary signature are then interpreted in the light of the classification of space-times in four or in higher dimensions.

scholarly journals Further geometry of the mean curvature one-form and the normal plane field one-form on a foliated Riemannian manifold

1997 ◽  
Vol 62 (1) ◽  
pp. 46-63 ◽  
Author(s):  
Grant Cairns ◽  
Richard H. Escobales
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Final Section ◽  
General Context ◽  
Topological Properties ◽  
Normal Plane ◽  
Plane Field ◽  
The Mean ◽  
Foliated Riemannian Manifold

AbstractFor foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

scholarly journals On Asymptotic Behavior of Zeta Singularities for Compact Locally Symmetric Spaces

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Number Theory ◽  
Asymptotic Behavior ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Compact Riemann Surface ◽  
Zeta Functions ◽  
Analytic Number Theory ◽  
Locally Symmetric Spaces ◽  
Higher Dimensional ◽  
Negative Sectional Curvature

We obtain precise estimates for the number of singularities of Selberg’s and Ruelle’s zeta functions for compact, higher-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature. The methods applied in this research represent a generalization of the methods described in the case of a compact Riemann surface. In particular, this includes an application of the Phragmen-Lindelof theorem, the variation of the argument of certain zeta functions, as well as the use of some classical analytic number theory techniques.

scholarly journals Smooth and PL-rigidity problems on locally symmetric spaces

2016 ◽  
Vol 7 (4) ◽  
pp. 205
Author(s):  
Ramesh Kasilingam
Keyword(s):  
Symmetric Spaces ◽  
Riemannian Metrics ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Topological Properties ◽  
Open Problems ◽  
Locally Symmetric Spaces ◽  
Negatively Curved ◽  
Rigidity Problems

This is a survey on known results and open problems about smooth and PL-rigidity problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved metrics on a manifold.

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 THE STRUCTURE OF HARMONIC MORPHISMS WITH TOTALLY GEODESIC FIBRES

2004 ◽  
Vol 06 (03) ◽  
pp. 419-430 ◽  
Author(s):  
M. T. MUSTAFA
Keyword(s):  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Positive Curvature ◽  
Compact Manifolds ◽  
Harmonic Morphisms ◽  
Compact Type ◽  
Riemannian Submersions ◽  
Locally Symmetric Spaces ◽  
Totally Geodesic ◽  
Negatively Curved

The structure of local and global harmonic morphisms between Riemannian manifolds, with totally fibres, is investigated. It is shown that non-positive curvature of the domain obstructs the existence of global harmonic morphisms with totally geodesic fibres and the only such maps from compact Riemannian manifolds of non-positive curvature are, up to a homothety, totally geodesic Riemannian submersions. Similar results are obtained for local harmonic morphisms with totally geodesic fibres from open subsets of non-negatively curved compact and non-compact manifolds. During the course, we prove non-existence of submersive harmonic morphisms with totally geodesic fibres from some important domains, for instance from compact locally symmetric spaces of non-compact type and open subsets of symmetric spaces of compact type.

scholarly journals $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

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