scholarly journals THE SMOOTHNESS OF RIEMANNIAN SUBMERSIONS WITH NON-NEGATIVE SECTIONAL CURVATURE

2005 ◽  
Vol 07 (01) ◽  
pp. 137-144
Author(s):  
JIANGUO CAO ◽  
MEI-CHI SHAW
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Submersion ◽  
Fundamental Theory ◽  
Riemannian Submersions ◽  
Flat Strip ◽  
Negative Sectional Curvature

Let Mn be a complete, non-compact and C∞-smooth Riemannian manifold with non-negative sectional curvature. Suppose that [Formula: see text] is a soul of Mn given by the fundamental theory of Cheeger and Gromoll, and suppose that [Formula: see text] is a distance non-increasing retraction from the whole manifold to the soul (e.g. the retraction given by Sharafutdinov). Then we show that the retraction Ψ above must give rise to a C∞-smooth Riemannian submersion from Mn to the soul [Formula: see text]. Moreover, we derive a new flat strip theorem associated with the Cheeger–Gromoll convex exhaustion for the manifold above.

scholarly journals Examples and classification of Riemannian submersions satisfying a basic equality

2005 ◽  
Vol 72 (3) ◽  
pp. 391-402 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Unit Sphere ◽  
Isometric Immersion ◽  
Riemannian Submersion ◽  
Sharp Inequality ◽  
Riemannian Submersions ◽  
Equality Case ◽  
Totally Geodesic ◽  
Basic Equality

In an earlier article we obtain a sharp inequality for an arbitrary isometric immersion from a Riemannian manifold admitting a Riemannian submersion with totally geodesic fibres into a unit sphere. In this article we investigate the immersions which satisfy the equality case of the inequality. As a by-product, we discover a new characterisation of Cartan hypersurface in S4.

scholarly journals Hölder Compactification for Some Manifolds with Pinched Negative Curvature Near Infinity

2008 ◽  
Vol 60 (6) ◽  
pp. 1201-1218 ◽  
Author(s):  
Eric Bahuaud ◽  
Tracey Marsh
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Exponential Map ◽  
Noncompact Riemannian Manifold ◽  
Compact Submanifold ◽  
Curvature Pinching ◽  
Negative Sectional Curvature

AbstractWe consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K ⊂ M so that the outward normal exponential map off the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and show that pinched negative sectional curvature outside K implies M has a compactification with a well-defined Hölder structure independent of K. The Hölder constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.

scholarly journals Geodesic flows on manifolds of negative curvature with smooth horospheric foliations

1991 ◽  
Vol 11 (4) ◽  
pp. 653-686 ◽  
Author(s):  
Renato Feres
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Negative Sectional Curvature

AbstractWe improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.

scholarly journals On the growth of fundamental groups of nonpositive curvature manifolds

1996 ◽  
Vol 54 (3) ◽  
pp. 483-487 ◽  
Author(s):  
Yi-Hu Yang
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Sectional Curvature ◽  
Exponential Growth ◽  
Compact Riemannian Manifold ◽  
Nonpositive Curvature ◽  
Fundamental Groups ◽  
Positive Sectional Curvature ◽  
Classic Result ◽  
Negative Sectional Curvature

Milnor's classic result that the fundamental group of a compact Riemannian manifold of negative sectional curvature has exponential growth is generalised to the case of negative Ricci curvature and non-positive sectional curvature.

On a conjecture of Green

1997 ◽  
Vol 17 (1) ◽  
pp. 247-252
Author(s):  
CHENGBO YUE
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Geodesic Flow ◽  
Rigidity Theorem ◽  
Locally Symmetric Space ◽  
Closed Riemannian Manifold ◽  
Rank One ◽  
The Mean ◽  
Negative Sectional Curvature

Green [5] conjectured that if $M$ is a closed Riemannian manifold of negative sectional curvature such that the mean curvatures of the horospheres through each point depend only on the point, then $V$ is a locally symmetric space of rank one. He proved this in dimension two. In this paper we prove that under Green's assumption, $M$ must be asymptotically harmonic and that the geodesic flow on $M$ is $C^{\infty}$ conjugate to that of a locally symmetric space of rank one. Combining this with the recent rigidity theorem of Besson–Courtois–Gallot [1], it follows that Green's conjecture is true for all dimensions.

Scalar curvature of Lagrangian Riemannian submersions and their harmonicity

2017 ◽  
Vol 14 (12) ◽  
pp. 1750171 ◽  
Author(s):  
Şemsi Eken Meri̇ç ◽  
Erol Kiliç ◽  
Yasemi̇n Sağiroğlu
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Sufficient Conditions ◽  
Riemannian Submersion ◽  
Hermitian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Necessary And Sufficient

In this paper, we consider a Lagrangian Riemannian submersion from a Hermitian manifold to a Riemannian manifold and establish some basic inequalities to obtain relationships between the intrinsic and extrinsic invariants for such a submersion. Indeed, using these inequalities, we provide necessary and sufficient conditions for which a Lagrangian Riemannian submersion [Formula: see text] has totally geodesic or totally umbilical fibers. Moreover, we study the harmonicity of Lagrangian Riemannian submersions and obtain a characterization for such submersions to be harmonic.

scholarly journals Semi-invariant Submersions from Almost Hermitian Manifolds

2013 ◽  
Vol 56 (1) ◽  
pp. 173-183 ◽  
Author(s):  
Bayram Ṣahin
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Sufficient Conditions ◽  
Riemannian Submersion ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Definition Of ◽  
Necessary And Sufficient

AbstractWe introduce semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations that arise from the definition of a Riemannian submersion, and find necessary sufficient conditions for total manifold to be a locally product Riemannian manifold. We also find necessary and sufficient conditions for a semi-invariant submersion to be totally geodesic. Moreover, we obtain a classification for semiinvariant submersions with totally umbilical fibers and show that such submersions put some restrictions on total manifolds.

On the Geometry of Submersions

2021 ◽  
pp. 199-208
Author(s):  
Abdigappar Narmanov ◽  
Xurshid Sharipov
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Submersion ◽  
Complete Riemannian Manifold ◽  
Complete Manifold ◽  
Geodesic Foliation ◽  
Metric Function

Subject of present paper is the geometry of foliation defined by submersions on complete Riemannian manifold. It is proven foliation defined by Riemannian submersion on the complete manifold of zero sectional curvature is total geodesic foliation with isometric leaves. Also it is shown level surfaces of metric function are conformally equivalent.

scholarly journals On the projections of Laplacians under Riemannian submersions

2001 ◽  
Vol 25 (1) ◽  
pp. 33-42
Author(s):  
Huiling Le
Keyword(s):  
Brownian Motion ◽  
Riemannian Manifold ◽  
Differential Operator ◽  
Riemannian Submersion ◽  
Beltrami Operator ◽  
Riemannian Submersions

We give a condition on Riemannian submersions from a Riemannian manifoldMto a Riemannian manifoldNwhich will ensure that it induces a differential operator onNfrom the Laplace-Beltrami operator onM. Equivalently, this condition ensures that a Riemannian submersion maps Brownian motion to a diffusion.

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