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 SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM

2009 ◽  
Vol 51 (3) ◽  
pp. 669-680 ◽  
Author(s):  
G. PACELLI BESSA ◽  
M. SILVANA COSTA
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
Complete Riemannian Manifold ◽  
Hadamard Manifold ◽  
Fundamental Tone ◽  
Finite Topology ◽  
Bounded Below

AbstractBased on the ideas of Bessa, Jorge and Montenegro (Comm. Anal. Geom., vol. 15, no. 4, 2007, pp. 725–732) we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 is proper (compact if N is compact). In addition, if N is Hadamard, then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realised as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.

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 Higher-order geometric flow of hypersurfaces in a Riemannian manifold

2019 ◽  
Vol 30 (13) ◽  
pp. 1940005
Author(s):  
Zonglin Jia ◽  
Youde Wang
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Sectional Curvature ◽  
Present Situation ◽  
Higher Order ◽  
Complete Riemannian Manifold ◽  
High Order ◽  
Geometric Flows ◽  
Integer Part ◽  
Geometric Flow

In this paper, we consider the high-order geometric flows of a compact submanifolds [Formula: see text] in a complete Riemannian manifold [Formula: see text] with [Formula: see text], which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on [Formula: see text]. Precisely, we show that if [Formula: see text] is strictly larger than the integer part of [Formula: see text] and [Formula: see text] is an immersion for all [Formula: see text] and if [Formula: see text] is bounded by a constant which relies on the injectivity radius [Formula: see text] and sectional curvature [Formula: see text] of [Formula: see text], then [Formula: see text] must be [Formula: see text].

HARNACK-TYPE INEQUALITIES FOR THE POROUS MEDIUM EQUATION ON A MANIFOLD WITH NON-NEGATIVE RICCI CURVATURE

2012 ◽  
Vol 23 (04) ◽  
pp. 1250009 ◽  
Author(s):  
JEONGWOOK CHANG ◽  
JINHO LEE
Keyword(s):  
Porous Medium ◽  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Porous Medium Equation ◽  
Complete Riemannian Manifold ◽  
Gradient Estimates ◽  
Medium Equation

We derive Harnack-type inequalities for non-negative solutions of the porous medium equation on a complete Riemannian manifold with non-negative Ricci curvature. Along with gradient estimates, reparametrization of a geodesic and time rescaling of a solution are key tools to get the results.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

scholarly journals Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

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 A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds

1998 ◽  
Vol 151 ◽  
pp. 25-36 ◽  
Author(s):  
Kensho Takegoshi
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Growth Condition ◽  
Riemannian Manifolds ◽  
Volume Growth ◽  
Complete Riemannian Manifold ◽  
Volume Estimate ◽  
Complete Riemannian Manifolds ◽  
Generalized Maximum Principle

Abstract.A generalized maximum principle on a complete Riemannian manifold (M, g) is shown under a certain volume growth condition of (M, g) and its geometric applications are given.

Sign in / Sign up

Export Citation Format

Share Document