Jacobi fields and geodesic spheres

1979 ◽  
Vol 82 (3-4) ◽  
pp. 233-240 ◽  
Author(s):  
L. Vanhecke ◽  
T. J. Willmore
Keyword(s):  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Holomorphic Sectional Curvature ◽  
Principal Curvatures ◽  
Jacobi Fields ◽  
Geodesic Spheres ◽  
Jacobi Vector ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds

SynopsisThis is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

scholarly journals On the Study of Jacobi Fields in Riemannian Manifolds

1970 ◽  
Vol 43 (4) ◽  
pp. 521-528
Author(s):  
Khondokar M Ahmed
Keyword(s):  
Field Equation ◽  
Key Words ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Jacobi Equation ◽  
Standard Form ◽  
Jacobi Field ◽  
Jacobi Fields ◽  
New Approach

A new approach of finding a Jacobi field equation with the relation between curvature and geodesics of a Riemanian manifold M has been derived. Using this derivation we have made an attempt to find a standard form of this equation involving sectional curvature K and other related objects. Key words: Riemanign curvature, Sectional curvature, Jacobi equation, Jacobifield.    doi: 10.3329/bjsir.v43i4.2242 Bangladesh J. Sci. Ind. Res. 43(4), 521-528, 2008

Geodesic spheres and Jacobi vector fields on Sasakian space forms

1987 ◽  
Vol 105 (1) ◽  
pp. 17-22 ◽  
Author(s):  
David E. Blair ◽  
Lieven Vanhecke
Keyword(s):  
Vector Fields ◽  
Space Form ◽  
Shape Operator ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Geodesic Spheres ◽  
Jacobi Vector

SynopsisUsing explicit equations for Jacobi vector fields on a Sasakian space form, we characterise such spaces by means of the shape operator of small geodesic spheres.

scholarly journals REILLY INEQUALITIES OF ELLIPTIC OPERATORS ON CLOSED SUBMANIFOLDS

2009 ◽  
Vol 80 (2) ◽  
pp. 335-346
Author(s):  
RUSHAN WANG
Keyword(s):  
Elliptic Operator ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Upper Bound ◽  
Vector Fields ◽  
Elliptic Operators ◽  
Position Vector ◽  
Space Forms ◽  
Nonzero Eigenvalue ◽  
Closed Hypersurfaces

AbstractUsing generalized position vector fields we obtain new upper bound estimates of the first nonzero eigenvalue of a kind of elliptic operator on closed submanifolds isometrically immersed in Riemannian manifolds of bounded sectional curvature. Applying these Reilly inequalities we improve a series of recent upper bound estimates of the first nonzero eigenvalue of the Lr operator on closed hypersurfaces in space forms.

scholarly journals CHARACTERIZATIONS OF LINEAR WEINGARTEN SPACELIKE HYPERSURFACES IN EINSTEIN SPACETIMES

2013 ◽  
Vol 55 (3) ◽  
pp. 567-579 ◽  
Author(s):  
HENRIQUE F. DE LIMA ◽  
JOSEÍLSON R. DE LIMA
Keyword(s):  
Maximum Principle ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Sufficient Conditions ◽  
Analytical Tool ◽  
Isoparametric Hypersurface ◽  
De Sitter ◽  
Principal Curvatures ◽  
Spacelike Hypersurfaces ◽  
Generalized Maximum Principle

AbstractOur purpose is to study the geometry of linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Einstein spacetime, whose sectional curvature is supposed to obey some standard restrictions. In this setting, by using as main analytical tool a generalized maximum principle for complete non-compact Riemannian manifolds, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Applications to the de Sitter space are given.

COMPLETE REAL HYPERSURFACES AND SPECIAL 𝕂-LINE BUNDLES IN 𝕂-HYPERBOLIC SPACES

2013 ◽  
Vol 24 (10) ◽  
pp. 1350082 ◽  
Author(s):  
NOBUHIRO INNAMI ◽  
YOE ITOKAWA ◽  
KATSUHIRO SHIOHAMA
Keyword(s):  
Sectional Curvature ◽  
Shape Operator ◽  
Hyperbolic Spaces ◽  
Real Hypersurfaces ◽  
Line Bundles ◽  
Holomorphic Sectional Curvature ◽  
Principal Curvatures ◽  
Totally Geodesic ◽  
Global Classification

Using the geometry of geodesics, we discuss the global aspects of complete real hypersurfaces in hyperbolic spaces of constant holomorphic sectional curvature [Formula: see text] over any division algebra 𝕂. Our assumption is that the shape operator and the curvature transformation with respect to the normal unit have the same eigenspaces. Note that we do not assume constancy of the principal curvatures. Under this assumption, we give a complete global classification of such hypersurfaces. Since the argument is purely geometric, we need not vary the argument for different base algebras. The foliations of [Formula: see text] with totally geodesic leaves called 𝕂-lines play an important role.

scholarly journals UNIT KILLING VECTOR FIELDS ON NEARLY KÄHLER MANIFOLDS

2005 ◽  
Vol 16 (03) ◽  
pp. 281-301 ◽  
Author(s):  
ANDREI MOROIANU ◽  
PAUL-ANDI NAGY ◽  
UWE SEMMELMANN
Keyword(s):  
Vector Fields ◽  
Unit Length ◽  
Killing Vector ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Finite Cover ◽  
Killing Vector Fields ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds ◽  
Compact Case

We study 6-dimensional nearly Kähler manifolds admitting a Killing vector field of unit length. In the compact case, it is shown that up to a finite cover there is only one geometry possible, that of the 3-symmetric space S3 × S3.

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