Geodesics and Jacobi Fields

Keyword(s):  
1970 ◽  
Vol 43 (4) ◽  
pp. 521-528
Author(s):  
Khondokar M Ahmed

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


2020 ◽  
pp. 45-57
Author(s):  
Walter Dittrich ◽  
Martin Reuter

Author(s):  
L. Vanhecke ◽  
T. J. Willmore

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.


2010 ◽  
Vol 57 (6) ◽  
pp. 1344-1349 ◽  
Author(s):  
Yong Seung Cho ◽  
Soon-Tae Hong
Keyword(s):  

1997 ◽  
Vol 40 (2) ◽  
pp. 293-308 ◽  
Author(s):  
Toshiaki Adachi

A scalar multiple of the Kähler form of a Kähler manifold is called a Kähler magnetic field. We are focused on trajectories of charged particles under this action. As a variation of trajectories we define a magnetic Jacobi field. In this paper we discuss a comparison theorem on magnetic Jacobi fields, which corresponds to the Rauch's comparison theorem.


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