Geodesic
Comparison Theory of Distance Spheres along Geodesics
Documenta mathematica, Tome 25 (2020), pp. 2241-2302.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Estimates for the principal curvature of distance spheres in Riemannian manifolds with sectional curvature bounded from below are well known. The same holds for the mean curvature of distance spheres in Riemannian manifolds with Ricci curvature bounded from below. \par In this article we present new estimates for convexity properties of the distance function of a point under different assumptions, for example for manifolds with lower bounds on the conjugate or on the focal radius in addition to these curvature conditions. \par The main idea is to introduce a new tensor field describing the differential of the exponential map and verifying a Riccati equation. This technique allows us to get new estimates for the volume form and for Jacobi fields in this context but also to gain new insights into well-known comparison theorems.
Classification : 53C21, 53B20, 53B21
Keywords: Riemannian geometry, comparison theorems, Ricci curvature, distance function, convexity, conjugate radius, focal radius, Jacobi field 
@article{DOCMA_2020__25__a9,
     author = {Brocks, Reinhard},
     title = {Comparison {Theory} of {Distance} {Spheres} along {Geodesics}},
     journal = {Documenta mathematica},
     pages = {2241--2302},
     publisher = {mathdoc},
     volume = {25},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a9/}
}
TY  - JOUR
AU  - Brocks, Reinhard
TI  - Comparison Theory of Distance Spheres along Geodesics
JO  - Documenta mathematica
PY  - 2020
SP  - 2241
EP  - 2302
VL  - 25
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a9/
LA  - en
ID  - DOCMA_2020__25__a9
ER  - 
%0 Journal Article
%A Brocks, Reinhard
%T Comparison Theory of Distance Spheres along Geodesics
%J Documenta mathematica
%D 2020
%P 2241-2302
%V 25
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a9/
%G en
%F DOCMA_2020__25__a9
Brocks, Reinhard. Comparison Theory of Distance Spheres along Geodesics. Documenta mathematica, Tome 25 (2020), pp. 2241-2302. http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a9/