Convex domains of Finsler and Riemannian manifolds

Details Convex domains of Finsler and Riemannian manifolds Convex domains of Finsler and Riemannian manifolds

2009-11-02

arxiv.org/abs/0911.0360v2

Calc. Var. Partial Differential Equations, 40 (2011) 335-356

Mathematics

Differential Geometry

22 pages, AMSLaTex. Typos corrected, references updated

Scientific paper

10.1007/s00526-010-0343-1

A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop's one in his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface.

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