A counterpart of Landau -- Hadamard type inequality for manifold-valued mappings

TL;DR

This paper extends classical Landau-Hadamard inequalities to mappings into Riemannian manifolds, establishing conditions where bounded covariant derivatives imply bounded tangent vectors, with an example on the 2D unit sphere.

Contribution

It introduces a Landau-Hadamard type inequality for manifold-valued mappings, linking covariant derivative boundedness to tangent vector boundedness using convex functions.

Findings

Established a Landau-Hadamard inequality for manifold-valued mappings.

Derived conditions for boundedness of tangent vectors from covariant derivatives.

Provided an example on the 2D unit sphere demonstrating the inequality.

Abstract

We obtain a Landau -- Hadamard type inequality for mappings defined on the whole real axis and taking values in Riemannian manifolds. In terms of an auxiliary convex function, we find conditions under which the boundedness of covariant derivative along the curve under consideration ensures the boundedness of the corresponding tangent vector field. As an example we obtain a Landau -- Hadamard type inequality for curves on 2D unite sphere.