Geodesic Sandwich Theorem with an Application

TL;DR

This paper extends the classical sandwich theorem to geodesic convex functions on Riemannian manifolds, deriving new inequalities and properties of gradients in this geometric context.

Contribution

It proves a geodesic sandwich theorem, establishes a related inequality on manifolds with bounded curvature, and analyzes the orthogonality of gradients to tangent vectors.

Findings

Established a geodesic sandwich theorem for convex functions.

Derived an inequality in manifolds with bounded sectional curvature.

Showed the gradient of convex functions is orthogonal to tangent vectors at some points.

Abstract

The main goal of the paper is to prove the sandwich theorem for geodesic convex functions in a complete Riemannian manifold. Then by using this theorem we have proved an inequality in a manifold with bounded sectional curvature. Finally, we have shown that the gradient of a convex function is orthogonal to the tangent vector at some point of any geodesic.