On the Kobayashi metrics on Riemannian manifolds

TL;DR

This paper introduces a new Kobayashi-Royden metric analog for Riemannian manifolds and explores its fundamental properties, bridging complex and Riemannian geometry.

Contribution

It presents the first definition and initial analysis of the Kobayashi-Royden metric in the context of Riemannian manifolds, expanding the scope of complex geometric tools.

Findings

Defined the Kobayashi-Royden metric analog for Riemannian manifolds

Established basic properties and invariance features of the metric

Laid groundwork for further geometric and analytic studies

Abstract

We introduce an analog of the Kobayashi-Royden metric on a Riemannian manifold and study its basic properties.