A generalization of Clairaut's formula and its applications

TL;DR

This paper generalizes Clairaut's formula to higher dimensions and explores conditions under which certain curves on submanifolds are geodesics, providing new theoretical insights into differential geometry.

Contribution

It introduces a generalized Clairaut's formula applicable to higher-dimensional submanifolds and establishes conditions for specific curves to be geodesics.

Findings

Generalized Clairaut's formula for higher dimensions

Conditions for curves to be geodesics on submanifolds

Theoretical framework for distance-based curve analysis

Abstract

The main purpose of this article is to study conditions for a curve on a submanifold $M\subset\mathbb{R}^n$, constructed in a particular way involving the Euclidean distance to $M$, to be a geodesic. We also present the naturally arising generalization of Clairaut's formula needed for the generalization of the main result to higher dimensions.