Riemannian Geodesics - an Illustration from the Calculus of Variations

TL;DR

This paper explores the properties of geodesics in Riemannian geometry, emphasizing their derivation via calculus of variations and providing explicit examples on hypersurfaces.

Contribution

It introduces a method for deriving geodesics from calculus of variations and demonstrates explicit computations on Riemannian hypersurfaces.

Findings

Geodesics can be obtained using calculus of variations.

Explicit geodesic formulas are derived for hypersurfaces.

The characteristics of geodesics in Euclidean motion are clarified.

Abstract

This paper sheds light on the essential characteristics of geodesics, which frequently occur in considerations from motion in Euclidean space. Focus is mainly on a method of obtaining them from the calculus of variations, and an explicit geodesic computation for a Riemannian hypersurface.