Some Modifications of the Theorem of Beltrami

TL;DR

This paper presents three modifications of Beltrami's theorem for specific geometric transformations and introduces an infinitesimal version of the theorem related to metric deformations preserving geodesics.

Contribution

It extends Beltrami's theorem to new contexts involving geodesic-preserving transformations and infinitesimal deformations of metrics.

Findings

Modified theorems for geodesic circle, hypersphere, and minimal surface preservation.

Defined infinitesimal metric deformations preserving geodesics.

Established an infinitesimal version of Beltrami's theorem.

Abstract

The two main topics of this text are as follows: Firstly, three modifications of the theorem of Beltrami will be presented for diffeomorphisms between Riemannian manifolds and a space form which preserve the geodesic circles, the geodesic hyperspheres, or the minimal surfaces, respectively. Secondly, it is defined what it means for an infinitesimal deformation of a metric to preserve the geodesics up to first order, and a corresponding infinitesimal version of Beltrami's theorem is given.