The Taylor Expansion of the Exponential Map and Geometric Applications

TL;DR

This paper develops a third-order Taylor expansion of the exponential map for submanifolds in Euclidean spaces, introducing deviation concepts and analyzing contact directions with spheres for surfaces in various dimensions.

Contribution

It provides explicit calculations of deviation directions and contact directions using Taylor expansion, extending geometric analysis of submanifolds in Euclidean spaces.

Findings

Computed extreme lateral and frontal deviation directions for surfaces in R^3.

Determined high contact directions with hyperspheres for surfaces in R^4.

Identified asymptotic directions for surfaces in R^5.

Abstract

In this work we consider the Taylor expansion of the exponential map of a submanifold immersed in R^n up to order three, in order to introduce the concepts of lateral and frontal deviation. We compute the directions of extreme lateral and frontal deviation for surfaces in R^3. Also we compute, by using the Taylor expansion, the directions of high contact with hyperspheres of a surface immersed in R^4 and the asymptotic directions of a surface immersed in R^5.