The Weingarten map and curvatures of SL(n, R)

TL;DR

This paper provides elementary derivations of the Gauss map, Weingarten map, and curvatures at the identity of the special linear group SL(n, R), filling a gap in explicit computations.

Contribution

It offers the first explicit elementary calculations of geometric maps and curvatures for SL(n, R), which were previously not documented.

Findings

Explicit formulas for the Gauss map and Weingarten map at identity.

Derivation of curvatures for SL(n, R) matrices.

Clarification of geometric properties of SL(n, R).

Abstract

In this article, we shall derive by elementary calculations the Gauss map, spherical image, Weingarten map and the curvatures at identity of the special linear group, that is, the matrices of determinant 1. We could not find any reference where this simple, nice computation has been explicitly put down.