Skew symmetric logarithms and geodesics on $O_n(\RR)$

TL;DR

This paper explores the geometric properties of the exponential map and geodesics on the special orthogonal group, linking skew symmetric matrices with Riemannian structures on orthogonal matrices.

Contribution

It provides new insights into the differential geometry of orthogonal groups using skew symmetric logarithms and geodesic analysis.

Findings

Characterization of exponential map properties on $O_n(\RR)$

Analysis of geodesic structures under Frobenius metric

Connections between skew symmetric matrices and Riemannian geometry

Abstract

We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric.