Symmetric spaces rolling on flat spaces

TL;DR

This paper clarifies different notions of rolling Riemannian manifolds and focuses on the specific case of symmetric spaces rolling on flat spaces, providing theoretical insights and illustrative examples.

Contribution

It distinguishes between intrinsic and extrinsic rolling notions and analyzes the specific case of symmetric spaces rolling on flat spaces with detailed explanations and examples.

Findings

Clarified the difference between intrinsic and extrinsic rolling.

Analyzed the case of symmetric spaces rolling on flat spaces.

Provided illustrative examples of the theoretical concepts.

Abstract

The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers with interest in applications. Secondly, we concentrate on rolling an important class of Riemannian manifolds. In the first part of the paper, the relation between intrinsic and extrinsic rollings is explained in detail, while in the second part we address rollings of symmetric spaces on flat spaces and complement the theoretical results with illustrative examples.