A fundamental differential system of 3-dimensional Riemannian geometry

TL;DR

This paper explores a fundamental exterior differential system in 3D Riemannian geometry, revealing new global tensors, invariants, and a Weingarten type equation for surfaces in hyperbolic space, enhancing understanding of intrinsic geometric structures.

Contribution

It introduces new global tensors and invariants for 3D Riemannian manifolds and provides a novel proof of structural equations, deepening the theoretical framework.

Findings

Discovery of new global tensors and invariants

Derivation of a Weingarten type equation for hyperbolic surfaces

Enhanced understanding of intrinsic exterior differential systems

Abstract

We briefly recall a fundamental exterior differential system introduced by the author and then apply it to the case of three dimensions. Here we find new global tensors and intrinsic invariants of oriented Riemaniann 3-manifolds. The system leads to a remarkable Weingarten type equation for surfaces on hyperbolic 3-space. An independent proof for low dimensions of the structural equations gives new insight on the intrinsic exterior differential system.