On the classical geometry of embedded manifolds in terms of Nambu brackets

TL;DR

This paper demonstrates that key differential geometric properties of embedded Riemannian manifolds can be expressed using Nambu brackets, providing algebraic formulations for classical geometric concepts.

Contribution

It introduces algebraic expressions for fundamental geometric quantities of embedded manifolds using Nambu brackets, linking differential geometry with multi-linear algebra.

Findings

Derived algebraic formulas for Weingarten's formula

Expressed Ricci curvature in algebraic terms

Reformulated Codazzi-Mainardi equations algebraically

Abstract

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of a multi-linear algebraic structure on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.