[Paper Review] On the classical geometry of embedded manifolds in terms of Nambu brackets

Joakim Arnlind, Jens Hoppe+1 more|arXiv (Cornell University)|Mar 31, 2010

Advanced Topics in Algebra2 references3 citations

TL;DR

This paper demonstrates that key geometric invariants of embedded Riemannian manifolds—such as the Ricci curvature, Weingarten's formula, and the Codazzi-Mainardi equations—can be expressed entirely in terms of Nambu brackets, a multi-linear algebraic structure on smooth functions. The central contribution is a derivation of the Codazzi-Mainardi equations using Poisson and Nambu bracket identities, showing that these equations reduce to algebraic identities in the Poisson algebra framework for surfaces in ℝ³.

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.

Motivation & Objective

• To reformulate classical differential geometry of embedded Riemannian manifolds in terms of algebraic structures on smooth functions.
• To demonstrate that geometric quantities like the Ricci curvature and Weingarten's formula can be expressed algebraically via Nambu brackets.
• To provide an algebraic derivation of the Codazzi-Mainardi equations using Poisson and Nambu bracket identities.
• To establish a framework for matrix regularization in Membrane Theory by expressing geometric invariants in terms of commutators and Poisson brackets.
• To show that for surfaces in ℝ³, the Codazzi-Mainardi equations are identities in Poisson algebras when the normal vector is defined via Nambu brackets.

Proposed method

• Utilizes Nambu brackets of order $ n $, defined as $ \{f_1, \dots, f_n\} = \frac{1}{\sqrt{g}} \varepsilon^{a_1\cdots a_n} \partial_{a_1}f_1 \cdots \partial_{a_n}f_n $, to encode geometric data.
• Applies the Jacobi identity and Leibniz rule for Nambu brackets to derive identities involving the normal vector and embedding coordinates.
• Constructs orthonormal normal frames using the Nambu bracket structure and the induced metric.
• Re-expresses the Weingarten formula and Ricci curvature using Nambu brackets and their derivatives.
• Derives the Codazzi-Mainardi equations as algebraic identities by manipulating Nambu bracket expressions and using the identity $ \sum_i \{x^i, n^i\} = 0 $.
• Demonstrates that for surfaces in ℝ³, the Codazzi-Mainardi equations reduce to a vanishing Nambu bracket expression: $ \sum_{j,k=1}^3 \big{\{} \gamma^{-2}(\mathcal{P}^2)^{ik}, n^k \big{\}} = 0 $.

Experimental results

Research questions

• RQ1 Can the Ricci curvature of an embedded Riemannian manifold be expressed purely in terms of Nambu brackets of smooth functions?
• RQ2 To what extent can the Weingarten formula and the Codazzi-Mainardi equations be reformulated as algebraic identities in the Poisson-Nambu algebra of smooth functions?
• RQ3 Is the Codazzi-Mainardi equation for surfaces in ℝ³ an identity in any Poisson algebra when the normal vector is defined via $ \frac{1}{2\gamma} \varepsilon_{ijk} \{x^j, x^k\} \partial_i $?
• RQ4 Can the classical differential geometry of hypersurfaces in ℝ^{n+1} be systematically reconstructed from Nambu bracket structures on the function algebra?
• RQ5 How do the geometric quantities of embedded manifolds behave under matrix regularization, and can they be consistently approximated using commutators and Poisson brackets?

Key findings

• The Gaussian curvature of a surface in ℝ³ is given algebraically by $ K = -\frac{1}{2} \sum_{i,j=1}^3 \{x^i, n^j\} \{x^j, n^i\} $, where $ x^i $ are embedding coordinates and $ n^i $ are components of the unit normal.
• The Codazzi-Mainardi equations for surfaces in ℝ³ are equivalent to the Nambu bracket identity $ \sum_{j,k=1}^3 \big{\{} \gamma^{-2}(\mathcal{P}^2)^{ik}, n^k \big{\}} = 0 $, which holds as a consequence of Poisson algebra identities.
• For any Poisson algebra, the identity $ \sum_{j,k,l,n=1}^3 \frac{1}{2} \varepsilon_{kln} \big{\{} \gamma^{-2} \{x^i,x^j\} \{x^j,x^k\}, \gamma^{-1} \{x^l,x^n\} \big{\}} = 0 $ holds, proving the Codazzi-Mainardi equations are algebraic identities.
• The classical form of the Codazzi-Mainardi equations is recovered via $ \sum_{i,j,k=1}^3 (\partial_c x^i) \big{\{} \gamma^{-2} (\mathcal{P}^2)^{ik}, n^k \big{\}} = \frac{1}{\rho} \varepsilon^{ab} \nabla_a h_{bc} $, linking algebraic brackets to standard geometric expressions.
• The Nambu bracket formulation generalizes to hypersurfaces in ℝ^{n+1}, where identities such as $ \varepsilon_{klL} \big{\{} \gamma^{-2} \{x^i,\vec{x}^J\} \{\vec{x}^J,x^k\}, \gamma^{-1} \{x^l,\vec{x}^L\} \big{\}}_f = 0 $ hold for arbitrary smooth functions.
• The Ricci curvature and Weingarten's formula are expressible in terms of Nambu brackets, showing that core geometric invariants emerge from the algebraic structure of the function algebra without explicit derivatives.