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On vector-valued multisymplectic forms

Tatyana Barron, Kai Boisvert, Noah Vale

Abstract

We obtain a standard local presentation for a vector-valued multisymplectic form on a smooth manifold, generalizing the known proof for polysymplectic forms. We show that vector-valued multisymplectic forms on a finite-dimensional real vector space form a non-unital operad. We prove an entropy inequality for partial compositions.

On vector-valued multisymplectic forms

Abstract

We obtain a standard local presentation for a vector-valued multisymplectic form on a smooth manifold, generalizing the known proof for polysymplectic forms. We show that vector-valued multisymplectic forms on a finite-dimensional real vector space form a non-unital operad. We prove an entropy inequality for partial compositions.
Paper Structure (5 sections, 4 theorems, 70 equations, 2 figures)

This paper contains 5 sections, 4 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Towards an analogue of the Darboux theorem
  3. The linear case
  4. Local canonical form
  5. Operads and entropy

Key Result

Lemma 2.2

Let $n,k,m\in {\mathbb{N}}$, $n\ge 2$, $1\le k\le n-1$. Let $V$ be an $n$-dimensional real vector space. Let $\omega$ be a multilinear map $\omega: {\bigwedge}^{k+1} V\to {\mathbb{R}}^m$. For each $i\in \{ 1,...,m\}$, let $\omega_i$ be the $i$-th component of $\omega$ and let $\bar{\omega}_i$ deno

Figures (2)

  • Figure 1: The curve (\ref{['eq:cure']}).
  • Figure 2: The curve (\ref{['eq:curd']}).

Theorems & Definitions (13)

  • Definition 2.1
  • Lemma 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Example 2.8
  • Example 2.9
  • Definition 3.1
  • ...and 3 more