Invariants for the Lagrangian Equivalence Problem

TL;DR

This paper investigates geometric invariants under automorphisms of a bundle related to the Lagrangian equivalence problem, identifying second-order and higher-order invariants for manifolds of dimension two or more.

Contribution

It provides a geometric characterization of second-order invariants and explores higher-order invariants in the context of Lagrangian equivalence.

Findings

Second-order invariants are explicitly determined.

Higher-order invariants are identified for manifolds with dimension ≥ 2.

The study links invariants to the automorphism group actions on jet spaces.

Abstract

Let $M$ be a connected smooth manifold, let $\operatorname{Aut}(p)$ be the group automorphisms of the bundle $p\colon \mathbb{R}\times M\to \mathbb{R}$, and let $q\colon J^1(\mathbb{R},M)\times \mathbb{R\to }J^1(\mathbb{R},M)$ be the canonical projection. Invariant functions on $J^r(q)$ under the natural action of $\operatorname{Aut}(p)$ are discussed in relationship with the Lagrangian equivalence problem. The second-order invariants are determined geometrically as well as some other higher-order invariants for $\dim M\geq 2$.