Geometric Structures in Field Theory

TL;DR

This review explores the generalizations of tangent and cotangent structures in field theory, comparing approaches like k-symplectic and multisymplectic geometries to unify the geometric framework of classical field theories.

Contribution

It systematically reviews and contrasts different geometric generalizations in field theory, highlighting their roles and relationships to unify the subject.

Findings

Identification of two main categories of geometric generalizations.

Comparison of axiomatic systems like k-symplectic and k-tangent structures.

Analysis of natural geometries on jet, cojet, and frame bundles.

Abstract

This review paper is concerned with the generalizations to field theory of the tangent and cotangent structures and bundles that play fundamental roles in the Lagrangian and Hamiltonian formulations of classical mechanics. The paper reviews, compares and constrasts the various generalizations in order to bring some unity to the field of study. The generalizations seem to fall into two categories. In one direction some have generalized the geometric structures of the bundles, arriving at the various axiomatic systems such as k-symplectic and k-tangent structures. The other direction was to fundamentally extend the bundles themselves and to then explore the natural geometry of the extensions. This latter direction gives us the multisymplectic geometry on jet and cojet bundles and n-symplectic geometry on frame bundles.