Abstract
This paper is dedicated to exhaustive structural analysis of the holonomy invariant foliated cocycles on the tangent bundle of an arbitrary ( m + n ) role=presentation> ( m + n ) ( m + n ) (m+n) -dimensional manifold. For this purpose, by applying Spencer theory of formal integrability, sufficient conditions for the metric associated with the semispray S role=presentation> S S S are determined to extend to a transverse metric for the lifted foliated cocycle on T M role=presentation> T M T M TM . Accordingly, this geometric structure converts to a holonomy invariant foliated cocycle on the tangent space, which is totally adapted to the Helmholtz conditions.
Highlights
Differential geometry of the total space of a manifold’s tangent bundle has its origins in diverse fields of study such as calculus of variations, differential equations, theoretical physics, and mechanics
We have comprehensively analyzed the structure of the holonomy invariant foliated cocycles on the tangent bundle of an arbitrary manifold via the notion of formal integrability, which was first introduced by Bucataru and Muzsnay in [10] and is fundamentally based on Frolicher–Nijenhuis formalism and is extensively fruitful since it provides a noteworthy setting to apply Spencer theory in order to investigate the formal integrability of Helmholtz conditions
This reformulation of the inverse problem of the calculus of variations enables us to apply Spencer theory in order to construct a transverse metric on the tangent bundle, which leads to the creation of the holonomy invariant foliated cocycles
Summary
Differential geometry of the total space of a manifold’s tangent bundle has its origins in diverse fields of study such as calculus of variations, differential equations, theoretical physics, and mechanics. We have comprehensively analyzed the structure of the holonomy invariant foliated cocycles on the tangent bundle of an arbitrary manifold via the notion of formal integrability, which was first introduced by Bucataru and Muzsnay in [10] and is fundamentally based on Frolicher–Nijenhuis formalism and is extensively fruitful since it provides a noteworthy setting to apply Spencer theory in order to investigate the formal integrability of Helmholtz conditions This reformulation of the inverse problem of the calculus of variations enables us to apply Spencer theory in order to construct a transverse metric on the tangent bundle, which leads to the creation of the holonomy invariant foliated cocycles. Some concluding remarks are mentioned at the end of the paper
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