Basic differential forms for actions of Lie groups

TL;DR

This paper explores the algebraic structure of invariant differential forms on Riemannian G-manifolds, establishing an isomorphism with forms on a special submanifold under certain conditions, advancing understanding of symmetry in geometric analysis.

Contribution

It introduces a new isomorphism between G-invariant horizontal forms on the manifold and Weyl group-invariant forms on a section, under specific conditions.

Findings

Algebra of G-invariant horizontal forms is isomorphic to Weyl group-invariant forms on a section.

Provides conditions under which the isomorphism holds.

Enhances understanding of symmetry and invariants in Riemannian G-manifolds.

Abstract

A section of a Riemannian $G$-manifold $M$ is a closed submanifold $\Sigma$ which meets each orbit orthogonally. It is shown that the algebra of $G$-invariant differential forms on $M$ which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on $\Sigma$ which are invariant with respect to the generalized Weyl group of this orbit, under some condition.