A generalization of Molino's theory and equivariant basic \^A-genus characters

TL;DR

This paper extends Molino's theory to pairs of Riemannian foliations, establishing an equivariant basic cohomological isomorphism and geometric realization via a-genus characters, advancing the understanding of elliptic operator indices.

Contribution

It introduces a novel generalization of Molino's theory for two Riemannian foliations and connects it to equivariant basic a-genus characters for index calculations.

Findings

Established an equivariant basic cohomological isomorphism for Killing foliations.

Provided a geometric realization through equivariant basic a-genus characters.

Extended Molino's theory to a broader class of Riemannian foliations.

Abstract

Molino's theory is a mathematical tool for studying Riemannian foliations. In this paper, we propose a generalization of Molino's theory with two Riemannian foliations. For this purpose, the projection of foliation with respect to a fibration is discussed. The generalization results in an equivariant basic cohomological isomorphism in case of Killing foliation. It is a generalization of results given by Goertsches and T\"oben. We also give a geometric realization of the cohomological isomorphism through equivariant basic \^A-genus characters, who play a prominent role in calculating the index of an elliptic operator by Atiyah-Singer's index formula.