Riemannian submersions and factorization of Dirac operators

TL;DR

This paper demonstrates how Dirac operators on Riemannian submersions of compact spin$^c$ manifolds can be factorized in unbounded KK-theory, revealing curvature obstructions and connecting to foundational work by Connes and Skandalis.

Contribution

It establishes a new factorization of Dirac operators in unbounded KK-theory for Riemannian submersions, including explicit curvature terms and KK-product representation.

Findings

Dirac operator on total space is unitarily equivalent to tensor sum with base space operator

Curvature term acts as an obstruction to factorization in unbounded KK-theory

Tensor sum corresponds to the KK-product of KK-cycles

Abstract

We establish the factorization of Dirac operators on Riemannian submersions of compact spin$^c$ manifolds in unbounded KK-theory. More precisely, we show that the Dirac operator on the total space of such a submersion is unitarily equivalent to the tensor sum of a family of Dirac operators with the Dirac operator on the base space, up to an explicit bounded curvature term. Thus, the latter is an obstruction to having a factorization in unbounded KK-theory. We show that our tensor sum represents the bounded KK-product of the corresponding KK-cycles and connect to the early work of Connes and Skandalis.