Smooth Fields of Hilbert Spaces, Hermitian bundles and Riemannian Direct Images

TL;DR

This paper explores the relationship between smooth structures on fields of Hilbert spaces and Hermitian bundles, applying the theory to Riemannian direct images to determine conditions for smoothness and bundle structures.

Contribution

It establishes a general framework linking geometric and analytical notions of smooth Hilbert fields and applies it to Riemannian direct images, deriving formulas for differentiating integral-defined functions.

Findings

Conditions for a field of Hilbert spaces to admit a smooth structure

Formulas for derivatives of functions defined as integrals over fibers

Application of theory to Riemannian submersions and vector bundles

Abstract

Given a field of Hilbert spaces there are two ways to endow it with a smooth structure: the standard and geometrical notion of Hilbert (or Hermitian) bundle and the analytical notion of smooth field of Hilbert spaces. We study the relationship between these concepts in a general framework. We apply our results in the following interesting example called Riemannian direct images: Let $M,N$ be Riemannian oriented manifolds, $\rho:M\to N$ be a submersion and $\pi:E\to M$ a finite dimensional vector bundle. Also, let $M_\lambda=\rho^{-1}(\lambda)$ and fix a suitable measure $\mu_\lambda$ in $M_\lambda$. Does the field of Hilbert spaces $\mathcal{H}(\lambda)=L^2(M_\lambda,E)$ admits a smooth field of Hilbert space structure? or a Hilbert bundle structure? In order to provide conditions to guarantee a positive answer for these questions, we develop an interesting formula to derivate functions defined on $N$ as a integral over $M_\lambda$.