An inverse problem for fractional connection Laplacians

TL;DR

This paper demonstrates that local data about a fractional connection Laplacian uniquely determines the global geometric and analytic structures of the underlying manifold, bundle, and operators.

Contribution

It extends inverse problem results from the fractional Laplace-Beltrami operator to fractional connection Laplacians on vector bundles.

Findings

Local knowledge of the fractional connection Laplacian determines global structures.

The result generalizes previous inverse problems for scalar fractional Laplacians.

The method applies to smooth Hermitian vector bundles over closed Riemannian manifolds.

Abstract

Consider a fractional operator $P^s$, $0<s<1$, for connection Laplacian $P:=\nabla^*\nabla+A$ on a smooth Hermitian vector bundle over a closed, connected Riemannian manifold of dimension $n\geq 2$. We show that local knowledge of the metric, Hermitian bundle, connection, potential, and source-to-solution map associated with $P^s$ determines these structures globally. This extends a result known for the fractional Laplace-Beltrami operator.