On spectral convergence of vector bundles and convergence of principal bundles
Kota Hattori

TL;DR
This paper investigates how the eigenvalues of the connection Laplacian on vector bundles vary continuously under geometric convergence, introducing a new topology for principal G-bundles with G-connections.
Contribution
It introduces the asymptotically G-equivariant measured Gromov-Hausdorff topology and applies it to analyze spectral convergence of vector and principal bundles.
Findings
Eigenvalues of the connection Laplacian are continuous under the new topology.
Established a framework for the convergence of principal G-bundles with G-connections.
Demonstrated the applicability of the topology to Riemannian manifolds with G-actions.
Abstract
In this article we consider the continuity of the eigenvalues of the connection Laplacian of -connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically -equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric -actions, and apply it to the total spaces of principal -bundles equipped with -connections over Riemannian manifolds.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric Analysis and Curvature Flows · Advanced Neuroimaging Techniques and Applications · Geometry and complex manifolds
