Overview
This research investigates the spectral properties of the horizontal Laplacian within the framework of a Riemannian submersion. The focus is specifically on submersions characterized by totally geodesic fibers and an integrable horizontal distribution. A primary finding establishes a unitary equivalence between the horizontal Laplacian and a twisted Laplacian, which acts upon sections of an infinite-rank flat vector bundle defined over the base manifold of the Riemannian submersion.
Research Context
The study of Laplacians is fundamental in geometric analysis, particularly in understanding the geometry and topology of manifolds. Riemannian submersions provide a structure to decompose a manifold into layers (fibers) over a base manifold. The horizontal Laplacian operates on functions that are constant along the fibers, representing a key operator in the context of these geometric structures. The specific conditions of totally geodesic fibers and an integrable horizontal distribution define a particular class of Riemannian submersions relevant to this analysis. The work builds upon and relates to concepts such as canonical variations introduced by Berard-Bergery and Bourguignon, and prior results concerning the essential spectrum on foliated manifolds by Kordyukov.
Approach
The core of the approach involves demonstrating a unitary equivalence. This equivalence transforms the problem of analyzing the horizontal Laplacian into studying a twisted Laplacian. The twisted Laplacian operates on the space of sections of a specific infinite-rank flat vector bundle where the bundle is defined over the base manifold of the Riemannian submersion. This transformation provides a different perspective on the spectral characteristics of the original operator. From this interpretation, the research proceeds to:
- Apply the interpretation to analyze the asymptotic behavior of the scaled first nonzero eigenvalue of the canonical variations.
- Compare the horizontal Laplacian with the usual Laplacian defined on a Riemannian covering over the base manifold.
- Examine the essential spectrum in cases where the holonomy group is infinite and amenable, leading to a comparison with established results in geometric analysis on foliated manifolds.
Findings
- The horizontal Laplacian of a Riemannian submersion, given its totally geodesic fibers and an integrable horizontal distribution, is unitarily equivalent to a twisted Laplacian. This twisted Laplacian acts on the space of sections of an infinite-rank flat vector bundle over the base manifold.
- This interpretation facilitates the study of the asymptotic behavior of the scaled first nonzero eigenvalue of the canonical variations, as introduced by Berard-Bergery and Bourguignon.
- The approach enables a comparison between the horizontal Laplacian and the usual Laplacian on a Riemannian covering over the base manifold.
- When the holonomy group is infinite and amenable, the essential spectrum of the horizontal Laplacian coincides with that of the usual Laplacian on a Riemannian covering, in the specific setup of this study. This finding strengthens a result by Kordyukov in the context of geometric analysis on foliated manifolds.
Why This Matters
The establishment of a unitary equivalence between the horizontal Laplacian and a twisted Laplacian provides a new analytical tool for understanding spectral properties in Riemannian submersions with specific geometric characteristics. This new interpretation may enable further advancements in the analysis of eigenvalue behavior and comparison with other Laplacian operators in related geometric settings.