Embedding Riemannian Manifolds by the Heat Kernel of the Connection Laplacian
This work addresses a foundational problem in differential geometry and mathematical analysis, with potential applications in manifold learning and geometric data processing, but appears incremental as it builds on existing heat kernel and Laplacian methods.
The authors tackled the problem of embedding closed Riemannian manifolds with specific geometric conditions into a Hilbert space, using the heat kernel of the Connection Laplacian, resulting in a distance construction and a pre-compactness theorem for the class.
Given a class of closed Riemannian manifolds with prescribed geometric conditions, we introduce an embedding of the manifolds into $\ell^2$ based on the heat kernel of the Connection Laplacian associated with the Levi-Civita connection on the tangent bundle. As a result, we can construct a distance in this class which leads to a pre-compactness theorem on the class under consideration.