Manifold embeddings by heat kernels of connection Laplacian

TL;DR

This paper demonstrates that closed manifolds can be embedded into Euclidean space using heat kernels of the connection Laplacian, with embeddings that approximate isometries and depend on geometric bounds.

Contribution

It introduces new embedding methods based on heat kernels of the connection Laplacian, with bounds on parameters ensuring near-isometric embeddings.

Findings

Embeddings depend on bounds of Ricci curvature and injectivity radius.

Both heat kernel-based maps can approximate isometries arbitrarily closely.

Embedding parameters are explicitly bounded by geometric quantities.

Abstract

We show that any closed $n$-dimensional manifold $(M,g)$ can be embedded by a map constructed using the heat kernels of the connection Laplacian as well as a maps constructed using truncated heat kernel at a certain time $t$ from a $\delta$-net $\{q_i\}_{i=1}^{N_0}$ via a rescaling trick. Both the time $t$ and $N_0$ are bounded in terms of the dimension, bounds on the Ricci curvature and its derivative, the injectivity radius, and the volume. Moreover, both maps can be made arbitrarily close to an isometry.