Manifold embeddings by heat kernels of connection Laplacian

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.