  Given an RCD$(K,N)$ space $({X},\mathsf{d},\mathfrak{m})$, one can use its
heat kernel $\rho$ to map it into the $L^2$ space by a locally Lipschitz map
$\Phi_t(x):=\rho(x,\cdot,t)$. The space $(X,\mathsf{d},\mathfrak{m})$ is said
to be an isometrically heat kernel immersing space, if each $\Phi_t$ is an
isometric immersion {}{after a normalization}. A main result states that any
compact isometrically heat kernel immersing RCD$(K,N)$ space is isometric to an
unweighted closed smooth Riemannian manifold. This is justified by a more
general result: if a compact non-collapsed RCD$(K, N)$ space has an
isometrically immersing eigenmap, then the space is isometric to an unweighted
closed Riemannian manifold, which greatly improves a regularity result in
\cite{H21} by Honda. As an application of these results, we give a
$C^\infty$-compactness theorem for a certain class of Riemannian manifolds with
a curvature-dimension-diameter bound and an isometrically immersing eigenmap.
