  We investigate the second principal term in the expansion of metrics
$c(n)t^{(n+2)/2}g_t$ induced by heat kernel embedding into $L^2$ on a compact
$RCD(K, N)$ space. We prove that the divergence free property of this term in
the weak, asymptotic sense if and only if the space is non-collapsed up to
multiplying a constant to the reference measure. This seems new even for
weighted Riemannian manifolds. Moreover an example tells us that the result
cannot be generalized to the noncompact case. In this sense, our result is
sharp.
