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We investigate the second principal term in the expansion of the metrics c(n)t (n+2)/2 g t induced by the heat kernel embedding into L 2 on a compact RCD(K,N) space. We prove that the divergence free property of this term holds in the weak, asymptotic sense if and only if the space is non-collapsed up to multiplying the reference measure by a constant. 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.
Honda et al. (2024) studied this question.