A characterization of non-collapsed RCD(K,N) spaces via Einstein tensors

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.