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

Abstract: 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.