In this paper we characterize the multivariate normal distribution through the mean vector of the distribution. In Theorem 3, we find that the multivariate normal distribution can also be characterized through its variance, even though unknown, provided the variance-covariance matrix is free of the parameter involved. We are able to determine the mean vector to within two unknowns, the variance-covariance matrix and a constant vector.
Mary R. Anderson. "A Characterization of the Multivariate Normal Distribution." Ann. Math. Statist. 42 (2) 824 - 827, April, 1971. https://doi.org/10.1214/aoms/1177693441