Distance induced by the Frobenius norm between two multivariate normals

There is a standard on covariance matrices defined by the Frobenious standard. http://mathworld.wolfram.com/FrobeniusNorm.html
Can this be used to define a valid distance between two multivariate Gaussian distributions? Something like $ d ^ {2} left ( mathcal {N} left ( mu_ {1}, Sigma_ {1} right), mathcal {N} left ( mu_ {2}, Sigma_ {2 } right) right) = || mu_ {1} – mu_ {2} || _ {F} ^ {2} + || Sigma_ {1} – Sigma_ {2} || _ {F} ^ {2} $. I am looking for references / articles.