Variance is the principal fundamental form in statistics. The model for the first fundamental form that of the length of a vector and is defined in terms of an inner product. A moment is basically the idea behind the balance which involves the weighted sum of distances. The moments used in Physics and Engineering are associated with exterior products. It also seems natural to associate moments with the deviations of statistics.
The condition for a minimum is associated with a nonchanging value of the variance in least squares fits. The numerical computations seem to suggest that there may be more general truths associated with variances and moments. It's something to ponder on.
Supplemental: Consider for example the process of normalization in statistics.