  We discover a geometric property of the space of tensors of fixed multilinear
(Tucker) rank. Namely, it is shown that real tensors of fixed multilinear rank
form a minimal submanifold of the Euclidean space of tensors endowed with the
Frobenius inner product. We also establish the absence of local extrema for
linear functionals restricted to the submanifold of rank-one tensors, finding
application in statistics.
