The problem of computing the best multilinear low-rank approximation of a tensor can be formulated as an opimization problem on a product of Grassmann manifolds (by multilinear low-rank approximation we understand an approximation in the sense of the Tucker model). In the Grassmann approach we want to find (bases of) subspaces that represent the low-rank approximation. We have recently derived a Newton algorithm for this problem, where a quadratic model on the tangent space of the manifold is used. From the Grassmann Hessian we derive conditions for a local optimum. We also discuss the sensitivity of the subspaces to perturbations of the tensor elements.