Second order optimality on orthogonal Stiefel manifolds

Laplace-Beltrami Operator on the Orthogonal Group in Cartesian Coordinates

“…x 2 i = R 2 } using Cartesian coordinates, we need to introduce a local frame on the sphere. Inspired by the construction of a local frame on the Stiefel manifold, see [9], page 11, we consider the local frame as follows b : for x ∈ S n−1 R , with x j = 0,…”

Characterization of the critical points for the shear-stretch strain energy of a Cosserat problem

“…We remind the embedded gradient vector field method for optimization of cost functions defined on constraint manifolds, see [10][11][12][13]. If S ⊂ M is a submanifold of a Riemannian manifold (M, g) described by a set of constraint functions, i.e.…”

“…A base for the tangent vector space T q S 3 can be computed in the following way, see [13]. For the point q we choose an index j ∈ {0, 1, 2, 3} such that q j = 0.…”

“…± , q (10) , and q (13) . Spec Hess G 1,0 q (16) = {0, −16λ(λ + 1)} , where the eigenvalue 0 has multiplicity 2.…”

“…In Section 3 we extend the initial optimization problem for the linear Cosserat model, that is defined on SO(3), to the general case of the special orthogonal group SO(n). We start by recalling the embedded gradient vector field method, [10][11][12][13], and apply it to obtain necessary and sufficient conditions for a rotation to be a critical point of the generalized free energy.…”