  We extend the class of SQP methods for equality constrained optimization to
the setting of differentiable manifolds. The use of retractions and
stratifications allows us to pull back the involved mappings to linear spaces.
We study local quadratic convergence to minimizers. In addition we present a
composite step method for globalization based on cubic regularization of the
objective function and affine covariant damped Newton method for feasibility.
We show transition to fast local convergence of this scheme. We test our method
on equilibrium problems in finite elasticity where the stable equilibrium
position of an inextensible transversely isotropic elastic rod under dead load
is sought.
