QuoteOriginally posted by: Noniusok, so I've convinced myself that geodesics are of the form Exp(tU)sId where sId is the nxk matrix with ones down the diag k times (n>k), and U is in the complement of the kernel of the projection at the identity. now, fix x different from mId in the Stiefel manifold. x=gH for some g and H=to the subroup of SO(N) that fixes the first k standard basis vectors. note that x=gsid by the commutivity of projection with left action. Also, SO(N) acts on left by isometries, so I think gExp(tU)mid are geodesics where U is again in the orth complement of the projection. question, do you agree that another description of the geodesics is Exp(tV)gsid where V is in the image of Ad(g) of the orth complement of the projection at the identity?alright, at least Chiral, help.....so, I've a smooth real valued function F on SO(N)/SO(K). Normally, I can define simply the gradient Grad(F) in the usual way....but, I want to push this up to the Lie Algebra of SO(N). In other words, I want to define GradNonius(F):SO(N)/SO(K)->U where U is an appropriate subset of the Lie Algebra of SO(N). The reason I want to do this is that, obviously, it is easy to work with stuff that lives in the Lie Algebra, even if you have to map it down to the tangent space of SO(N)/SO(K) at sid, and then move the projected vector back to the tangent space of some point in SO(N)/SO(K). I can't believe I've been stuck on this.