August 12th, 2003, 11:25 am
QuoteOriginally posted by: Nanyway, the construction is pretty easy in fact......If G is a Lie Group and G/H is a homogeneous space, consider f:G/H->R smooth. Then, G acts on G/H on the left through isometries. Let Lam be an element of the Lie Algebra of G. Consider the curve f(Exp(Lam t)m) where m is in G/H. Differentiation and setting t = yields, d/dtf(Exp(Lam t)m). This defines a map from G/H to linear functionals defined on the Lie Algebra of G. Using the metric, there is a natural identification between linear functionals and elements of the Lie Algebra. Define GradNonius(f) to be that element of the Lie Algebra such that <GradNonius(f)(m),Lam>=d/dt(t=0)f(Exp(Lam t)m).I claim that GradNonius(f) is orthogonal to the space that represents a vertical fiber over m in the bundle G->H. A vertical fiber over m is by definition the elements V in the tangent space of TG at any g such that p(g)=m such that dp(g)V=0. This tangent space is naturally identified with the Lie Algebra via the adjoint rep. If dp(g)V=0, then Exp(tV) fixes m, which means Exp(tV) is in the fixed group of m. Thus, d/dt(t=0)f(Exp(V t)m)=0. Since GradNonius lives in a space that is orthogonal to the vertical fibers, it is in a horizontal lift, which means that it lives in a space that is naturally identified with the tangent space of m. It also means that Exp(GradNonius(f)(m)t)m is a geodesic in G/H passing through m.Anyway, all of this means that grad descent is relatively easy. Nonius,What a bunch of crap. Your manifold ain't smooth.RockyLifeless Rock,why do you communicate to a living organism this information? Yes, I agree, rocks are never smooth. Rocks, like you, have sharp edges n shit. I never ever ever met a rock that understood math. Are you communicating to me that you, Rock, actually understand the math behind this? Is there a seperate realm of math that is applicable to the universe of rocks? Perhaps I am naive. I have an open mind though. Teach me about the way rocks, stones, dirt, and such understand the logic and beauty of mathematics.