CGC Foliation
Let
be a -dimensional Riemannian manifold. Then the CGC foliation of is a foliation such that each leaf is a ( -dimensional) curve with constant geodesic curvature.
CGC
Action The CGC
action is a smooth group action on a smooth Riemannian -manifold , say , such that where denotes the arc length of in the CGC leaf, and is the canonical chart for the Lie group .
Remark
Geometrically, it means that we find the CGC leaf containing
, then if , then we move a distance of along the leaf; otherwise, is in but not in , then go in the opposite direction.
Theorem
Suppose
is a Riemannian -manifold. Then is rotationally-invariant with respect to some group action if and only if it is rotationally-invariant with respect to the CGC action.