Almost invariant submanifolds for compact group actions

We define a C^1 distance between submanifolds of a riemannian manifold M andshow that, if a compact submanifold N is not moved too much under the isometricaction of a compact group G, there is a G-invariant submanifold C^1-close to N.The proof involves a procedure of averaging nearby submanifolds of riemannianmanifolds in a symmetric way. The procedure combines averaging techniques ofCartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney's idea of realizingsubmanifolds as zeros of sections of extended normal bundles.