  We define a C^1 distance between submanifolds of a riemannian manifold M and
show that, if a compact submanifold N is not moved too much under the isometric
action 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 riemannian
manifolds in a symmetric way. The procedure combines averaging techniques of
Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney's idea of realizing
submanifolds as zeros of sections of extended normal bundles.
