Abstract. We define a C1 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 C1-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.
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Alan Weinstein, Almost invariant submanifolds for compact group actions. J. Eur. Math. Soc. 2 (2000), no. 1, pp. 53–86DOI 10.1007/S100970050014