Reflections in submanifolds
The main topic of this paper is a study of local geodesic reflections of a Riemannian manifold (M,g) with respect to its submanifold \(P\subset M\). For instance, it is proved that if the reflection is volume-preserving, then the ``mirror'' P is necessarily a minimal submanifold. This result holds in both directions if (M,g) is a space of constant curvature and \(P\subset M\) is two-dimensional. Further, similar characterizations are given for those reflections of (M,g) in P which preserve the mean curvature of all tubes about P. The most interesting problem, namely to determine all minimal submanifolds just as mirrors of reflections with simple geometrical properties remains still open.
scientific article; zbMATH DE number 4071849