Abstract

In 1979, Ferus [2] introduced the notion of a symmetric submanifold of the Euclidean space E d in the following sense: Let M C E d be a Riemannian submanifold. For each x E M, let tr x be the reflection of E d with respect to the normal space Nx of M at x. Then M is an (extrinsic) symmetric submanifold of E d if a~(M) ( M holds for each x ~ M. Ferus proved that each symmetric submanifold of E d is a direct product of an affine subspace and a standard embedded symmetric R-space, and in this way he obtained the full classification (see also [3]). A more direct proof of this result was given by Striibing [16]. The present authors studied the analogous problem for "generalized symmetric submanifolds" of Euclidean spacesmost of the results have been presented in the CSc. Thesis [12] of the second author from 1981. These results remained unpublished up to now. Recently, S~mchez [15] has given a full classification of compact k-symmetric submanifolds of E ~ with odd k he proved that all these submanifolds are in fact symmetric. In our original research we used a different definition of generalized symmetric submanifold, which was less restrictive than that by Shnchez, and which led to a completely different theory. For this reason we decided to publish, with some delay, a part of our earlier results, hoping that they might still attract some interest.

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