k-symmetric submanifolds of \(R^ N\)
This paper studies submanifolds \(M^ n\) of \(R^ N\) which are k- symmetric and such that the corresponding isometry \(\Theta_ p\) at each point extends to an isometry \(\Omega_ p: R^ N\to R^ N\) which is the identity on the normal space to \(M^ n\) at the point. The case \(k=2\) was considered by \textit{D. Ferus} [Math. Ann. 247, 81-93 (1980; Zbl 0446.53041)]. He obtains a complete classification by using the fact that these submanifolds have parallel second fundamental form. But for \(k\neq 2\) these submanifolds do not have parallel second fundamental form and then quite different methods are required. The purpose of this paper is to study the case k odd. It contains a complete description of the compact k-symmetric spaces which admit imbeddings of this type as well as a description of the imbeddings. The methods are those of the theory of representations of Lie groups and algebras. The paper also contains results about the total absolute curvature of these submanifolds.
scientific article; zbMATH DE number 3851963
scientific article