Symmetric R-Spaces

Submanifolds with parallel second fundamental form are defincd as extrinsic analogue of locally symmetric manifolds [6, 7]. It follovvs that ali of them are locally invariant under the reflection in the normal space of an arbitrary point. These type of submanifolds are also cailed symmetric submamfolds [7 ]. Examples are symmetric jR-spaces. Submanifolds with pointwise planar normal sections (P2-PNS) are introduced in [3, 4, 5 ]. It bas shown tliat spherical submanifolds have P2-PNS property if and only if they must be parellel submanifolds. In [İJ the present author and A. West showed that non-parallel submanifold M has P2-PNS property if and only if It is a hypersurface. In this article we prove that if M is a symmetric R-space then it must be tlıe orbit of the element A such that ad (A))^ = ad{ We also show that the imbeddings of the symmetric K-spaces of tlıe form f: M = Kİ Ko —■ P by f ([fc]) = (k) have P2-PNS.

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  • Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics-Cover
  • ISSN: 1303-5991
  • Yayın Aralığı: Yılda 4 Sayı
  • Başlangıç: 1948
  • Yayıncı: Ankara Üniversitesi