Symmetric R-Spaces

Year 1994, Volume: 43 , 0 - 0, 01.01.1994

Kadri Aslan

https://doi.org/10.1501/Commua1_0000000474

Abstract

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.

Keywords

Symmetric, R-Spaces, parallel second fundamental

References

• Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi