Introduction The curvature tensor R of a locally symmetric Riemannian space satisfies R(X, Y) R — 0 for all tangent vectors X and 7, where the linear endomorphism R(X, Y) acts on R as a derivation. This identity holds in a space of recurrent curvature also. The spaces with R(X9 Y) R = 0 have been investigated first by E. Cartan [2] as these spaces can be considered as a direct generalization of the notion of symmetric spaces. Further on remarkable results were obtained by the authors A. Lichnerowicz [13], R. S. Couty [3], [4] and N. S. Sinjukov [19], [20], [21]. In one of his papers K. Nomizu [15] conjectered that an irreducible, complete Riemannian space with dim > 3 and with the above symmetric property of the curvature tensor is always a locally symmetric space. But this conjecture was refuted by H. Takagi [22] who constructed 3-dimensional complete irreducible nonlocally-symmetric hypersurfaces with R(X, Y) R — 0. These two papers were very stimulating for the further investigations. We also have to mention the following authors in this field: S. Tanno [23], [24], [25], K. Sekigawa [16], [17] and P. I. Kovaljev [9], [10], [11]. In the following we call a space satisfying R(X, Y) R = 0 a semi-symmetric space. The main purpose of this paper is to determine all semi-symmetric spaces in a structure theorem. In §1 we give local decomposition theorems using the infinitesimal holonomy group, and in §2 we give some basic formulas. We would like to make it perfectly clear that the results of these chapters are concerning general Riemannian spaces, and not only semi-symmetric spaces. In §3 we construct several nonsymmetric semi-symmetric spaces and in §4 we show that every semi-symmetric space can be decomposed locally on an everywhere dense open subset into the direct product of locally symmetric spaces and of the spaces constructed in §3.

Structure theorems on riemannian spaces satisfying R(X, Y) · R =0,

The purpose of this paper is to study a Kenmotsu manifold which is derived from the almost contact Riemannian manifold with some special conditions. In general, we have some relations about semi-symmetric, Ricci semi-symmetric or Weyl semisymmetric conditions in Riemannian manifolds. In this paper, we partially classify the Kenmotsu manifold and consider the manifold admitting a transformation which keeps Riemannian curvature tensor and Ricci tensor invariant.

https://www.koreascience.or.kr:443/article/JAKO200520828091676.pdf

On kenmotsu manifolds

Semi-parallel immersions are defined as extrinsic analogue for semi-symmetric spaces and as a direct generalization of parallel immersions. Using results of Backes on Euclidean Jordan triple systems, the totally geodesic immersions are shown to be the only minimal semi-parallel immersions into a Euclidean space. Semi-parallel immersions of surfaces into Em are studied and a classification of semi-parallel immersions with pointwise planar normal sections of surfaces in Em is given.

Semi-parallel surfaces in Euclidean space

Based on Schouten’s interpretation of the Riemann–Christoffel curvature tensor R, a geometrical meaning for the tensor R·R is presented. It follows that the condition of semi-symmetry, i.e. R·R = 0, can be interpreted as the invariance of the sectional curvature of every plane after parallel transport around an infinitesimal parallelogram. Using the tensor R· R, and in analogy with the definition of the sectional curvature K(p,π) of a plane π, a scalar curvature invariant L(p,π, \({\overline{\pi}}\)) is constructed which in general depends on two planes π and \({\overline{\pi}}\) at the same point p. This invariant can be geometrically interpreted in terms of the parallelogramoids of Levi–Civita and it is shown that it completely determines the tensor R· R. Further it is demonstrated that the isotropy of this new scalar curvature invariant L(p,π, \({\overline{\pi}}\)) with respect to both the planes π and \({\overline{\pi}}\) amounts to the Riemannian manifold to be pseudo-symmetric in the sense of Deszcz.

Properties of a scalar curvature invariant depending on two planes

The symmetry of warped product space-times

/pdf/structure-theorems-on-riemannian-spaces-satisfying-r-x-y-16ofy64qj1.pdf

Z. I. Szabó

Papers

Structure theorems on Riemannian spaces satisfying $R(X,\,Y)\cdot R=0$. I. The local version