Structure theorems on Riemannian spaces satisfying $R(X,\,Y)\cdot R=0$. I. The local version
About: This article is published in Journal of Differential Geometry.The article was published on 1982-01-01 and is currently open access. It has received 376 citations till now. The article focuses on the topics: Structure (category theory).
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TL;DR: In this paper, Tanno et al. showed that 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.
Abstract: 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.
266 citations
TL;DR: In this paper, the authors studied the Ricci tensor invariance of the Riemannian curvature tensor of the Kenmotsu manifold, which is derived from the almost contact Ricci manifold with some special conditions.
Abstract: 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.
90 citations
TL;DR: Semi-parallel immersions are defined as extrinsic analogue for semi-symmetric spaces and as a direct generalization of parallel immersion as mentioned in this paper, using results of Backes on Euclidean Jordan triple systems.
Abstract: 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.
75 citations
TL;DR: In this paper, a geometrical interpretation of the Riemann-Christoffel curvature tensor R·R is presented, which can be interpreted as the invariance of the sectional curvature of every plane after parallel transport around an infinitesimal parallelogram.
Abstract: 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.
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01 Jan 1962TL;DR: In this article, the classification of symmetric spaces has been studied in the context of Lie groups and Lie algebras, and a list of notational conventions has been proposed.
Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces On the classification of symmetric spaces Functions on symmetric spaces Bibliography List of notational conventions Symbols frequently used Author index Subject index Reviews for the first edition.
3,013 citations
TL;DR: In this paper, the lower bound of the dimension of the Euclidean space in which a compact Riemann manifold can be imbedded isometrically, if its curvatures satisfy certain conditions.
Abstract: This paper will be concerned with some estimates on the lower bound of the dimension of the Euclidean space in which a compact Riemann manifold can be imbedded isometrically, if its curvatures satisfy certain conditions. Our basic geometrical idea is a very simple one. Denote by M a compact Riemann manifold of dimension n in an Euclidean space E of dimension n + N, the Riemann metric on M being induced by the imbedding. Let 0 be a fixed point of E. The distance OP, P e M, is a continuous function in M and attains a maximum at a point P0 e M, since M is compact. It is intuitively clear that M will be "concave toward 0" at Po, so that there will be some restrictions on the Riemann curvature of M at P0. If M is given abstractly, the imbedding will not be possible, if these restrictions are not fulfilled by the given Riemann metric at any of the points of M. Actually, however, if the difference N of the dimensions of M and E is greater than one, the implication of this geometrical fact on the Riemann curvature of M is not very simple. The question leads to algebraic problems which probably do not have simple answers. We propose to give in this paper a few conclusions which can be drawn. It should be mentioned that the above geometrical idea has been used by Tompkins1 to prove that a locally flat compact Riemann manifold of dimension n cannot be isometrically imbedded in an Euclidean space of dimension 2n 1. Among other things this theorem will be generalized and the invariants entering in the problem geometrically interpreted. As for the differentiability assumptions we suppose our manifold and the imbedding to be of class > 4.
173 citations
TL;DR: In this paper, the authors consider foliations with certain properties common to several geometrically defined ones and show that the standard imbedding of S" into S ~÷p is rigid for certain codimensions p, see also [7].
Abstract: In this article we consider foliations with certain properties common to several geometrically defined ones. We show that their codimension is either large or zero. As an application we obtain characterisations of the totally geodesic, complete submanifolds of S N and C pN which for most dimensions are sharper than those given by Nomizu [6], Abe [1], and the author [3]. Using this characterisation and a slightly generalized formula from [2], we prove that the standard imbedding of S" into S ~÷p is rigid for certain codimensions p, see also [7]. 2. Statement of Results Let 0(t) denote the largest integer such that the fibration
84 citations