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Journal ArticleDOI

On conformally recurrent spaces of second order

01 Aug 1969-Journal of The Australian Mathematical Society (Cambridge University Press)-Vol. 10, pp 155-161
TL;DR: In this article, it was shown that a Riemannian space whose curvature tensor satisfies R%kitm = dlmR h ljk was called a Recurrent Space of Second Order (RSO) space and such a space was denoted by Kn.
Abstract: A; is a non-zero vector and comma denotes covariant differentiation with respect to the metric tensor ga. The present paper is concerned with nonflat Riemannian spaces Vn(n > 3) defined by (2) c ; , M m = «IMc*ft where alm is a tensor not identically zero. We shall call a Riemannian space defined by (2) a conformally recurrent space of second order and shall denote an w-space of this kind by C(Kn). A Riemannian space whose curvature tensor satisfies R%kitm = dlmR h ljk was called a Recurrent space of second order by A. Lichnerowicz [2]. Such an w-space shall be denoted by Kn. Evidently every Kn is a C ( Kn) but the converse is not necessarily true. Sections 2 and 3 of this paper deal with Einstein and 2-Ricci-recurrent C(Kn) respectively while section 4 deals with C( Kn) admitting a parallel vector field. In the last section it will be shown that a Riemannian space satisfying Wijkjm = a'lmW^jk where W%k is Weyl's projective curvature tensor is a C(Kn).

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Citations
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Journal ArticleDOI
TL;DR: The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor are characterized and classified in this paper, and the main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkmann's class in n dimensions.
Abstract: The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor—2–symmetric spacetimes—are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkmann's class in n dimensions. Related issues and applications are considered, and new open questions presented.

39 citations

Journal ArticleDOI
TL;DR: In this article, the structure of recurrently related operators of the curvature tensor and of its covariant derivatives for n-dimensional Riemannian spaces with arbitrary signature is examined.
Abstract: The structure of recurrently related operators of the curvature tensor and of its covariant derivatives for n-dimensional Riemannian spaces with arbitrary signature is examined. Applications to Einstein's theory of gravitation are given.

18 citations

Dissertation
01 Jan 2017
TL;DR: In this paper, a tensor by combining Riemann-Christoffel curvature tensor, Ricci tensor and the metric tensor is presented, and with the help of algebraic classification, the equivalence of different geometric structures is proved.
Abstract: In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann–Christoffel curvature tensor, the same type structures given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann–Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases. Then with the help of this generalized tensor and using algebraic classification we prove the equivalency of different geometric structures (see Theorems 6.3, 6.4, 6.5, 6.6 and 6.7; Tables 1 and 2). Mathematics Subject Classification (2010). 53C15, 53C21, 53C25, 53C35.

11 citations


Additional excerpts

  • ..., conformally recurent ([6], [20], [40], [62], [176], [223], [294], [304]), projectively recurrent [7], concircularly recurrent ([123], [131]), conharmonically recurrent [132] and quasi-conformally recurrent [65]) manifold....

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Posted Content
TL;DR: The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor (2-symmetric spacetimes) are characterized and classified in this article.
Abstract: The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor (2-symmetric spacetimes) are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkmann's class in n dimensions. Related issues and applications are considered, and new open questions presented.

4 citations


Cites background from "On conformally recurrent spaces of ..."

  • ...Similarly for conformally (or Ricci) recurrent [2, 59, 67, 57], conformally (or Ricci) 2-recurrent [20, 55, 68], and k-recurrent....

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References
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Journal ArticleDOI

204 citations


"On conformally recurrent spaces of ..." refers background in this paper

  • ...(2-3) « , » * } » + « , » K } « + « * » K J M = 0 Multiplying (2....

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  • ...kj = " reduces to (2-7) blm Thm+bhi Tmm+bjk Tlmhi = 0 where (2-8) blm = alm-aml....

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  • ...„ = 4, [nti+ 2{n^ R ){n_2) iu] = « £ ^ Hence (3-8) Ru....

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  • ...11) reduces to (3-12) c'lmDhm+c'hiDmm+c'jkDlmM = 0....

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  • ...3) by a\ where a\ = gavt we have (2-4) a\almR\ik+a\aimR\kl+a\akmR% = 0 Rtj = 0 implies ahmR h m = 0 by contracting h and k in (2....

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01 Jan 1967

76 citations


"On conformally recurrent spaces of ..." refers background in this paper

  • ...Again, if in a C((2)Kn), Rtj = 0, then from (1) and (2) it follows that Rm>lm = almR h iik, that is, the space is a (2)Kn....

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  • ...1) as well as (1) and (2) we have ((2)-(5)) hiik,lm = <*lmThiik where...

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