DECOMPOSITION OF RECURRENT CURVATURE TENSOR FIELDS IN A KAEHLERIAN MANIFOLD OF FIRST ORDER. Manoj Singh Bisht 1 and U.S.Negi 2

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1 DECOMPOSITION OF RECURRENT CURVATURE TENSOR FIELDS IN A KAEHLERIAN MANIFOLD OF FIRST ORDER Manoj Sing Bist 1 and U.S.Negi 2 1, 2 Department of Matematics, H.N.B. Garwal (A Central) University, SRT Campus Badsai Taul, Teri Garwal , Uttarakand, India. ABSTRACT: In tis paper, we ave defined and studied te decomposition of recurrent curvature tensor field in a Kaelerian manifold of first order by considering te decomposition of recurrent curvature tensor in terms of a nonzero vector and a tensor fields. Also, several oter teorems ave been derived. KEY WORDS: Riemannian space, Kaelerian manifold, H-projective recurrent, curvature tensors MSC: 46A13, 53B35, 32C15, 53C INTRODUCTION. A 2n- dimensional Kaelerian manifold K n is a Riemannian space, if it admits a structure tensor F i satisfying (Yano, 1965): (1.1) F i F j = - j i, (1.2) F ij = -F ji, (F ij = F a i g aj ) and (1.3) F i, j = 0, Were te comma (,) followed an index denotes covariant differentiation wit respect to te metric tensor g ij of te Riemannian space. Te Riemannian curvature tensor field R ijk, is given by (1.4) R ijk = i { j k } - j { i k } + { i l } { j l k } { j l } { i l k }, Wereas te Ricci tensor and te scalar curvature are respectively given by R ij = R a aij and R = R ij g ij. Te Ricci tensor satisfies te following identities: (1.5) F a i R aj = - R ia F a j, (1.6) F a i R j a = R a i F j a. (1.7) F i a R a b F b j = - R i j. (1.8) F i a R a i = -R j a F a j. (1.9) F a i R i a = 0. If we now define a curvature tensor S ij by (1.10) S ij = - F a i R aj. Ten we ave (1.11) S ij = - S ji. And (1.12) F a i S aj = -S ia F a j. It is well known tat tese tensors satisfy te identity given by (Tacibana 1967) Page 489 Researc Guru: Online Journal of Multidisciplinary Subjects (Peer Reviewed)

2 Researc Guru: Volume-12, Issue-4, Marc-2019 (ISSN: X) (1.13) R a ijk, a = R jk, i R ik, j Te Kaelerian olomorpically projective recurrent curvature tensor P ijk, are given by (Sina 1973) 1 (1.14) P ijk = R ijk (R ik j R jk i + S ik F j S jk F i + 2S ij F k), n + 2 Te Bianci identities in K n are given by (1.15) R ijk + R jki + R kij = 0 and (1.16) R ijk,a + R ika,j + R iaj,k = 0. Definition (1.1): A Kaelerian space is said to be recurrent, if we ave (Sing 1971) (1.17) R ijk, a - a R ijk = 0, for some non-zero recurrence vector a, and is called semi-recurrent (or Riccirecurrent), if it satisfies (1.18) R ij, a - a R ij = 0. Multiplying te above equation by g ij, we get (1.19) R, a - a R = 0 Remark (1.1): From (1.2) it follows tat every Kaelerian recurrent space is Kaelerian Ricci-recurrent space but te converse is not necessarily true. 2. DECOMPOSITION OF RECURRENT CURVATURE TENSOR FIELDS IN A KAEHLERIAN MANIFOLD OF FIRST ORDER. We consider te decomposition of recurrent curvature tensor field R ijk in te following form (Sing, 1982): (2.1) R ijk = X l v l Φ ijk, Were v l is a non-zero vector field and X l, Φ ijk are two non-zero tensor fields suc tat (Negi and Gairola, 2010): (2.2) X l λ = P l and (2.3) λ v = 1 P 1 is called decomposed vector field and tis is a non-zero vector field. Definition (2.1): Te vector field λ a and te tensor field X l given by equations (1.17) and (2.1) beave like recurrent vector and recurrent tensor fields and teir recurrent forms are given by (Sing 1982) (2.4) λ a,m = μ m λ a and (2.5) X l,m = v m X l, Page 490 Researc Guru: Online Journal of Multidisciplinary Subjects (Peer Reviewed)

3 Researc Guru: Volume-12, Issue-4, Marc-2019 (ISSN: X) Were μ m and v m are non-zero recurrence vector fields. Definition (2.2): Under te decomposition (2.1), te decomposed vector fields P l is ave like a recurrent vector field and its recurrent form is given by (Sing 1982) (2.6) P l, m = (μ m + v m ) P l We sall prove te following: Teorem (2.1): Under te decomposition (2.1), te Bianci identities for R ijk take te forms (Takano, 1967) (2.7) Φ ijk + Φ jki + Φ kij = 0, (Φ ijk = - Φ ikj ) and (2.8) λ a Φ ijk + λ j Φ ika + λ k Φ iaj = 0. Proof: From equations (1.15), (1.16), (1.17) and (2.1), we get (2.9) X l v l (Φ ijk + Φ jki + Φ kij ) = 0 and (2.10) X l v l ( λ a Φ ijk + λ j Φ ika + λ k Φ iaj ) = 0. Te identities (2.7) and (2.8) follow immediately from tese equations and te fact X l v l 0. Teorem (2.2): Under te decomposition (2.1), te vector field v l and te tensor field Φ ijk be ave like recurrent vector and recurrent tensor fields. obtain Proof: Multiplying equation (2.8) by v a and using relation (2.3), we (2.11) Φ ijk = λ k Φ ij λ j Φ ik Were Φ ijk v k = Φ ij is a tensor fields. Terefore, te relation (2.1) takes te form (2.12) R ijk = X l v l ( λ k Φ ij λ j Φ ik ) Differentiating equation (2.12) covariantly wit respect to x m and using equations (1.17), (2.4), (2.5), (2.12), we get (2.13) ( λ k Φ ij λ j Φ ik ) v l, m = v l [v m ( λ j Φ i k λ k Φ ij ) +μ m ( λ j Φ i k λ k Φ ij )] Multiplying tis equation by v a, we obtain Page 491 Researc Guru: Online Journal of Multidisciplinary Subjects (Peer Reviewed)

4 Researc Guru: Volume-12, Issue-4, Marc-2019 (ISSN: X) (2.14) ( λ k Φ ij λ j Φ ik ) v a v l, m = v l v a [v m ( λ j Φ i k λ k Φ ij ) +μ m ( λ j Φ i k λ k Φ ij )] Tis yield (2.15) v a, v l, m = v l v a,m Since v l 0, tere exists a proportional non-zero vector field π m suc tat (2.16) v l, m = π m v l Terefore, v l is recurrent vector field. Furter, differentiating equation (2.1) covariantly wit respect to x m and using equations (1.17), (2.1), (2.4), (2.5), (2.11), we obtain (2.17) Φ ijk, m = ( λ m v m π m ) Φ ijk Hence, Φ ijk is recurrent tensor field. If v m + π m 0, we ave Teorem (2.3): Under te decomposition (2.1), te vector field X l v l is recurrent wit te recurrence vector field (v m + π m ). Proof: Differentiating te vector field X l v l covariantly wit respect to x m and using equation (2.5) and (2.16), we get te proof. On te oter and, if v m + π m = 0, we ave Teorem (2.4): Under te decomposition (2.1), Φ ijk will be recurrent wit te same recurrence vector λ m as te curvature tensor field R ijk. Proof: Te proof follows immediately from equation (2.17). Teorem (2.5): Under te decomposition (2.1), te vector field v l and tensor fields R ijk, R ij, Φ ijk satisfying te relations (2.18) λ R ijk = λ i R jk λ j R ik = P l v l Φ ijk Proof: Wit te elp of equations (1.13), (1.17) and (1.18), we obtain (2.19) λ R ijk = λ i R jk λ j R ik Multiplying equations (2.1) by λ and using relation (2.2), we obtain (2.20) λ R ijk = P l v l Φ ijk From equations (2.19) and (2.20), we get te relations (2.18). Page 492 Researc Guru: Online Journal of Multidisciplinary Subjects (Peer Reviewed)

5 Researc Guru: Volume-12, Issue-4, Marc-2019 (ISSN: X) Teorem (2.6): Under te decomposition (2.1), te curvature tensor R ijk and Kaelerian olomorpically projective curvature tensor fields are equal iff (2.21) δ j Φ ik δ i Φ jk + Φ ak (F j F a i F i F a j ) + 2 F kf a i Φ aj = 0. Proof: Equation (1.14) may be expressed in te form (2.22) P ijk = R ijk + D ijk were 1 (2.23) D ijk = ( R ik j R jk i + S ik F j S jk F i + 2S ij F k), n+2 Contracting indices and k in (2.1), we obtain (2.24) R ij = X k l v l Φ ijk, Wit te elp of equation (2.24), we ave (2.25) S ij = F a i R aj = F a i X r l v l Φ ajr, Making use of equations (2.24) and (2.25) in (2.23), we obtain X r l v l (2.26) D ijk = [Φ ikr j Φ jkr i + Φ akr (F j F a i F i F a j) (n+2) + 2 F k F a i Φ ajr ] From equation (2.23), it is clear tat P ijk = R ijk, iff D ijk =0, wic in view of equation (2.26) becomes (2.27) Φ ikr j Φ jkr i + Φ akr (F j F a i F i F a j) + 2 F kf a i Φ ajr = 0. Multiplying tis equation by v r and using te relation Φ ijk v k = Φ ij, We ave te required equation. Page 493 Researc Guru: Online Journal of Multidisciplinary Subjects (Peer Reviewed)

6 Researc Guru: Volume-12, Issue-4, Marc-2019 (ISSN: X) REFERENCES [1]. Yano, K., Differential Geometry on Complex and Almost Complex Spaces. Pergramon Press, London (1965). [2]. Takano, K., Decomposition of curvature tensor in a recurrent space, Tensor (N.S.), Vol.18, No.3, (1967), pp [3]. Sing, S.S., On Kaelerian spaces wit recurrent Bocner curvature. Acc. Naz. Dei Lincei, Rend, Vol. 51, No. (3, 4), (1971), pp [4]. Sina, B.B., On H-curvature tensors in Kaeler manifold. Kyungpook Mat. J., 13, No.2, (1973), pp [5]. Sing, A.K., On decomposition of recurrent curvature tensor fields in a Kaelerian space. Journ. Ind. Mat. Soc., Vol. 46, (1982), pp [6]. Negi,U.S. and Kailas Gairola, On decomposition of curvature tensor fields in a Kaelerian recurrent space, International Trans. in Matematical Sciences and Computer, Volume 3, No.1, (2010), pp [7] Negi.U.S., Te Study of Structure and Operators on Almost Kaelerian Spaces, Global Journal of Teoretical and Applied Matematics Sciences (GJTAM), Volume 6, Number 2 (2016), pp [8] Negi, U.S., On te Topology of Almost Kaelerian Manifolds, Ultra Scientist, Vol. 28(1) A, (2016), pp [9] Negi, U.S., On Kaelerian conarmonic * bi-recurrent spaces, Jnanaba, Vol. 47(II), (2017), pp [10] Negi, U.S., Geometrical interpretation of an analytic HP-transformation in almost Kaelerian spaces, International Journal of Matematical Arcive- 9(1), (2018), pp Page 494 Researc Guru: Online Journal of Multidisciplinary Subjects (Peer Reviewed)