The L-sectional curvature of S-manifolds

Konuralp Journal of Mathematics Volume 4 No. 1 pp. 246–253 (2016) c KJM THE L-SECTIONAL CURVATURE OF S-MANIFOLDS MEHMET AKIF AKYOL, LUIS M. FERN´ ANDEZ, AND ALICIA PRIETO-MART´ IN Abstract. We investigate L-sectional curvature of S-manifolds with respect to the Riemannian connection and to certain semi-symmetric metric and nonmetric connections naturally related with the structure, obtaining conditions for them to be constant and giving examples of S-manifolds in such conditions. Moreover, we calculate the scalar curvature in all the cases. 1. Introduction. In 1963, Yano [13] introduced the notion of f-structure on a C∞(2n+s)- dimensional manifold M, as a non-vanishing tensor field fof type (1,1) on Mwhich satisfies f3+f= 0 and has constant rank r= 2n. Almost complex (s= 0) and almost contact (s= 1) are well-known examples of f-structures. The case s= 2 appeared in the study of hypersurfaces in almost contact manifolds [5, 8] and it motivated that, in 1970, Goldberg and Yano [9] defined globally framed f-structures (also called f.pk-structures), for which the subbundle ker fis parallelizable. Then, there exists a global frame {ξ1, . . . , ξs}for the subbundle ker f(the vector fields ξ1, . . . , ξsare called the structure vector fields), with dual 1-forms η1, . . . , ηs. Thus, we can consider a Riemannian metric gon M, associated with a globally framed f-structure, such that g(fX, fY ) = g(X, Y )−Ps α=1 ηα(X)ηα(Y), for any vector fields X, Y in Mand then, the structure is called a metric f-structure. Therefore, TM splits into two complementary subbundles Imf(whose differentiable distribution is usually denoted by L) and ker fand, moreover, the restriction of f to Imfdetermines a complex structure. A wider class of globally framed f-manifolds (that is, manifolds endowed with a globally framed f-structure) was introduced in [3] by Blair according to the following definition: a metric f-structure is said to be a K-structure if the fundamental 2-form Φ, given by Φ(X, Y ) = g(X, fY ), for any vector fields Xand Yon M, is Date: January 1, 2013 and, in revised form, February 2, 2013. 2000 Mathematics Subject Classification. 53C05, 53C15. Key words and phrases. S-manifold, Semi-Symmetric Metric Connection, Semi-Symmetric non-Metric Connection, L-Sectional Curvature, Scalar Curvature. The second author is partially supported by the MEC projects MTM 2011-22621 (MEC, Spain, 2011) and MTM 2014-52197-P (MEC, Spain, 2014). 246 THE L-SECTIONAL CURVATURE OF S-MANIFOLDS 247 closed and the normality condition holds, that is, [f, f]+2 Ps α=1 dηα⊗ξα= 0, where [f, f] denotes the Nijenhuis torsion of f. A K-manifold is called an S-manifold if dηα= Φ, for all α= 1, . . . , s. If s= 1, an S-manifold is a Sasakian manifold. Furthermore, S-manifolds have been studied by several authors (see, for example, [4, 6, 10, 12]). It is well known that there are not exist S-manifolds (s≥2) of constant sectional curvature and, for Sasakian manifolds, the unit sphere is the only one. This is due to the fact that K(X, ξα) = 1 and K(ξα, ξβ) = 0, for any unit vector field X∈ L and any α, β = 1 ...,s. For this reason, it is interesting to study the sectional curvature of planar sections spanned by vector fields of L(called L-sectional curvature) and to obtain conditions for this sectional curvature to be constant. Further, in 1924 Friedmann and Schouten [7] introduced semi-symmetric linear connections on a differentiable manifold. Later, Hayden [11] defined the notion of metric connection with torsion on a Riemannian manifold. More precisely, if ∇is a linear connection in a differentiable manifold M, the torsion tensor Tof ∇is given by T(X, Y ) = ∇XY− ∇YX−[X, Y ], for any vector fields Xand Yon M. The connection ∇is said to be symmetric if the torsion tensor Tvanishes, otherwise it is said to be non-symmetric. In this case, ∇is said to be a semi-symmetric connection if T(X, Y ) = η(Y)X−η(X)Y, for any X, Y , where ηis a 1-form on M. Moreover, if gis a Riemannian metric on M,∇is called a metric connection if ∇g= 0, otherwise it is called non-metric. It is well known that the Riemannian connection is the unique metric and symmetric linear connection on a Riemannian manifold. Recently, S-manifolds endowed with a semi-symmetric either metric or non-metric connection naturally related with the S-structure have been studied in [1, 2]. In this paper, we investigate L-sectional curvature of S-manifolds with respect to the Riemannian connection and to the semi-symmetric metric and non-metric connections introduced in [1, 2], obtaining conditions for them to be constant and giving examples of S-manifolds in such conditions. Moreover, we calculate the scalar curvature in all the cases. 2. Preliminaries on S-manifolds. A (2n+s)−dimensional differentiable manifold Mis called a metric f-manifold if there exist a (1,1) type tensor field f,svector fields ξ1, . . . , ξs, called structure vector fields,s1-forms η1, . . . , ηsand a Riemannian metric gon Msuch that (2.1) f2=−I+ s X α=1 ηα⊗ξα, ηα(ξβ) = δαβ, fξα= 0, ηα◦f= 0, (2.2) g(fX, fY ) = g(X, Y )− s X α=1 ηα(X)ηα(Y), for any X, Y ∈ X(M), α, β ∈ {1, . . . , s}. In addition: (2.3) ηα(X) = g(X, ξα), g(X, fY ) = −g(fX, Y ). Then, a 2-form Φ is defined by Φ(X, Y ) = g(X, fY ), for any X, Y ∈ X(M), called the fundamental 2-form. In what follows, we denote by Mthe distribution spanned by the structure vector fields ξ1, . . . , ξsand by Lits orthogonal complementary 248 MEHMET AKIF AKYOL, LUIS M. FERN´ ANDEZ, AND ALICIA PRIETO-MART´ IN distribution. Then, X(M) = L⊕M. If X∈ M, then fX = 0 and if X∈ L, then ηα(X) = 0, for any α∈ {1, . . . , s}, that is, f2X=−X. In a metric f-manifold, special local orthonormal basis of vector fields can be considered: let Ube a coordinate neighborhood and E1a unit vector field on Uorthogonal to the structure vector fields. Then, from (2.1)-(2.3), fE1is also a unit vector field on Uorthogonal to E1and the structure vector fields. Next, if it is possible, let E2be a unit vector field on Uorthogonal to E1,fE1and the structure vector fields and so on. The local orthonormal basis {E1, . . . , En, fE1, . . . , fEn, ξ1, . . . , ξs},, so obtained is called an f-basis. Moreover, a metric f-manifold is normal if [f, f]+2 s X α=1 dηα⊗ξα= 0, where [f, f] denotes the Nijenhuis tensor field associated to f. A metric f-manifold is said to be an S-manifold if it is normal and η1∧ · · · ∧ ηs∧(dηα)n6= 0 and Φ = dηα,1≤α≤s. Observe that, if s= 1, an S-manifold is a Sasakian manifold. For s≥2, examples of S-manifolds can be found in [3, 4, 10]. If ∇is a linear connection on an S-manifold and Kdenotes the sectional curvature associated with ∇, the L-sectional curvature KLof ∇is defined as KL(X, Y ) = K(X, Y ), for any X, Y ∈ L. The scalar curvature of the S-manifold with respect to ∇is given by (2.4) τ=1 2 2n+s X i,j=1 K(ei, ej), for any local orthonormal frame {e1, . . . , e2n+s}of tangent vector fields to M. 3. The L-sectional curvature of S-manifolds. From now on, let Mdenote an S-manifold (M, f, ξ1, . . . , ξs, η1, . . . , ηs, g) of dimension 2n+s. We are going to study the sectional curvature of Mwith respect to different types of connections on M. 3.1. The case of the Riemannian connection. First, let ∇denote the Riemannian connection of g. For the sectional curvature Kof ∇, in [6] it is proved that (3.1) K(ξα, X) = R(ξα, X, X, ξα) = g(fX, fX), for any X∈ X(M) and α∈ {1, . . . , s}. Consequently, if s= 1, the unit sphere is the only Sasakian manifold of constant (sectional) curvature. If s≥2, from (3.1), we deduce that Mcannot have constant sectional curvature. For this reason, it is necessary to introduce a more restrictive curvature. In general, a plane section π on a metric f-manifold Mis said to be an f-section if it is determined by a unit vector X, normal to the structure vector fields and fX. The sectional curvature of πis called an f-sectional curvature. An S-manifold is said to be an S-space-form THE L-SECTIONAL CURVATURE OF S-MANIFOLDS 249 if it has constant f-sectional curvature cand then, it is denoted by M(c). The curvature tensor field Rof M(c) satisfies [12]: (3.2) R(X, Y, Z, W) = s X α,β=1 {g(fX, fW )ηα(Y)ηβ(Z) −g(fX, fZ)ηα(Y)ηβ(W) + g(fY, fZ)ηα(X)ηβ(W) −g(fY, fW)ηα(X)ηβ(Z)} +c+ 3s 4{g(fX, fW )g(fY, fZ)−g(fX, fZ)g(fY, fW )} +c−s 4{Φ(X, W )Φ(Y, Z)−Φ(X, Z)Φ(Y, W)−2Φ(X, Y )Φ(Z, W )}, for any X, Y, Z, W ∈ X (M). Therefore, if Mis an S-space-form of constant f-sectional curvature cand considering an f-basis, from (3.1) and (3.2), we deduce that the scalar curvature of M with respect to the curvature tensor field of the Riemanian connection ∇satisfies: τ=n(n−1)(c+ 3s) 2+n(c+ 2s). Now, in view of (3.1) it is interesting to investigate the conditions for KLto be constant. In this context, we observe that, if n= 1, KLis actually the f-sectional curvature. Moreover, for n≥2, we can prove the following theorem. Theorem 3.1. Let Mbe a (2n+s)-dimensional S-manifold with n≥2. If the L-sectional curvature KLwith respect to the Riemannian connection ∇is constant equal to c, then c=s. In this case, the scalar curvature of Mis: τ=ns(2n+ 1). Proof. It is clear that if KLis constant equal to c, then Mis an S-space-form M(c). Consequently, from (3.2), we have (3.3) KL(X, Y ) = c+ 3s 4+3(c−s) 4g(X, fY )2, for any orthonormal vector fields X, Y ∈ L. Now, since n≥2, we can choose X and Ysuch that g(X, fY ) = 0. Thus, from (3.3) we deduce c+ 3s 4=c, that is, c=s. Now, considering a local orthonormal frame of tangent vector fields such that e2n+α=ξα, for any α= 1, . . . , s, since K(ei, ej) = KL(ei, ej) = s,i, j = 1,...,2n, i6=j, and using (3.1) and (2.4), we get the desired result for the scalar curvature.  By using (3.2) and (3.3), we have: Corollary 3.2. Let M(c)be an S-space-form of constant f-sectional curvature c. Then, Mis of constant L-sectional curvature (equal to c) if and only if c=s 250 MEHMET AKIF AKYOL, LUIS M. FERN´ ANDEZ, AND ALICIA PRIETO-MART´ IN Example 3.3. Let us consider R2n+2+(s−1) with coordinates (x1...,xn+1, y1, . . . , yn+1, z1, . . . , zs−1) and with its standard S-structure of constant f-sectional curvature −3(s−1), given by (see [10]): ξα= 2 ∂ ∂zα , ηα=1 2 dzα− n+1 X i=1 yidxi!, α = 1, . . . , s −1, g= s−1 X α=1 ηα⊗ηα+1 4 n+1 X i=1 (dxi⊗dxi+dyi⊗dyi), fX = n+1 X i=1 (Yi ∂ ∂xi −Xi ∂ ∂yi ) + s−1 X α=1 n+1 X i=1 Yiyi ∂ ∂zα , where X= n+1 X i=1 (Xi ∂ ∂xi +Yi ∂ ∂yi ) + s−1 X α=1 Zα ∂ ∂zα is any vector field tangent to R2n+2+(s−1). Now, let S2n+1(2) be a (2n+ 1)-dimensional ordinary sphere of radius 2 and M=S2n+1(2) ×Rs−1a hypersurface of R2n+2+(s−1). Let ξs= n+1 X i=1 −yi ∂ ∂xi +xi ∂ ∂yi− n+1 X i=1 s−1 X α=1 y2 i ∂ ∂zα and ηs(X) = g(X, ξs), for any vector field Xtangent to M. Then, if we put e ξα=sξα;eηα=1 sηα;α= 1, . . . , s; e f=f;eg=1 sg+1−s s2 s X α=1 ηα⊗ηα, it is known ([10]) that (M, e f, e ξ1,...,e ξs,eη1,...,eηs,eg) is an S-space-form of constant f-sectional curvature c=s. Moreover, from (3.2), it is easy to show that the L-sectional curvature KLis also constant and equal to s. 3.2. The case of a semi-symmetric metric connection. In [1], a semi-symmetric metric connection on M, naturally related to the S-structure, is defined by (3.4) ∇∗ XY=∇XY+ s X j=1 ηj(Y)X− s X j=1 g(X, Y )ξj, for any X, Y ∈ X (M). For the sectional curvature K∗of ∇∗, the following theorem was proved in [1]: Theorem 3.4. Let Mbe an S-manifold. Then, the sectional curvature of ∇∗ satisfies (i) K∗(X, Y ) = K(X, Y )−s; (ii) K∗(X, ξα) = K∗(ξα, X)=2−s; (iii) K∗(ξα, ξβ) = K∗(ξβ, ξα)=2−s, for any orthonormal vector fields X, Y ∈ L and α, β ∈ {1, . . . , s},α6=β. THE L-SECTIONAL CURVATURE OF S-MANIFOLDS 251 Therefore, from Theorem 3.1, if s6= 2, an S-manifold cannot have constant sectional curvature with respect to the semi-symmetric metric connection defined in (3.4). For s= 2, M=S2n+1(2) ×Rendowed with the connection ∇∗and the S-structure given in Example 3.3 is an S-manifold of constant sectional curvature (equal to 0) with respect to ∇∗. Moreover, for any s, by using Theorem 3.1 again and (i) of Theorem 3.4, if the L-sectional curvature associated with ∇∗is constant equal to c, then c= 0 and examples of such a situation are given in Example 3.3. In this case, the scalar curvature is given by: τ∗=(4ns +s(s−1))(2 −s) 2. Regarding the f-sectional curvature of ∇∗, from Theorem 4.5 in [1], we know that it is constant if and only if the f-sectional curvature associated with the Riemannian connection is constant too. In this case, if cdenotes the constant fsectional curvature of the Riemannian connection, c−sis the constant f-sectional curvature of ∇∗. Furthermore, from (i) of Theorem 3.4 and (3.3) it is easy to show that K∗ L(X, Y ) = c−s 4(1 + 3g(X, fY )2), for any orthonormal vector fields X, Y ∈ L. Therefore, considering an f-basis, we deduce that the scalar curvature of a (2n+s)-dimensional S-manifold of constant f-sectional curvature cwith respect to ∇∗satisfies: τ∗=n(n+ 1)(c−s) + (4ns +s(s−1))(2 −s) 2. 3.3. The case of a semi-symmetric non-metric connection. In [2], a semisymmetric non-metric connection on M, naturally related to the S-structure, is defined by e ∇XY=∇XY+ s X j=1 ηj(Y)X, for any X, Y ∈ X(M). To consider the sectional curvature of e ∇has no sense because e R(ξα, X, X, ξα) = 1, while e R(X, ξα, ξα, X) = 2, for any unit vector field X∈ L and any α∈ {1, . . . , s}(see [2] for the details). However, for the L-sectional curvature e KL, we have that e KL(X, Y ) = KL(X, Y ), for any orthogonal vector fields X, Y ∈ L. Consequently, Theorem 3.3 and Example 3.3 can be applied here. In the case of constant L-sectional curvature (equal to s) and since e R(ξα, ξβ, ξβ, ξα) = 1, for any α, β ∈ {1, . . . , s},α6=β, the scalar curvature is given by: eτ= 2ns(n+ 1) + s(s−1) 2. Regarding the f-sectional curvature of e ∇, in [2] it is proved that it is constant if and only if the f-sectional curvature associated with the Riemannian connection is constant too. In this case, both constant are the same and the curvature tensor field of ∇is completely determined by c. Furthermore, since from (3.3), e KL(X, Y ) = c+ 3s 4+3(c−s) 4g(X, fY )2, for any orthonormal vector fields X, Y ∈ L, considering an f-basis, we deduce that the scalar curvature of a (2n+s)-dimensional S-manifold of constant f-sectional 252 MEHMET AKIF AKYOL, LUIS M. FERN´ ANDEZ, AND ALICIA PRIETO-MART´ IN curvature cwith respect to e ∇satisfies: eτ=n(n+ 1)(c+ 3s) + s(s−1) 2. References [1] M. Akif Akyol, A. Turgut Vanli and L.M. Fern´andez, Curvature properties of a semi-symmetric metric connection defined on S-manifolds, Ann. Polonici Math., Vol:107, No.1 (2013), 71-86. [2] M. Akif Akyol, A. Turgut Vanli and L.M. Fern´andez, Semi-symmetry properties of S-manifolds endowed with a semi-symmetric non metric connection, to appear in An. Sti. Univ. “Al. I. Cuza” (Iasi) (2015). [3] D.E. Blair, Geometry of manifolds with structural group U(n)×O(s), J. Differ. Geom., Vol:4 (1970), 155-167. [4] D.E. Blair, On a generalization of the Hopf fibration, An. Sti. Univ. “Al. I. Cuza” (Iasi), Vol:17 (1971), 171-177. [5] D.E. Blair and G.D. Ludden, Hypersurfaces in almost contact manifolds, Tˆohoku Math. J., Vol:21 (1969), 354-362. [6] J.L. Cabrerizo, L.M. Fern´andez and M. Fern´andez, The curvature tensor fields on f-manifolds with complemented frames, An. Sti. Univ. “Al. I. Cuza” (Iasi), Vol:36 (1990), 151-161. [7] A. Friedmann and J.A. Schouten, ¨ Uber die Geometrie der halbsymmetrischen ¨ Ubertragung, Math. Zeitschr., Vol:21 (1924), 211-223. [8] S.I. Goldberg and K.Yano, Globally framed f-manifolds, Illinois J. Math., Vol:15 (1971), 456-474. [9] S.I. Goldberg and K.Yano, On normal globally framed manifolds, Tˆohoku Math. J., Vol:22 (1970), 362-370. [10] I. Hasegawa, Y. Okuyama and T. Abe, On p-th Sasakian manifolds, J. Hokkaido Univ. of Education (Section II A), Vol:37 (1986), 1-16. [11] H.A. Hayden, Subspaces of a space with torsion, Proc. London Math. Soc., Vol 34 (1932), 27-50. [12] M. Kobayashi and S. Tsuchiya, Invariant submanifolds of an f-manifold with complemented frames, Kodai Math. Sem. Rep., Vol:24 (1972), 430-450. [13] K. Yano, On a structure defined by a tensor field fof type (1,1) satisfying f3+f= 0, Tensor, N. S., Vol:14 (1963), 99-109. THE L-SECTIONAL CURVATURE OF S-MANIFOLDS 253 Bing¨ ol University, Faculty of Science and Art, Department of Mathematics, 12000, Bing¨ ol-TURKEY E-mail address:[email protected] Universidad de Sevilla, Facultad de Matem´ aticas, Departmento de Geometr´ ıa y Topolog´ ıa, C./ Tarfia s.n., 41012 Sevilla-SPAIN E-mail address:[email protected] Universidad de Sevilla, Facultad de Matem´ aticas, Departmento de Geometr´ ıa y Topolog´ ıa, C./ Tarfia s.n., 41012 Sevilla-SPAIN E-mail address:[email protected]