CR-submanifolds of Lorentzian manifolds

International Electronic Journal of Geometry Volume 2 No. 1 pp. 95-103 (2009) c °IEJG CR-SUBMANIFOLDS OF LORENTZIAN MANIFOLDS PABLO ALEGRE (Communicated by Cihan ¨ OZG¨ UR) Abstract. This paper is about the talk given in the VIth Geometry Symposium in Bursa, Turkey, on July 2008. We present the notion of CR-submanifolds of a Lorentzian almost contact manifold, study their principal characteristics and the particular cases in which the manifold is a Lorentzian Sasakian manifold or a Lorentzian Sasakian space form. We also present some new results for CR-submanifolds of a generalized Lorentzian Sasakian space form. 1. Introduction CR-submanifolds were introduced by A. Bejancu in [5]. Given a submanifold, M, of a Kaehler manifold, e N(J, g), for every tangent vector, X∈TM, let us write JX =TX +NX, where TX and NX are the tangent and normal component respectively. Mis called a CR-submanifold if TM admits a decomposition in two distributions, TM =D⊕D⊥, where Dis invariant, JD ⊆D, and D⊥is anti-invariant, JD⊥⊆T⊥M. These submanifolds generalized both invariant and anti-invariant ones. D.E. Blair and B.Y. Chen introduced the notion of CR-submanifold of an Hermitian manifold, and many other authors have amplified this study by considering different structures on the ambient manifold, as M. Barros and F. Urbano dealing with generalized complex space forms, [4], or considering other decompositions of the tangent bundle as generic and skew CR submanifolds, [17]. For a contact Riemannian manifold, f M(φ, η, ξ, g), A. Bejancu and N. Papaghiuc, [6], consider a submanifold Mtangent to ξ, they called it semi-invariant if TM = D⊕D⊥⊕< ξ >, where the first distribution is invariant and the second antiinvariant. M. Kobayashi defined a CR-submanifold of a Sasakian manifold if T M = D⊕D⊥, [14]. Obviously both definitions are the same for submanifolds tangent to ξ. From now on we only consider submanifolds tangent to ξ. 2000 Mathematics Subject Classification. 53C40, 53C50. Key words and phrases. Lorentzian metric, contact and almost contact Lorentzian manifolds, Lorentzian Sasakian, Lorentzian Sasakian space form, generalized Lorentzian Sasakian space form, invariant, anti-invariant and CR-submanifolds. The author is supported by the PAI project (Junta de Andaluc´ıa, Spain, 2008). 95 96 PABLO ALEGRE Also A. Bejancu and N. Papaghiuc studied CR-submanifolds of a Sasakian space form, [7] while other authors have leaded with other decompositions of the tangent bundle like semi-slant submanifolds and generalized CR-submanifolds, [9] and [15]. On semi-defined geometry, Kalpana and G. Guha initiated the study of semiinvariant submanifolds of a Lorentzian para Sasakian manifold, [12]. Furthermore, H. Gill and K.K. Dube have recently introduced generalized CR-submanifolds of a trans Lorentzian Sasakian manifold,[11], and of a trans hyperbolic Sasakian manifold, [10]. Let us remember this structures. First, an odd dimensional manifold f M2n+1 is called a Lorentzian almost para contact manifold if it is doted with a structure (φ, ξ, η, g) with φ2X=X+η(X)ξ, η(ξ) = −1, g(φX, φY ) = g(X, Y ) + η(X)η(Y, )η(X) = g(X, ξ), and it is called Lorentzian para Sasakian if ( e ∇Xφ)Y=g(X, Y )+η(Y)X+2η(X)η(Y)ξ. (f M2n+1, φ, ξ, η, g) is called a hyperbolic almost contact manifold if φ2X=X+η(X)ξ, η(ξ) = −1, g(φX, φY ) = −g(X, Y )−η(X)η(Y), η(X) = g(X, ξ). Our purpose is to define CR-submanifolds of a Lorentzian almost contact manifold, that is a manifold with an almost contact structure and a compatible Lorentzian metric, (f M2n+1, φ, ξ, η, g), (1.1) φ2X=−X+η(X)ξ, η(ξ)=1, g(φX, φY ) = g(X, Y ) + η(X)η(Y), η(X) = −g(X, ξ). It is Lorentzian Sasakian if (1.2) (e ∇Xφ)Y=−g(X, Y )ξ−η(Y)X. After the introduction, Section 2 contains some preliminaries results, in Section 3, we define CR-submanifolds of a Lorentzian almost contact manifold, and study the integrability of some distributions. In Section 4, we characterized CR-submanifolds of Lorentzian Sasakian space forms, and we do the same with generalized Lorentzian Sasakian space forms in Section 5. Most of the results of Section 3 and 4 appear in an earlier paper [1]. 2. Preliminaries Let us consider a submanifold Mof a Lorentzian almost contact metric manifold (f M, φ, ξ, η, g), tangent to the structure vector field ξ. Put φX =TX +NX for any tangent vector field X, where TX (resp. NX) denotes the tangential (resp. normal) component of φX. Similarly, φV =tV +nV for any normal vector field Vwith tV tangent and nV normal to M. Given a submanifold Mof a Lorentzian almost contact manifold (f M, φ, ξ, η, g), we also use gfor the induced metric on M. We denote by e ∇the Levi-Civita connection on f Mand by ∇the induced Levi- Civita connection on M. Thus, the Gauss and Weingarten formulas are respectively given by (2.1) e ∇XY=∇XY+h(X, Y ), (2.2) e ∇XV=−AVX+∇⊥ XV, CR-SUBMANIFOLDS OF LORENTZIAN MANIFOLDS 97 for vector fields X, Y tangent to Mand a vector field Vnormal to M, where hdenotes the second fundamental form, ∇⊥the normal connection and AVthe shape operator in the direction of V. The second fundamental form and the shape operator are related by (2.3) g(h(X, Y ), V ) = g(AVX, Y ). The submanifold Mis said to be totally geodesic if hvanishes identically. 3. CR-submanifolds, example and integrability conditions In this section, we introduce the notion of CR-submanifold of a Lorentzian almost contact manifold. This definition generalizes the notions of both invariant and anti-invariant submanifolds and it is the equivalent one to CR-submanifolds on the Riemannian setting. We characterize the anti-invariant case and present different examples of CR-submanifolds in a Lorentzian almost contact manifold. Finally, we study the integrability of all the distributions involved in the definition of CR- submanifolds. A submanifold, M, of a Lorentzian almost contact manifold, (f M2n+1, φ, η, ξ, g), is called a CR-submanifold if TM =D⊕D⊥⊕< ξ >, with Dan invariant distribution and D⊥an anti-invariant distribution. Note that ξis a timelike vector field and all vector fields in D⊕D⊥are spacelike. A CR-submanifold is also characterized by the decomposition of the normal bundle. Proposition 3.1. [1] Let Mbe a submanifold of a Lorentzian almost contact manifold (f M, φ, ξ, η, g). Then, Mis a CR-submanifold if and only if T⊥M=D⊕D⊥; where Dis the maximal anti-invariant space in T⊥Mand D⊥is its orthogonal complement on T⊥M. A CR-submanifold is known to be invariant,anti-invariant and proper if D⊥= 0, D= 0 and D6= 0 6=D⊥respectively. And it is called vertical proper if D6= 0 6= φD⊥. We can prove the next proposition about the distribution D. Proposition 3.2. [1] Let Mbe a CR-submanifold of a normal Lorentzian Sasakian manifold. Then, Dis even dimensional. The anti-invariant case is characterized in the next result. Theorem 3.1. [1] Let Mbe a submanifold of a normal Lorentzian Sasakian manifold. Mis anti-invariant if and only if ∇T= 0. The next theorem, a Lorentzian version of the ones offered in [9], helps us to construct a number of examples of CR-submanifolds. Theorem 3.2. [1] Let Mbe a CR-submanifold of an almost Hermitian manifold (e N2n, G, J). Then, M×Ris a CR-submanifold of (N2n×R, φ, ξ, η, g). But we can also present some other direct examples, they are based on semi-slant submanifolds of a Riemannian Sasakian manifold given on [9]. Let us consider on R2m+1 the following normal Lorentzian Sasakian structure (φ0, ξ, η, g), given by η=1 2Ãdz − m X i=1 yidxi!, ξ =∂ ∂z , 98 PABLO ALEGRE g=−η⊗η+1 4 m X i=1 (dxi⊗dxi+dyi⊗dyi), φ0µXi ∂ ∂xi+Yi ∂ ∂yi+Z∂ ∂z ¶= m X i=1 µYi ∂ ∂xi−Xi ∂ ∂yi¶+ m X i=1 Yiyi∂ ∂z , where {xi, yi, z},i= 1, ..., m are the cartesian coordinates. The first example is with odd dimensional D⊥: Example 3.1. [1] The equation x(u, v, w, s, t) = 2(u, 0, w, 0, v, 0,0, s, t), defines a CR-submanifold in R9with its normal Lorentzian Sasakian structure, (φ0, ξ, η, g). To prove this fact, we take the orthogonal basis e1= 2 µ∂ ∂x1 +y1 ∂ ∂z ¶, e2= 2 ∂ ∂y1 , e3= 2 µ∂ ∂x3 +y3 ∂ ∂z ¶, e4= 2 ∂ ∂y4 , e5= 2 ∂ ∂z =ξ, and define the distributions D=< e1, e2>, and D⊥=< e3, e4> . It is clear that TM =D⊕D⊥⊕< ξ >. And the second one with even dimD⊥: Example 3.2. [1] The equation x(u, v, s, t) = 2(u, v, s, 0, t),defines a CR-submani- fold in R5with its normal Lorentzian Sasakian structure (φ0, ξ, η, g). In this case, TM =D⊕D⊥⊕< ξ >, just taking D= Span ½2µ∂ ∂x1 +y1 ∂ ∂z ¶,2∂ ∂y1¾and D⊥= Span ½2µ∂ ∂x2 +y2 ∂ ∂z ¶¾. We can state the following theorems about the integrability of the distributions involved in the definition of a CR-submanifold. Theorem 3.3. [1] Let Mbe a CR-submanifold of a Lorentzian almost contact manifold with D6= 0. Then Dand D⊕D⊥are never integrable. Theorem 3.4. [1] Let Mbe a CR-submanifold of a normal contact Lorentzian manifold. Then, i) D⊥is always integrable, ii) D⊥⊕< ξ > is always integrable, iii) D⊕< ξ > is integrable if and only if h(X, TY )−h(Y, TX) = 0 for all X, Y ∈D⊕< ξ >. Studying the last condition on the above theorem we get this result, which was suggested by Prof. B.Y. Chen during the talk. Theorem 3.5. Let Mbe a CR-submanifold of a normal Lorentzian Sasakian manifold. If h(X, TY )−h(Y, T X)=0for all X, Y ∈D⊕< ξ >, then D⊕< ξ > is minimal. Proof. As we prove in Theorem 3.2, Dis even dimensional. For every X∈D, φX =TX ∈D, so we can choose an orthonormal basis on D⊕< ξ >:{e1, ..., e2r+1} with er+i=Teifor i= 1, ..., r and e2r+1 =ξ. Because of the condition, h(X, TY ) = h(Y, TX), so h(TX, TY ) = h(Y, T 2X) = −h(Y, X). CR-SUBMANIFOLDS OF LORENTZIAN MANIFOLDS 99 And for a Lorentzian Sasakian manifold, e ∇Xξ=φX, so h(X, ξ) = NX and h(ξ, ξ) = Nξ = 0. So we have prove that h(Tei, Tei) = −h(ei, ei) and h(ξ, ξ) = 0. So 2r+1 X i=1 h(ei, ei) = 0 and D⊕< ξ > is minimal. ¤ 4. CR-submanifolds of Lorentzian Sasakian space forms ALorentzian Sasakian space form is a Lorentzian Sasakian manifold with constant φ-sectional curvature. We can characterize proper CR-submanifolds of a normal contact Lorentzian manifold with constant φ-sectional curvature. The following theorems are equivalent for normal Lorentzian Sasakian space forms to the ones proved by M. M. Tripathi, [18], for generalized complex space forms and [16] for Sasakian space foms. We use the expression of the Riemann curvature tensor of a Lorentzian contact manifold with constant φ-sectional curvature kgiven by T. Ikawa in [13]: e R(X, Y )Z=k−3 4{g(Y, Z)X−g(X, Z)Y}+ +k+ 1 4{g(X, φZ)φY −g(Y, φZ)φX + 2g(X, φY )φZ}+ (4.1) +k+ 1 4{η(Y)η(Z)X−η(X)η(Z)Y+g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξ}. Theorem 4.1. [1] Let Mbe a submanifold tangent to ξof a Lorentzian Sasakian space form f M(k)with k6=−1. Then, Mis a vertical proper CR-submanifold if and only if the maximal invariant subspaces Dx=TxM∩φ(TxM),x∈M, define a non-trivial subbundle Dof TM such that e R(D, D, D⊥, D⊥) = 0, where D⊥denotes the non-trivial orthogonal complementary subbundle of Din TM. Theorem 4.2. [1] Let Mbe a submanifold tangent to ξof a Lorentzian Sasakian space form f M(k)with k6= 0. Then, Mis a proper CR-submanifold if and only if the maximal anti-invariant subspaces D⊥ xof TxM,x∈M, define a non-trivial subbundle D⊥of TM such that e R(D, φD, D, D) = 0, where Ddenotes the orthogonal complementary subbundle of D⊥⊕< ξ > in TM, and Ddenotes the orthogonal complementary of φD⊥in T⊥M. Using the same reasoning but for distributions on the normal bundle we get some results. Theorem 4.3. [1] Let Mbe a submanifold tangent to ξof a Lorentzian Sasakian space form f M(k)with k6=−1. Then Mis a vertical proper CR-submanifold if and only if the maximal invariant subspaces Dx=T⊥ xM∩φ(T⊥ xM),x∈M, of T⊥ xM define a non-trivial subbundle Dof T⊥Msuch that e R(D, D, D⊥, D⊥) = 0, 100 PABLO ALEGRE where D⊥denotes the non-trivial orthogonal complementary subbundle of Din T⊥M. And also, Theorem 4.4. [1] Let Mbe a submanifold tangent to ξof a Lorentzian Sasakian space form f M(k)with k6= 0. Then, Mis a proper CR-submanifold if and only if the maximal anti-invariant subspaces D⊥ xof T⊥ xM,x∈M, define a non-trivial subbundle D⊥of T⊥Msuch that e R(D, φD, D, D) = 0, where Ddenotes the non-trivial orthogonal complementary subbundle of φD⊥⊕< ξ > in T M, and Ddenotes the non-trivial orthogonal complementary of D⊥in T⊥M. 5. CR-submanifolds of generalized Lorentzian Sasakian space forms A Lorentzian almost contact manifold is called generalized Lorentzian Sasakian space form if it has got pointwise constant φ-sectional curvature, [2]. Such a submanifold is characterized by the expression of the Riemann curvature tensor: e R(X, Y )Z=f1{g(Y, Z)X−g(X, Z)Y}+ +f2{g(X, φZ)φY −g(Y, φZ)φX + 2g(X, φY )φZ}+ (5.1) +f3{η(Y)η(Z)X−η(X)η(Z)Y+g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξ}, for certain functions f1, f2and f3. We denote this submanifolds by f M(f1, f2, f3). As we did for Lorentzian Sasakian space form we characterized CR-submanifolds of generalized Lorentzian Sasakian space forms because of the Riemann curvature tensor and its action over some distributions. Theorem 5.1. Let Mbe a submanifold tangent to ξof a generalized normal Lorentzian Sasakian space form f M(f1, f2, f3)with f26= 0. Then, Mis a vertical proper CR-submanifold if and only if the maximal invariant subspaces Dx= TxM∩φ(TxM),x∈M, define a non-trivial subbundle Dof TM such that e R(D, D, D⊥, D⊥) = 0, where D⊥denotes the non-trivial orthogonal complementary subbundle of Din TM. Proof. Let Mbe a CR-submanifold, in virtue of (5.1), for every X, Y ∈Dand Z, V ∈D⊥we get R(X, Y, Z, W) = 0. For the converse, let suppose that the maximal invariant subspace and its complementary subbundle in T M verify the equation above. For all X∈Dand Z, W ∈D⊥, e R(X, φX, Z, W) = −2f2g(φX, φX)g(φZ, W) = −2f2g(X, X)g(φZ, W )=0. Since Xis space-like and f26= 0, g(φZ, W) = 0 and φD⊥is orthogonal to D⊥. Moreover φD⊥is orthogonal to D,g(X, φZ) = −g(φX, Z) = 0 because Dis an invariant distribution, and of course φD⊥is orthogonal to < ξ >. Therefore, φD⊥⊆TM⊥, and then D⊥is anti-invariant. So, TM admits a decomposition D⊕D⊥⊕< ξ >, with both an invariant and an anti-invariant distribution, and M is a CR-submanifold. ¤ CR-SUBMANIFOLDS OF LORENTZIAN MANIFOLDS 101 Theorem 5.2. Let Mbe a submanifold tangent to ξof a generalized normal Lorentzian Sasakian space form f M(f1, f2, f3)with f16=−3f2. Then, Mis a proper CR-submanifold if and only if the maximal anti-invariant subspaces D⊥ xof TxM, x∈M, define a non-trivial subbundle D⊥of TM such that e R(D, φD, D, D) = 0, where Ddenotes the orthogonal complementary subbundle of D⊥⊕< ξ > in TM, and Ddenotes the orthogonal complementary of φD⊥in T⊥M. Proof. The direct assertion it is a straightforward computation. For the converse, let suppose that the maximal anti-invariant subbundle of TxMand its complementary verify the equation above. Then, 0 = e R(X, φX, V, X) = (3f2+f1)g(X, X)g(φX, V ), for X∈D,V∈D. First, as 3f2+f16= 0, g(φX, V ) = 0 for all X∈Dand V∈D, that is φD is orthogonal to D. Since D⊥is anti-invariant, for every X∈D,Z∈D⊥, g(φX, Z) = −g(X, φZ) = 0, so φD is orthogonal to D⊥. And by hypothesis, Dis orthogonal to ξ. Therefore, we deduce that φD ⊂TM, and then φD =D. That is, Dis invariant and Mis a CR-submanifold. ¤ In virtue of Proposition 3.1 a CR-submanifold is characterized by the decomposition of the normal bundle. So again studying the relation between e Rand the distributions of the normal bundle we get some results. Theorem 5.3. Let Mbe a submanifold tangent to ξof a generalized normal Lorentzian Sasakian space form f M(f1, f2, f3)with f26= 0. Then Mis a vertical proper CR-submanifold if and only if the maximal invariant subspaces Dx= T⊥ xM∩φ(T⊥ xM),x∈M, of T⊥ xMdefine a non-trivial subbundle Dof T⊥Msuch that e R(D, D, D⊥, D⊥) = 0, where D⊥denotes the non-trivial orthogonal complementary subbundle of Din T⊥M. Proof. If Mis a CR-submanifold, let be N, U ∈Dand V, W ∈D⊥. Then R(N, U, V, W) = =f2(g(φU, V )g(φN, W )−g(N, φV )g(φU, W)+2g(N, φU)g(φV, W)) = 0, because g(φU, V ) = g(φN, V ) = g(φV, W) = 0. Conversely, let be Dand D⊥be on the hypothesis. For N∈Dand U, V ∈D⊥ using (5.1): 0 = e R(N, φN, U, V ) = −f2g(N, N)g(φU, V ). Since f26= 0 and Nis space-like, it must be g(φU, V ) = 0, so φD⊥is orthogonal to D⊥. But always φD⊥is orthogonal to D, because g(φU, N) = −g(U, φN) = 0. Then φD⊥⊂TM and it is anti-invariant. Therefore, T⊥Madmits a decomposition D⊕D⊥where one is an invariant distribution and the other is an anti-invariant one, and that implies, by virtue of Proposition 3.1, that Mis a CR-submanifold. ¤ 102 PABLO ALEGRE Theorem 5.4. Let Mbe a submanifold tangent to ξof a generalized normal Lorentzian Sasakian space form f M(f1, f2, f3)with f16=−3f2. Then, Mis a proper CR-submanifold if and only if the maximal anti-invariant subspaces D⊥ xof T⊥ xM, x∈M, define a non-trivial subbundle D⊥of T⊥Msuch that e R(D, φD, D, D) = 0, where Ddenotes the non-trivial orthogonal complementary subbundle of φD⊥⊕< ξ > in T M, and Ddenotes the non-trivial orthogonal complementary of D⊥in T⊥M. Proof. It is just a simple computation to prove the direct assertion. Conversely, suppose that both distributions, Dand D⊥, satisfy the equation above, let us prove that D⊥is anti-invariant. Then, in virtue of Proposition 3.1, Mis a CR- submanifold. Let be X∈Dand V∈D, 0 = e R(V, φV, X, V ) = (f1+ 3f2)g(V, V )g(φV, X). As f16=−3f2and Vis a space-like vector field, this implies g(φV, X) = 0 and then φD ⊥D. Of course, φD is orthogonal to Dand ξ. Finally, it is also orthogonal to φD⊥, because, given W∈D⊥, it is g(φV, φW ) = g(V, W )=0.Thus, φD ⊆D, and then invariant. Therefore, T⊥M=D⊕D⊥, one is invariant and the other anti-invariant, which the proof finishes. ¤ Finally, we present an example of a CR-submanifold of a generalized Lorentzian Sasakian space form. The construction is similar to the one made in Theorem 3.2 but now using warped products. First we remember how to obtain an example of generalized Lorentzian Sasakian space form. Theorem 5.5. [2] Let e N2n(F1, F2)be a generalized complex space form. Then the warped product f M2n+1 =R×fe N, endowed with the Lorentzian almost contact structure (φ, ξ, η, gf), is a generalized Lorentzian Sasakian space form, f M(f1, f2, f3), with functions f1=(F1◦π) + f02 f2, f2=(F2◦π) f2, f3=(F1◦π) + f02 f2−f00 f. Theorem 5.6. Let Mbe a CR-submanifold of an almost Hermitian manifold (e N2n, G, J). Then, M×fRis a CR-submanifold of (e N2n×fR, φ, ξ, η, g). Proof. TM can be decompose in D⊕D⊥with Dinvariant and D⊥anti-invariant. Therefore, T(M×fR) = D⊕D⊥⊕< ξ > with D={(U, 0)/U ∈D}and D⊥= {(V, 0)/V ∈D⊥}. It only rests proving that Dis invariant and D⊥is anti-invariant. First, for any (U, V )∈D φ(U, 0) = (JU, 0) ∈D so Dis invariant. Moreover, φ(V, 0) = (JV, 0) is normal to T(M×fR) so D⊥is anti-invariant. ¤ CR-SUBMANIFOLDS OF LORENTZIAN MANIFOLDS 103 References [1] Alegre, P., CR-submanifolds of normal Lorentzian Sasakian manifolds, Submitted. [2] Alegre, P. and Carriazo, A., Lorentzian Sasakian manifolds with constant pointwise φsectional curvature, Submitted. [3] Alegre, P., Blair, D.E. and Carriazo, A., Generalized Sasakian-space-forms, Israel J. Math., 141(2004), 157-183. [4] Barros, M. and Urbano, F., CR-submanifolds of generalized complex space forms, An. S¸ti. Univ. “Al. I. Cuza” Ia¸si Sect¸. 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Department of Economics, Cuantitative Methods and Economic History. Statistic and Operations Research Area, University Pablo de Olavide, Sevilla-Spain E-mail address:[email protected]