CUBO A Mathematical Journal Vol.18, No¯ 01, (59–68). December 2016 Submanifolds of a (k, µ)-Contact Manifold M.S. Siddesha, C.S. Bagewadi Department of Mathematics, Kuvempu University, Shankaraghatta - 577 451, Shimoga, Karnataka, INDIA. mssiddesha@gmail.com, prof bagewadi@yahoo.co.in ABSTRACT The object of the present paper is to study submanifolds of (k, µ)-contact manifolds. We find the necessary and sufficient conditions for a submanifolds of (k, µ)-contact manifolds to be invariant and anti-invariant. Also, we examine the integrability of the distributions involved in the definition of CR-submanifolds of (k, µ)-contact manifolds. RESUMEN El objeto del presente art́ıculo es estudiar subvariedades de variedades (k, µ)-contacto. Encontramos las condiciones necesarias y suficientes para que subvariedades de var- iedades (k, µ)-contacto sean invariantes y anti-invariantes. También examinamos la integrabilidad de las distribuciones involucradas en la definición de subvariedades CR de variedades (k, µ)-contacto. Keywords and Phrases: (k, µ)-contact manifold; invariant submanifold; anti-invariant subman- ifold. 2010 AMS Mathematics Subject Classification: 53C15, 53C40. 60 M.S. Siddesha, C.S. Bagewadi CUBO 18, 1 (2016) 1 Introduction In 1995 Blair, Koufogiorgos and Papantoniou [4] introduced the notion of contact metric manifolds with characteristic vector field ξ belonging to the (k, µ)-nullity distribution and such type of manifolds are called (k, µ)-contact manifolds. To study the geometry of an unknown manifold, it is sometime convenient and yet interesting to first imbed it into a rather known manifold and then study its geometry side by side that of the ambient manifold. This approach gave birth to the introduction of submanifold theory. The study of complex submanifolds of a Kähler manifold from differential geometric points of view was initiated by Calabi [7] in the early 1950’s and it was continued by several geometers like Blair, Ogiue [5], Chen, Verheyen [8], Yano, Kon [19], Yano, Ishihara [18], Kon [11] and many others. From then onwards submanifolds of a contact manifold have been major area of research and these submanifolds are divided into several types, mainly invariant, anti-invariant and semi-invariant. The study of submanifolds of different contact manifolds is carried out from 1970 onwards by several authors, for example [9]-[12], while the study of submanifolds of (k, µ)-contact manifold have been done by Montano et al [13], Avjit Sarkar et al [1], Tripathi et al [16], Siddesha and Bagewadi [14] and others. In [13], the authors have shown that invariant submanifolds of (k, µ)-contact manifold carries a (k, µ) structure and prove the totally geodesicity of invariant submanifolds when the second fundamental form is parallel. Later authors of [2] and [15] continued the work of above authors and they proved the totally geodesicity of recurrent, generalized recurrent of second fundamental form and semiparallel, pseudoparallel, Ricci-generalized pseudoparallel submanifolds. Motivated by these studies of the above authors [2, 13, 16], in the present paper we find the necessary and sufficient conditions for the submanifolds to be invariant and anti-invariant. Also we study CR-submanifolds of (k, µ)-contact manifold and examine the integrability of the horizontal and vertical distributions involved in the definition of CR-submanifolds of (k, µ)-contact manifold. The paper is organized as follows: In section 2, we give a brief account of (k, µ)-contact manifolds and necessary details about submanifolds. In section 3, we show the existence of an invariant and anti-invariant submanifold, while, the section 4 deals with non-existence of an anti-invariant submanifold. Lastly in section 5, we consider CR-submanifolds of (k, µ)-contact manifold with distributions D and D⊥, we find the conditions under which D⊥ is integrable or totally geodesic. 2 Preliminaries A contact manifold is a C∞-(2n + 1) manifold M̃2n+1 equipped with a global 1-form η such that η ∧ (dη)n ̸= 0 everywhere on M̃2n+1. Given a contact form η it is well known that there exists a unique vector field ξ, called the characteristic vector field of η, such that η(ξ) = 1 and dη(X, ξ) = 0 for every vector field X on M̃2n+1. A Riemannian metric g is said to be associated CUBO 18, 1 (2016) Submanifolds of a (k, µ)-Contact Manifold 61 metric if there exists a tensor field φ of type (1,1) such that φ2X = −X + η(X)ξ, η(ξ) = 1, η ◦ φ = 0, φξ = 0, (2.1) g(φX, φY) = g(X, Y) − η(X)η(Y), g(X, ξ) = η(X), (2.2) for all vector fields X, Y on M̃. Then the structure (φ, ξ, η, g) on M̃ is called a contact metric structure and the manifold equipped with such a structure is called a contact metric manifold [3]. We now define a (1, 1) tensor field h by h = 1 2 Lξφ, where L denotes the Lie differentiation, then h is symmetric and satisfies hφ = −φh. Further, a q-dimensional distribution on a manifold M is defined as a mapping D on M which assigns to each point p ∈ M, a q-dimensional subspace Dp of TpM. The (k, µ)-nullity distribution of a contact metric manifold M̃(φ, ξ, η, g) is a distribution N(k, µ) : p → Np(k, µ) = {Z ∈ TpM : R̃(X, Y)Z = k[g̃(Y, Z)X − g̃(X, Z)Y] + µ[g̃(Y, Z)hX − g̃(X, Z)hY]}, for all X, Y ∈ TM̃. Hence if the characteristic vector field ξ belongs to the (k, µ) nullity distribution, then we have R̃(X, Y)ξ = k[η(Y)X − η(X)Y] + µ[η(Y)hX − η(X)hY]. (2.3) The contact metric manifold satisfying the relation (2.3) is called (k, µ) contact metric manifold [4]. It consists of both k-nullity distribution for µ = 0 and Sasakian for k = 1. A (k, µ)-contact metric manifold M̃(φ, ξ, η, g) satisfies (∇̃Xφ)Y = g(X + hX, Y)ξ − η(Y)(X + hX), (2.4) for all X, Y ∈ TM̃, where ∇̃ denotes the Riemannian connection with respect to g. From (2.4), we have ∇̃Xξ = −φX − φhX, (2.5) for all X, Y ∈ TM̃. Again, if we put Ω(X, Y) = g(X, φY), then Ω is a skew-symmetric (0, 2) tensor field [4]. Thus we have from (2.5) Ω(X + hX, Y) = (∇̃Xη)(Y). (2.6) Also from (2.4), it follows that (∇̃ZΩ)(X, Y) = g(X, (∇̃Zφ)Y) = −g((∇̃Zφ)X, Y), (2.7) (∇̃ZΩ)(X, Y) = g(Z + hZ, Y)η(X) − η(Y)g(X, Z + hZ), (2.8) for any X, Y ∈ TM̃ Let M be a Riemannian submanifold of a (k, µ)-contact manifold M̃. Then the Gauss and Wein- 62 M.S. Siddesha, C.S. Bagewadi CUBO 18, 1 (2016) garten formulae are given by ∇̃XY = ∇XY + σ(X, Y), (2.9) ∇̃XN = −ANX + ∇ ⊥ XN, (2.10) for all X, Y ∈ TM and each N ∈ T⊥M, where ∇ is the Levi-Civita connection on M, ∇⊥ is the normal connection on the normal bundle T⊥M, σ is the second fundamental form of M and A is the shape operator with respect to the normal connection N. Then the shape operator A and the second fundamental form σ are related by g(σ(X, Y), N) = g(ANX, Y), (2.11) for all X, Y ∈ TM and N ∈ T⊥M. We denote by the same symbols g both metrics on M̃ and M. Definition 1. A submanifold M is said to be (i) totally geodesic in M̃ if σ = 0 or equivalently AN = 0 (2.12) for each N ∈ T⊥M. (ii) Minimal in M̃ if the curvature vector H satisfies H = Tr(σ) dimM = 0 (2.13) and (iii) totally umbilical if σ(X, Y) = g(X, Y)H. (2.14) 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 V with tV tangent and nV normal to M. Then from straightforward calculation and using (2.4), (2.9) and (2.10), we obtain Lemma 2.1. Let M be a submanifold of a (k, µ)-contact manifold (M̃, φ, ξ, η, g), then (∇XT)Y − tσ(X, Y) − ANYX = g(X + hX, Y)ξ − η(Y)(X + hX), (2.15) (∇XN)Y + σ(X, TY) − nσ(X, Y) = 0, (2.16) for any vector fields X, Y ∈ TM. 3 Submanifolds of a (k, µ)-contact manifold In this section, we define invariant and anti-invariant submanifolds of (k, µ)-contact manifold and prove the existence. CUBO 18, 1 (2016) Submanifolds of a (k, µ)-Contact Manifold 63 A submanifold M of a (k, µ)-contact manifold M̃ is said to be invariant (resp. anti-invariant) submanifold of M̃ if for each x ∈ M, φ(TxM) ⊂ TxM (resp. φ(TxM) ⊂ T ⊥ x M), here TxM and T⊥x M are the tangent and normal bundles. We first prove the following lemma: Lemma 3.1. For a submanifold M of a (k, µ)-contact manifold M̃, we have −φX − φhX = ∇Xξ + σ(X, ξ), ξ ∈ TM, (3.1) −φX − φhX = −AξX + ∇ ⊥ Xξ, ξ ∈ T ⊥M (3.2) η(ANX) = 0, ξ ∈ T ⊥M (3.3) η(ANX) = −g(φX + φhX, N), ξ ∈ TM (3.4) for each X ∈ TM and N ∈ T⊥M. Proof. From (2.5) and (2.9), we get (3.1). Also from (2.5) and (2.10), we obtain (3.2). Again, in view of (2.2), (3.3) is obvious. Now for ξ ∈ TM, and in view of (2.2), (2.5), (2.10) we get η(ANX) = g(ξ, ANX) = −g(ξ, ∇̃XN) = g(∇̃Xξ, N) = −g(φX + φhX, N). This completes the proof of our lemma. Theorem 3.1. Let M be a submanifold of a (k, µ)-contact manifold M̃ such that the structure vector field ξ is tangent to M. Then M is invariant if and only if σ(X, ξ) = 0, and M is anti- invariant if and only if ∇Xξ = 0. Since it is trivial from Lemma 3.2., we omit to prove our theorem. Theorem 3.2. If M is a totally umbilical submanifold of a (k, µ)-contact manifold M̃ such that the structure vector field ξ is tangent to M, then (i) M is necessarily minimal and consequently totally geodesic and (ii) M is an invariant submanifold of M̃ and ∇Xξ ̸= 0. Proof. Let M be a totally umbilical. Using (2.1), (2.2) and (3.1) in (2.14), we get 0 = σ(ξ, ξ) = g(ξ, ξ)H = H. Hence in view of (2.13) and (2.14), we obtain (i). The second part follow from Theorem 3.1. and the above (i). Theorem 3.3. A submanifold M of a (k, µ)-contact manifold M̃ with structure vector field ξ normal to M is anti-invariant in M̃ if and only if AξX = 0. Consequently, if M is totally geodesic, then it is anti-invariant. 64 M.S. Siddesha, C.S. Bagewadi CUBO 18, 1 (2016) Proof. Since ξ is normal to M, by virtue of (2.10) and (3.2) yields g(−φX − φhX, Y) = g(AξX, Y) = g(σ(X, Y), ξ), X, Y ∈ TM, which provides the proof of our theorem. 4 Non-existence of an anti-invariant distribution This section is devoted to study of anti-invariant distribution. A distribution D on a manifold M is said to be invariant under φ if φDx ⊂ Dx for each x ∈ M and orthogonal complementary distribution D⊥ on M is said to be anti-invariant under φ if φD⊥x ⊂ D ⊥ x for each x ∈ M. Now we define semi-invariant submanifold as follows: A submanifold M of a (k, µ)-contact manifold M̃ is said to be semi-invariant submanifold [6], if the following conditions are satisfied (i) TM = D⊕D⊥⊕{ξ}, where D, D⊥ are orthogonal distributions on M and {ξ} is the 1-dimensional distribution spanned by ξ, (ii) The distribution D is invariant by φ, (iii) The distribution D⊥ is anti-invariant under φ. The distribution D(resp. D⊥) is called the horizontal (resp. vertical) distribution. If both the distribution D and D⊥ are non-zero then the semi-invariant submanifold is called a proper semi- invariant submanifold. To prove the main result of this section first we prove the following lemmas: Lemma 4.1. For a submanifold M of a (k, µ)-contact manifold M̃, we have (∇̃ZΩ)(X, Y) = g(AφYX, Z) − Ω(X, ∇ZY) − Ω(X, σ(Z, Y)) (4.1) for Y ∈ D⊥, X, Z ∈ TM. (∇̃ZΩ)(X, Y) = g(AφXY + AφYX, Z) (4.2) for all X, Y ∈ D⊥, Z ∈ TM. Proof. Let Y ∈ D⊥, Z ∈ TM. Then, by virtue of (2.11) and the fact φY ∈ T⊥M, we get (∇̃Zφ)Y = −AφYZ + ∇ ⊥ ZφY − φ(∇̃ZY). (4.3) Using this equation in (2.7), we can easily derive (4.1). Next, in the special case of X ∈ D⊥, since φX ∈ T⊥M, (4.1) in view of (2.5) and (2.11) yields (4.2). Lemma 4.2. Let M be a submanifold of a (k, µ)-contact manifold M̃ and D⊥ ⊥ {ξ}. Then we get (∇̃ZΩ)(X, X) = 0, (4.4) CUBO 18, 1 (2016) Submanifolds of a (k, µ)-Contact Manifold 65 for X ∈ D⊥ and Z ∈ TM,and consequently AφXX = 0, (4.5) for X ∈ D⊥. Proof. Since D⊥ ⊥ {ξ}, we have η(X) = 0 for any X ∈ D⊥ and hence in view of (2.8), we get (4.4). Again (4.5) follows from (4.2) and (4.4). Theorem 4.1. There does not exist any anti-invariant distribution D⊥ on a submanifold M of a (k, µ)-contact manifold M̃ if ξ is tangent to M and D⊥ ⊥ {ξ}. Proof. Since D⊥ ⊥ {ξ}, we get η(X) = 0 for any X ∈ D⊥. Thus, from (2.2), (3.4) and (4.5), we have 0 = η(AφXX) = g(AφXX, ξ) = −g(φX, φX + φhX) = −g(X, X + hX), for any X ∈ D⊥. This implies g(X, X) = 0. Hence, X must be zero vector. Thus, if X is any arbitrary vector in D⊥ then we have X = 0. Therefore, D⊥ = 0. This proves the theorem. Hence by virtue of Theorem 3.1, we have the following: Corolary 1. A (k, µ)-contact manifold does not admit any proper semi-invariant submanifold. 5 CR-submanifolds of (k, µ)-contact manifold In this section, we shall see the integrability conditions of the involved distributions D and D⊥ in the definition of CR-submanifold M of a (k, µ)-contact manifold. A submanifold M is said to be CR-submanifold in M̃ if there exist two orthogonal comple- mentary distributions D and D⊥ of TM such that ξ ∈ TM and (1) D is invariant by φ, i.e. φ(Dp) ⊂ Dp, ∀p ∈ M, (2) D⊥ is anti-invariant by φ, i.e. φ(D⊥p ) ⊂ T ⊥ p M, ∀p ∈ M. Proposition 1. Let M be a CR-submanifold of a (k, µ)-contact manifold M̃. Then, D, D⊥ and D ⊕ D⊥ are ξ-parallel. Proof. For any X ∈ D and Y ∈ D⊥ g(∇ξX, ξ) = ξg(X, ξ) − g(X, ∇ξξ) = 0, g(∇ξX, Y) = ξg(X, Y) − g(X, ∇ξY) = g(T 2X, ∇ξY) = −g(TX, T∇ξY) = g(TX, ∇ξTY) = 0, 66 M.S. Siddesha, C.S. Bagewadi CUBO 18, 1 (2016) so ∇ξX ∈ D, that is D is ξ-parallel. Similarly, we can proceed for D⊥. Finally, if D and D⊥ are ξ-parallel, D ⊕ D⊥ also is. Lemma 5.1. Let M be a submanifold of a (k, µ)-contact manifold. Then, 2g(X, TY) = η([X, Y]) for all X, Y orthogonal to ξ. Proof. For a (k, µ)-contact manifold it holds that dη = Φ. So 2g(X, TY) = 2Φ(X, Y) = 2dη(X, Y) = η[X, Y]. Lemma 5.2. Let M be a CR-submanifold of a (k, µ)-contact manifold. Then, D⊥ is integrable if and only if dΦ(X, Y, Z) = 0, for any X tangent to M, Y, Z ∈ D⊥. Proof. Consider X tangent to M and Y, Z ∈ D⊥. Then, 3dΦ(X, Y, Z) = X(Φ(Y, Z)) + Y(Φ(Z, X)) + Z(Φ(X, Y)) −Φ([X, Y], Z) − Φ([Z, X], Y) − Φ([Y, Z], X) = −g([Y, Z], φX) = g(φ[Y, Z], X), so dΦ(X, Y, Z) = 0 if and only if [Y, Z] ∈ KerT = D⊥⊕ < ξ >. This is equivalent to [Y, Z] + η[Y, Z]ξ ∈ D⊥, but, using Lemma 5.5., η([Y, Z]) = 2g(X, TY) = 0. Now we can state the following theorem: Theorem 5.1. Let M be a CR-submanifold of a (k, µ)-contact manifold. Then, D⊥ is always integrable. Proof. If M is a contact metric manifold, dΦ = d2η = 0 and so, the result follows from Lemma 5.6. Lemma 5.3. Let M be a CR-submanifold of a (k, µ)-contact manifold. Then, D⊥⊕ < ξ > is integrable if and only if dΦ(X, Y, Z) = 0, for any X tangent to M, Y, Z ∈ D⊥⊕ < ξ >. Proof. Given X ∈ TM, Y, Z ∈ D ⊕ D⊥, we have TY = TZ = 0 and 3dΦ(X, Y, Z) = −g([Y, Z], φX) = g(φ[Y, Z], X). So [Y, Z] is normal if and only if dΦ(X, Y, Z) = 0, for all X tangent to M. Again, from this lemma, we deduce: Theorem 5.2. Let M be a CR-submanifold of a (k, µ)-contact manifold. Then, D⊥⊕ < ξ > is always integrable. CUBO 18, 1 (2016) Submanifolds of a (k, µ)-Contact Manifold 67 Finally, we characterize the integrability of D⊕ < ξ > Theorem 5.3. Let M be a CR-submanifold of a (k, µ)-contact manifold. Then, D⊕ < ξ > is integrable if and only if σ(X, TY) − σ(Y, TX) = 0 for all X, Y ∈ D⊕ < ξ >. Proof. Given X, Y ∈ D⊕ < ξ >, [X, Y] belongs to D⊕ < ξ > if and only if N[X, Y] = 0. Using (2.16), N[X, Y] = N∇XY − N∇YX = ∇XNY + σ(X, TY) − nσ(X, Y) − ∇YNX − σ(Y, TX) + nσ(X, Y) = σ(X, TY) − σ(Y, TX), from which the proof follows. Theorem 5.4. Let M be a CR-submanifold of a (k, µ)-contact manifold. Then, M is locally the product M1 × M2, where M1 is a leaf of D⊕ < ξ > and M2 is a leaf of D ⊥ if and only if σ(X, TY) ∈ TM̃, for all X tangent to M, Y ∈ D⊥. Proof. We shall prove that both D⊕ < ξ > and D⊥ are involutive and their leaves are totally geodesic immersed in M, so M is locally the product of these leaves. For Y ∈ D⊕ < ξ >, Z ∈ D⊥, by virtue of (2.4) and (2.9), g(∇XY, Z) = g(∇̃XY, Z) = g(φ∇̃XY, φZ) = g(∇̃XφY + g(X + hX, Y)ξ − η(Y)(X + hX), NZ) = g(∇̃XTY, NZ) = g(σ(X, TY), NZ). (5.1) So if X ∈ D⊕ < ξ >, ∇XY ∈ D⊕ < ξ > if and only if σ(X, TY) ∈ D̃. Then D⊕ < ξ > is involutive and its leaf is totally geodesic immersed in M. Similarly, from (5.1), if X ∈ D⊥, as g(∇XZ, Y) = −g(Z, ∇XY) = −g(σ(X, TY), NZ), we have that ∇XY ∈ D⊕ < ξ > if and only if σ(X, TY) ∈ D̃. In this case, we obtain that D⊕ < ξ > is also involutive and its leaf is totally geodesic immersed in M. Acknowledgement: The authors are thankful to the referee for his/her valuable suggestions in improvement of the paper. References [1] Avijit Sarkar, Md. Showkat Ali and Dipankar Biswas, On submanifolds of (k, µ)-contact metric manifolds, Dhaka Univ. J. 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