Submanifolds of a conformal Sasakian manifold

In the present paper first, we define the conformal Sasakian manifolds and then we study geometry of invariant, anti-invariant and CR-submanifolds of conformal Sasakian manifolds.


Introduction
A (2n + 1)-dimensional Riemannian manifold (M, g) said to be a Sasakian manifold if it admits an endomorphism φ of its tangent bundle T M , a vector field ξ and a 1-form η satisfying φ 2 = −Id + η ⊗ ξ, η(ξ) = 1, g(φX, φY ) = g(X, Y ) − η(X)η(Y ), for all vector fields X, Y on M , where ∇ denotes the Riemannian connection [1]. The close relationship between Kaehler manifolds and Sasakian manifolds naturally leads to the question which objects, methods and theorems can be transfered from one to the other. The locally conformal Kaehler manifold is one of the sixteen classes of almost Hermitian manifolds [7]. Libermann did the first study on locally conformal Kaehler manifolds [3]. Vaisman, put down some geometrical conditions for locally conformal Kaehler manifolds [4], and Tricerri mentioned different examples about the locally conformal Kaehler manifolds [5]. 24 E. Abedi and M. Ilmakchi We introduced conformal Sasakian manifolds by using an idea of conformal Kaehler manifolds in [2]. The paper is organized as follows. In Section 2, we recall som preliminary definitions about conformal Sasakian manifolds. Furthermore, we give some basic results on conformal Sasakian manifolds and their submanifolds. In Section 3, we obtain a necessary and sufficient condition for the invariant submanifolds of a conformal Sasakian manifold to be minimal. In Section 4, we study anti-invariant submanifolds of a conformal Sasakian manifold and obtain the conditions under which these type submanifolds have a flat normal connection. Section 5 considers CRsubmanifolds of a conformal Sasakian manifold with distributions D and D ⊥ . we find the conditions under which D ⊥ is integrable or totally geodesic. In the final section, we give an example of a three-dimensional conformal Sasakian manifold that is not Sasakian.

Riemannian geometry of conformal Sasakian manifolds
A differentiable manifold M 2n+1 is said an almost contact manifold if it admits a vector field ξ, a one-form η and a (1,1)-tensor field ϕ with the following properties Furthermore, if M be a Riemannian manifold with the Riemannian metric g such that for all X, Y on M , then (ϕ, ξ, η, g) is called an almost contact metric structure on M and M is said an almost contact metric manifold. A Sasakian manifold is a normal contact metric manifold, that is, an almost contact metric manifold such that dϕ = 0 and [ϕ, ϕ](X, Y ) = −2dη(X, Y )ξ for all X, Y on M , where [ϕ, ϕ] is the Nijenhuis torsion of ϕ. An almost contact metric manifold (M 2n+1 , ϕ, ξ, η, g) is said a Sasakian manifold if and only if for all vector fields X, Y on M , where ∇ denotes the Levi-Civita connection with respect to g [1]. Let (Ḿ m ,ǵ) be a Riemannian (sub)manifold into Riemannian manifold (M n , g), m < n, with isometric immersion ι : (Ḿ ,ǵ) −→ (M, g). Then the Gauss and Weingarten formulas are given by for all X, Y tangent toḾ and normal vector field N onḾ , where∇ and ∇ are the Levi-Civita connections ofḾ and M , respectively, also h and A N are the second fundamental form and the shape operator corresponding to N , respectively and ∇ ⊥ is the normal connection on T ⊥Ḿ . LetŔ and R denote the curvature tensors onḾ and M , respectively, then the Gauss and Codazzi equations are given by for all X, Y, Z, W ∈ TḾ and N a ∈ T ⊥Ḿ , where A a is the shape operator with respect to N a , a : 1, · · · , p = n − m and the s ab are the coefficients of the third fundamental form ofḾ in M . Also, let R ⊥ be the normal curvature tensor ofḾ then we will have the Ricci equation by following 5) where N 1 , N 2 are unit normal vector fields onḾ and A 1 , A 2 are the shape operators with respect to N 1 , N 2 . A smooth manifold M 2n+1 with an almost contact metric structure (ϕ, η, ξ, g) is called a conformal Sasakian manifold if there is a positive smooth function f : is a Sasakian structure on M [6]. Let ∇ and ∇ denote connections of M related to metrics g and g, respectively.
Using Koszul formula, we derive the following relation between the connections ∇ and ∇ for all vector fields X, Y on M , so that ω(X) = X(f ) and ω ♯ is vector field of metrically equavalente to one form of ω, that is, g(ω ♯ , X) = ω(X). Vector field ω ♯ = gradf is called the Lee vector field of conformal Sasakian manifold M . Then with a straightforward computation we will have for all vector fields X, Y, Z, W on M , where B := ∇ω − 1 2 ω ⊗ ω and R, R are the curvature tensors of M related to connections of ∇ and ∇, respectively. Also, from (2.2) and (2.6) we have . Suppose∇ andŔ are the Levi-Civita connection and curvature tensor oń M m , respectively. We set for each X ∈ T M ′ and N ∈ T M ′⊥ . Then from (2.9) we get for all vector field X, Y onḾ . Separating the tangential and normal parts from the above equation we will have for all X, Y, Z ∈ TḾ and N a ∈ T ⊥Ḿ , where A a is the shape operator with respect to N a , a : 1, · · · , p = 2n − m + 1.

Invariant submanifolds
A submanifoldḾ of a conformal Sasakian manifold M is called an invariant submanifold of M if ϕTḾ ⊂ TḾ . Hence, ϕN ∈ T ⊥Ḿ for each N ∈ T ⊥Ḿ , that is, tN ≡ 0.
Proof: By relation (2.9) and the Gauss formula we have for all X, Y ∈ TḾ . SinceḾ is invariant, comparing tangential and normal parts we get Since ξ ∈ TḾ , taking X = ϕX in (3.2) we obtain for all X, Y onḾ . Again, since ξ ∈ TḾ, we put X = Y = ξ in (3.2) then we find Let {E α , ϕE α , ξ|α = 1, ..., n = m−1 2 } be an orthonormal frame onḾ and suppose H is the mean curvature vector. Then from the above relation we have Thus the theorem is proved. ✷

Anti-invariant submanifolds
A submanifoldḾ m of a conformal Sasakian manifold M is called an antiinvariant of M if ϕTḾ ⊂ T ⊥Ḿ . Then ϕX ∈ T ⊥Ḿ , for each X ∈ TḾ , that is, P ≡ 0.

4)
for all X, Y, Z ∈ TḾ. Replacing (2.8) in (4.4) we can write Taking the inner product from (4.5) with ϕW and using the Ricci and Gauss equations, we obtain for all X, Y, Z, W ∈ TḾ . From (4.1) we get for all X, Y, Z, W ∈ TḾ . Putting (4.2) into (4.6) and using (4.7), we find for all X, Y ∈ TḾ and N ∈ TḾ ⊥ holds onḾ , where θ is a 2-form onḾ . Proof: Since R ⊥ is reccurent, by (4.9) and using (4.3) in Proposition 4.2 we obtaiń for all X, Y, Z ∈ TḾ . Since ξ ∈ T ⊥Ḿ , taking the inner product from the above equation with each vector field W ∈ TḾ and Contracting it over Z and W we get mθ(X, Y ) = 0, (4.11) for all X, Y onḾ . Then (4.9) results R ⊥ = 0. Thus,Ḿ has a flat normal connection. ✷

Distribution on submanifolds
Let M 2n+1 be a conformal Sasakian manifold. ThenḾ m is said a CR-submanifold in M if there exist two orthogonal complementray distributions D and D ⊥ of TḾ such that ξ ∈ TḾ and (1) D is invariant by ϕ, i.e. ϕ(D p ) ⊂ D p , ∀p ∈Ḿ . Proof: Since Φ(X, Y ) = g(X, ϕY ) for all X, Y ∈ TḾ , we get Φ(X, Y ) = 0 and Φ(Z, W ) = 0 for all X ∈ D and Y, Z ∈ D ⊥ . Since (M 2n+1 , ϕ, ξ, η, g) is a Sasakian manifold, we have d Φ = 0, where Φ(X, Y ) = g(X, ϕY ). Thus, we find Using (Φ ∧ ω)(X, Y, Z) = 0 for all X ∈ D and Y, Z ∈ D ⊥ in the above equation we can write for all Y, Z ∈ T (S) = D ⊥ and X ∈ D, where h S is the second fundamental form of S inḾ . Hence we have Since ω ♯ is normal toḾ , in view of (5.2) we get for all Y, Z ∈ T (S) = D ⊥ and X ∈ D. So (5.1) and (5.3) complete the proof of the theorem. ✷