On Sectional Curvatures of ( )-Sasakian Manifolds

The index of a metric plays significant roles in differential geometry as it generates variety of vector fields such as space-like, time-like, and light-like fileds. With the help of these vector fields, we establish interesting properties on ( )-Sasakian manifolds, which was introduced by Bejancu and Duggal [1] and further investigated by Xufeng and Xiaoli [2]. Since Sasakian manifolds with indefinite metrics play crucial roles in physics [3], hence the study of these manifolds becomes the central theme in present scenario. Here the next section is concerned with the basic results of Riemannian curvature tensor of ( )-Sasakian manifolds. In Section 3, these results will be used to obtain the equivalent relations among φ-sectional curvature, totally real sectional curvature, and totally real bisectional curvature. In [1], authors defined the ( )-Sasakian manifold as follows. Let M be a real (2n+ 1)-dimensional differentiable manifold endowed with an almost contact structure (φ,η,ξ), where φ is a tensor field of type (1,1), η is a 1-form, and ξ is a vector field on M satisfying


Introduction
The index of a metric plays significant roles in differential geometry as it generates variety of vector fields such as space-like, time-like, and light-like fileds.With the help of these vector fields, we establish interesting properties on ( )-Sasakian manifolds, which was introduced by Bejancu and Duggal [1] and further investigated by Xufeng and Xiaoli [2].Since Sasakian manifolds with indefinite metrics play crucial roles in physics [3], hence the study of these manifolds becomes the central theme in present scenario.Here the next section is concerned with the basic results of Riemannian curvature tensor of ( )-Sasakian manifolds.In Section 3, these results will be used to obtain the equivalent relations among φ-sectional curvature, totally real sectional curvature, and totally real bisectional curvature.In [1], authors defined the ( )-Sasakian manifold as follows.
Let M be a real (2n + 1)-dimensional differentiable manifold endowed with an almost contact structure (φ,η,ξ), where φ is a tensor field of type (1,1), η is a 1-form, and ξ is a vector field on M satisfying It follows that then M is called an almost contact manifold.If there exists a semi-Riemannian metric g satisfying where = ±1, then (φ,η,ξ,g) is called an ( ) almost contact metric structure and M is known as an ( ) almost contact manifold.For an ( ) almost contact manifold we also have hence ξ is never a light-like vector field on M, and according to the casual character of ξ, we have two classes of ( )-Sasakian manifolds.When = −1 and the index of g is an odd number (v = 2s + 1), then M is a time-like Sasakian manifold and M is a space-like Sasakian manifold when = −1 and v = 2s.For = 1 and v = 0, we obtain usual Sasakian manifold and for then the ( )-contact metric structure is called an ( )-Sasakian structure, and manifold endowed with this structure is called an ( )-Sasakian manifold.Now, let σ be a plane section in tangent space T p (M) at a point p of M, and let it be spanned by vectors X and Y , then the sectional curvature of σ is given by (1.6) A plane {X, Y }, where X and Y are orthonormal to ξ and satisfy φ({X,Y }) ⊥ {X, Y }, is called totally real section, and sectional curvature associated with this section is called a totally real sectional curvature.The totally real bisectional curvature B(X,Y ) is defined as A plane section {X, φX}, where X is orthonormal to ξ, is called φ-section, and the curvature associated with this is called φ-sectional curvature which is denoted by H(X), where If a Sasakian manifold M has constant φ-sectional curvature c, then it is called a Sasakian space form and denoted by M 2n+1 (c).

Riemannian curvature tensor
Theorem 2.1 [1].An ( ) almost contact metric structure (φ,η,ξ,g) is ( )-Sasakian if and only if where ∇ is the Levi-Civita connection with respect to g.Also one has For an ( )-Sasakian manifold, using (2.1) we have where R denotes the Riemannian curvature tensor on M, and also from above we have Using (2.1) and (2.2), we have And by using (2.5), we obtain the following set of equations: (2.8) Now, we can write (2.5) as where (2.11) Clearly P(X,Y ;Z,W) = −P(Z, W;X,Y ), and if {X, Y } is an orthonormal pair orthogonal to ξ, and if we set g(φX,Y (2.12) If we put D(X) = Q(X,φX) for any vector X orthogonal to ξ and Q(X,Y ) = g (R(X,Y )Y ,X) for any vectors X and Y , then we have the following lemma.Lemma 2.2.For any vectors X and Y orthogonal to ξ, one obtains (2.13) Proof.For X, Y orthogonal to ξ, we have and using (2.8), we have + 2R(X,φY ,Y ,φX) + 2Q(X,φY ) . (2.16) Replacing Y by φY in (2.16), we get (2.17) Rakesh Kumar et al. 5 Using (2.16) and (2.17), we have (2.21) Thus using the last four equations, we have the result.Now, it should be noted that D(X) = H(X) if and only if X is a unit vector, and Q(X,Y ) = K(X,Y ) if and only if {X, Y } is an orthonormal pair.Then, as an application of lemma, we have the following lemma.
Lemma 2.3.Let {X, Y } be an orthonormal pair of the tangent space of an ( )-Sasakian manifold M orthogonal to ξ.If one puts g(X,φY Proof.It follows from Lemma (2.2).Since the φ-sectional curvature determines the curvature of a Sasakian manifold, then it can be easily verified that if the φ-sectional curvature H(X) is independent of the choice of a vector X at any point and has value c, then c is constant on M and the curvature tensor + 2g(X,φY )g(φZ,W) .
(2.23) Now, our next aim of this paper is as follows.
(i) M has constant φ-sectional curvature c; that is, H(X) is constant.
(ii) M has constant totally real sectional curvature; that is, for any totally real section {X, Y }, K(X,Y ) is constant.(iii) M has constant totally real bisectional curvature; that is, B(X,Y ) is constant.

Proof of the main Theorem 2.4
In the proof, we assume that X, Y , and Z are unit vector fields.
If H(X) is constant and equal to c, then for a totally real section {X, Y }, (2.23) gives K(X,Y ) = −(c + 3 )/4 and B(X,Y ) = −(c + 7 )/2; this gives (i)⇒(ii) and (i)⇒(iii) respectively.Now, let {X, Y } be a totally real section, then 2} is also a totally real section, and assume that M has constant totally real sectional curvature (say k); then or Since the dimension of M is (2n + 1),n = 3, therefore there exists a unit vector Z orthonormal to Hence, the result is given.