Some Conditions on Trans-Sasakian Manifolds to Be Homothetic to Sasakian Manifolds

: In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and ﬁnd necessary and sufﬁcient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with ﬁnding necessary and sufﬁcient conditions on a compact and connected trans-Sasakian manifold to be homothetic to a compact and connected Sasakian manifold, and the ﬁfth result deals with ﬁnding necessary and sufﬁcient condition on a connected trans-Sasakian manifold to be homothetic to a connected Sasakian manifold. Finally, we ﬁnd necessary and sufﬁcient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to a compact and simply connected Einstein Sasakian manifold.


Introduction
It is well known that the product M = M × R of a (2n + 1)-dimensional almost contact metric manifold (M, F, t, u, g) (cf. Reference [1]) has an almost complex structure J, which with product metric g makes (M, g) an almost Hermitian manifold. The geometry of the almost contact metric manifold (M, F, t, u, g) depends on the geometry of the almost Hermitian manifold (M, J, g) and gives several structures on M like a Sasakian structure, a quasi-Sasakian structure, and others (cf. References [1][2][3]). There are sixteen different types of structures on the almost Hermitian manifold (M, J, g) (cf. Reference [4]), and the structure in the class W 4 on (M, J, g) gives a structure (F, t, u, g, α, β) on M known as trans-Sasakian structure (cf. Reference [5]), which generalizes a Sasakian structure, a Kenmotsu structure, and a cosymplectic structure on a contact metric manifold (cf. References [2,3]), where α, β are smooth functions defined on M. Here, the class W 4 should not be confused with Stiefel-Whitney characteristic class, but it is one of the sixteen classes specified by different combinations of covariant derivatives of the almost complex structure J on the almost Hermitian manifold.
An interesting question is to seek conditions under which a closed (compact without boundary) TRS-manifold is homothetic to a Sasakian manifold. The geometry of TRSmanifold is important because of Thurston's conjecture (cf. Reference [7]), now known as Geometrization-Conjecture, which gave eight geometries on a 3-dimensional manifold, namely Spherical geometry S 3 , Euclidean geometry E 3 , Hyperbolic geometry H 3 , the geometry of S 2 × R, the geometry of H 2 × R, the geometry of universal cover of SL(2, R), the Nil geometry, and the Sol geometry (for details on this topic, see Reference [8]). In addition, we know that 3-dimensional Sasakian manifolds are in abundance, for example, the unit sphere S 3 , the Euclidean space E 3 , the unit tangent bundle T 1 S 2 of the sphere S 2 , the special unitary group SU(2), the Heisenberg group H 3 , and the special linear group SL(2, R) (cf. Reference [9]). Thus, the geometry of TRS-manifolds, in matching them with Thurston's eight geometries on 3-dimensional closed Riemannian manifolds, becomes more interesting, and, as we see, many in the list of Thurston's geometries are included in the list of Sasakian manifolds.
In addition, in References [19][20][21], interesting results on the geometry of TRS-manifolds are obtained, where the authors (W. Wang, X. Liu, Y. Wang, Y. Zhao) considered other aspects in Thurston's eight geometries. In Reference [13], a question was asked whether the function β on a compact TRS-manifold satisfying the differential equation gradβ = t(β)t necessitates the TRS-manifold to be homothetic to a Sasakian manifold. It is shown that this question has negative answer (cf. Reference [21]). However, with additional restrictions, such as positivity of sectional curvatures, and certain differential inequality satisfied by the function, β gives an affirmative answer to this question (cf. Theorem 3.5, [15]).
Owing to Thurston's geometrization conjecture, geometry of TRS-manifolds (being 3-dimensional Riemannian manifolds) have become an important subject. Moreover, Sasakian geometry picks up many important geometries in Thurston's eight geometries; hence, the question of finding conditions under which a TRS-manifold is homothetic to a Sasakian manifold has considerable importance. In Section 3 of this paper, the first four results deal with finding necessary and sufficient conditions on a compact connected TRS-manifold (M, F, t, u, g, α, β) to be homothetic to a compact and connected Sasakian manifold, and the fifth deals with finding necessary and sufficient conditions on a connected TRS-manifold (M, F, t, u, g, α, β) to be homothetic to a connected Sasakian manifold.
In the first result, we consider a compact connected TRS-manifold (M, F, t, u, g, α, β) of constant scalar curvature τ satisfying the inequality τ ≤ 6 α 2 + β 2 and the Ricci operator T satisfying T(t) = τ 3 t, and we give necessary and sufficient conditions for M to be homothetic to a compact and connected Sasakian manifold (see Theorem 2). In the second result, we show that a compact and connected TRS-manifold (M, F, t, u, g, α, β) with Ricci curvature S(t, t) a non-zero constant and satisfying S(t, t) ≤ 2 α 2 + β 2 give necessary and sufficient conditions for M to be homothetic to a compact and connected Sasakian manifold (see Theorem 3). Similarly, in the third result, we show that conditions S(t, t) = 0 and F(gradα) = gradβ on a compact and connected TRS-manifold (M, F, t, u, g, α, β) are necessary and sufficient for M to be homothetic to a compact and connected Sasakian manifold (see Theorem 4). In addition, the fourth result deals with conditions α(p) = 0 for a point p ∈ M and F(gradβ) = −gradα on a compact and connected TRS-manifold (M, F, t, u, g, α, β) to reach a similar conclusion (see Theorem 5). Finally, in the fifth result, we show that compactness could be dropped with the conditions α(p) = 0 for a point p ∈ M and gradα 2 = 4α 2 β 2 on a connected TRS-manifold (M, F, t, u, g, α, β) to reach a similar conclusion (see Theorem 6).
Among Sasakian manifolds, Einstein Sasakian manifolds play an important role because of their elegant geometry, as well as their important applications in theoretical physics (see the excellent monograph in Reference [9]). In the last section of this paper, we find necessary and sufficient conditions on a compact and simply connected TRSmanifold (M, F, t, u, g, α, β) to be homothetic to a compact simply connected Einstein Sasakian manifold (see Theorem 7).

Preliminaries
Let (M, F, t, u, g) be a 3-dimensional almost contact metric manifold, where F is a (1, 1)-tensor field, t a unit vector field, and u a smooth 1-form dual to t with respect to the Riemannian metric g, satisfying is the Lie algebra of smooth vector fields on M (cf. Reference [1]). If there are smooth functions α, β on an almost contact metric manifold (M, F, t, u, g) satisfying , and ∇ is the Levi-Civita connection with respect to the metric g (cf. References [7,[10][11][12][13][14][15]20]). We shall abbreviate the 3-dimensional trans-Sasakian manifold (M, F, t, u, g, α, β) by TRS-manifold. Using Equations (1) and (2), we get Let S be the Ricci tensor of a Riemannian manifold (M, g). Then, the Ricci operator T is defined by S(U, V) = g(TU, V), U, V ∈ X(M). On a TRS-manifold, we have the following: Note that Equation (3) implies and, using this equation, together with Equation (4) Thus, on a compact TRS-manifold, using Equation (6) and the above equation, we have Now, we state the following result of Okumura.
Theorem 1 (Reference [18]). Let (M, g) be a Riemannian manifold. If M admits a Killing vector field t of constant length satisfying for non-zero constant α and any vector fields U and V, then M is homothetic to a Sasakian manifold.
Given two Riemannian manifolds (M 1 , g 1 ), (M 2 , g 2 ), a diffeomorphism f : M 1 → M 2 is said to be a conformal transformation if the pullback f * (g 2 ) of the metric g 2 satisfies where σ is a smooth function on M 1 . If the function σ is a constant, then the diffeomorphism f is said to be a homothety, and, in this situation, the Riemannian manifold (M 1 , g 1 ) is said to be homothetic to the Riemannian manifold (M 2 , g 2 ). Thus, Theorem 1 gives a condition under which a Riemannian manifold (M, g) is homothetic to a Sasakian manifold.
For a smooth function f on a Riemannian manifold (M, g), the Hessian operator A f of f is defined by and the Laplace operator ∆ is defined by ∆ f = div(grad f ), and it satisfies

TRS-Manifolds Homothetic to Sasakian Manifolds
In this section, we find necessary and sufficient conditions on a TRS-manifold (M, F, t, u, g, α, β) to be homothetic to a Sasakian manifold. Proof. Suppose T(t) = τ 3 t holds, then, using Equation (5), we have Taking the inner product in the above equation with t, we get Using Equation (6), we have div(βt) = t(β) + 2β 2 , and, by (8), we have Integrating the above equation, we conclude Using the inequality in the statement, we get Since, τ is a constant, we get M is homothetic to a Sasakian manifold of constant scalar curvature (cf. Theorem 3.1, in Reference [14]). The converse is trivial.
Integrating the above equation, we conclude that β = 0. Consequently, Equation (9) implies S(t, t) = 2α 2 , and, as the Ricci curvature S(t, t) is non-zero constant, we conclude α is a non-zero constant. The Equation (3) now takes the form and we get i.e., the unit vector field t is Killing. Moreover, using Equation (11), we get where α is a non-zero constant. Hence, by Theorem 1, we conclude that M is homothetic to a Sasakian manifold. The converse is trivial as for a Sasakian manifold S(t, t) = 2. Proof. Suppose F(gradα) = gradβ. Then, we have ∆β = divF(gradα) and where {e 1 , e 2 , e 3 } is a local orthonormal frame on M, and A α is the Hessian operator of α.
We observe that Equation (4) implies g(t, gradα) 2 = 4α 2 β 2 , and, with t being a unit vector field, it implies 4α 2 β 2 ≤ gradα 2 . Naturally, one feels prompted to ask what happens in case of the equality. Interestingly, the answer is the TRS-manifold which, in this case, is homothetic to a Sasakian manifold without imposition of compactness, as seen in the following result.
Using Equations (4) and (13) in the above equation, we arrive at Note that, as in the statement α = 0 and, accordingly, on connected M, the above equation implies gradβ = t(β)t.
Integrating the above equation confirms β = 0. Then, Equations (4) and (17) imply t(α) = 0 and F(gradα) = 0, and, operating F on the second equation, we get gradα = 0, i.e., α is a constant. Now, M being simply connected, we claim that α is a non-zero constant. For, if α = 0, then, by Equation (3), the vector field t is parallel; therefore, u is closed and has to be exact, and there exists a smooth function ϕ such that u = dϕ. Thus, t = gradϕ, and, as M is compact, there exists a point x ∈ M such that (gradϕ)(x) = 0, i.e., t(x) = 0, which is a contradiction to the fact that t is a unit vector field. Hence, constant α = 0 and, with β = 0, Equations (2) imply the Lie derivative This proves that t is a Killing vector field, and the flow of t consists of isometries of M, and, as such, we have (£ t T)(U) = 0, U ∈ X(M), Now, with β = 0 and α a constant, we have T(t) = 2α 2 t, and, taking covariant derivative in this equation while using Equation (3), we have (∇T)(U, t) + T(−αFU) = −2α 3 FU, U ∈ X(M).
Combining Equations (18) and (19) with constant α = 0, we get F(TU) = 2α 2 FU, U ∈ X(M), and, operating F on the above equation, and keeping in mind the equation T(t) = 2α 2 t, we have TU = 2α 2 U, U ∈ X(M), i.e., M is an Einstein manifold. Now, with α, a non-zero constant, and β = 0, using Equations (2) and (3), we conclude Hence, by virtue of Theorem 1, we get M is homothetic to a compact simply connected Einstein Sasakian manifold. The converse is trivial.