Research Article On Compact Trans-Sasakian Manifolds

We study 3-dimensional compact and simply connected trans-Sasakian manifolds and ﬁ nd necessary and su ﬃ cient conditions under which these manifolds are homothetic to Sasakian manifolds. The ﬁ rst two results deal with ﬁ nding necessary and su ﬃ cient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to an Einstein Sasakian manifold and in the third result deals with ﬁ nding necessary and su ﬃ cient condition on a compact and simply connected trans-Sasakian manifold to be homothetic to a Sasakian manifold.


Introduction
It is well known that for an almost contact metric manifold ðM, F, ζ, η, gÞ (cf. [1]), the product M = M × R has an almost complex structure J, which with product metric g makes ð M, gÞ an almost Hermitian manifold. The properties of the almost Hermitian manifold ð M, J, gÞ control the properties of the almost contact metric manifold ðM, F, ζ, η, gÞ and provide several structures on M such as a Sasakian structure and a quasi-Sasakian structure (cf. [1][2][3]). There are known sixteen different types of structures on ð M, J, gÞ (cf. [4]), and using the structure in the class W 4 on ð M, J, gÞ, a structure ðF, ζ, η, g, α, βÞ was introduced on M, which is called trans-Sasakian structure (cf. [5]), that generalizes Sasakian structure, Kenmotsu structure, and cosymplectic structure on a contact metric manifold (cf. [2,3]), where α and β being the real functions defined on M.
Recall that a trans-Sasakian manifold ðM, F, ζ, η, g, α, βÞ is called a trans-Sasakain manifold of type ðα, βÞ, and trans-Sasakian manifolds of type ð0, 0Þ, ðα, 0Þ, and ð0, βÞ are called a cosymplectic, a α-Sasakian, and a β-Kenmotsu manifolds, respectively. It is on account of a result proved in [6] that a trans-Sasakian manifold of dimension five or greater than five reduces to a cosymplectic manifold, a α-Sasakian manifold, or a β-Kenmotsu manifold, so there is an emphasis on studying three-dimensional trans-Sasakian manifolds.
Among other questions, finding conditions under which a compact 3-dimensional trans-Sasakian manifold ðM, F, ζ , η, gÞ is homothetic to a Sasakian manifold is of prime importance. The geometry of 3-dimensional trans-Sasakian manifold is also important owing to Thurston's conjecture (cf. [7]), and fetching conditions on a 3-dimensional trans-Sasakian manifold ðM, F, ζ, η, gÞ in matching it among Thurston's eight geometries becomes more interesting. It is worth noting that in Thurston's eight geometries, the first place is occupied by the spherical geometry S 3 .
An interesting work on 3-dimensional trans-Sasakian manifolds is found in [14,15], where the authors have considered other aspects in Thurston's eight geometries. In [10], it is asked whether the function β on a 3-dimensional compact trans-Sasakian manifold ðM, F, ζ, η, g, α, βÞ satisfying grad β = ζðβÞζ necessitates the trans-Sasakian manifold to be homothetic to a Sasakian manifold. In [15], it is shown that this question has negative answer.
Einstein Sasakian manifolds are very important due to their geometric importance (cf. [16]). In this paper, in our first two results, we find necessary and sufficient conditions on a compact simply connected 3-dimensional trans-Sasakian manifold ðM, F, ζ, η, g, α, βÞ to be homothetic to an Einstein Sasakian manifold, and in the third, we find a necessary and sufficient condition on a compact simply connected 3-dimensional trans-Sasakian ðM, F, ζ, η, g, α, βÞ to be homothetic to a Sasakian manifold.
In the first result, we consider a compact and simply connected trans-Sasakian manifold ðM, F, ζ, η, g, α, βÞ of positive constant scalar curvature τ, the function β satisfying Fischer-Marsden equation shows that the functions α and β are related to τ by the inequality βðα 2 − β 2 − τ/4Þ ≥ 0, and the Ricci operator Q satisfying Codazzi-type equation with respect to vector field ζ necessarily implies that ðM, F, ζ, η, g, α, βÞ is homothetic to an Einstein Sasakian manifold. In the second result, we show that a compact simply connected trans-Sasakian manifold with function α constant along the integral curves of ζ, scalar curvature τ satisfying the inequality αð6α 2 − τÞ ≥ 0, and the Ricci operator Q satisfying Codazzi-type equation with respect to vector field ζ necessarily imply that ðM, F, ζ, η, g, α, βÞ is homothetic to an Einstein Sasakian manifold. Finally, in the last result, we show that on a compact and simply connected trans-Sasakian manifold, the function β satisfies the differential inequality ζðβ 2 Þ ≤ −2β 3 , and vector fields ð∇QÞðgradα, ζÞ, ζ are orthogonal, which necessarily imply that ðM, F, ζ, η, g, α, βÞ is homothetic to a Sasakian manifold, where the covariant derivative ð∇QÞðU, ζÞ = ∇ U Qζ − Qð∇ U ζÞ for a smooth vector field U on M.
For a smooth function h on the Riemannian manifold ð M, gÞ, then the operator A h defined by is called the Hessian operator of h, and it is a symmetric operator. Moreover, the Hessian HessðhÞ of h is defined by

Advances in Mathematical Physics
The Laplace operator Δ on ðM, gÞ is defined by Δh = div ðgrad hÞ, and we also have Fischer-Marsden differential equation on a Riemannian manifold ðM, gÞ is (cf. [18])

Trans-Sasakian Manifolds Homothetic to Einstein Sasakian Manifolds
In this section, we find necessary and sufficient conditions for a compact and simply connected 3-dimensional trans-Sasakian manifold ðM, F, ζ, η, g, α, βÞ to be homothetic to an Einstein Sasakian manifold.
is homothetic to an Einstein Sasakian manifold of positive scalar curvature, if and only if, the Ricci operator Q satisfies Proof. Suppose ðM, F, ζ, η, g, α, βÞ is a compact simply connected 3-dimensional trans-Sasakian manifold satisfying the hypothesis. Then, equation (13) gives and taking trace in above equation and using equation (12), we have Note that by equation (3), we have ∇ ζ ζ = 0, and therefore, HessðβÞðζ, ζÞ = ζζðβÞ. Using this equation and equation (17) in equation (16), we get Now, using equation (5), we have Ricðζ, ζÞ = 2ðα 2 − β 2 − ζðβÞÞ. Thus, the above equation becomes Using equation (6), we have div ðζðβÞζÞ = ζζðβÞ + 2βζð βÞ, and inserting it in the above equation, we conclude Integrating the above equation, we get Using the inequality in the statement, we conclude Since M is simply connected, it is connected, and therefore equation (22) implies either (i) β = 0 or (ii) α 2 − β 2 − τ /4 = 0. Suppose (ii) holds, then as τ is a constant, we get ζð α 2 Þ = ζðβ 2 Þ, which in view of equation (4) implies βζðβÞ = −2α 2 β; that is, 3β 2 ζðβÞ = −6α 2 β 2 . Thus, we have Using equation (6), we have div ðβ 3 ζÞ = ζðβ 3 Þ + 2β 4 , and inserting it in above equation, we get Integrating the above equation, we get Now, using (ii) in above integral, we have and since the scalar curvature τ > 0, through above integral, we conclude that β = 0. Thus, using equations (2), (3), (4), and (5), take the forms ζ α ð Þ = 0, Taking the covariant derivative in the second equation of equation (28), we get 3 Advances in Mathematical Physics and using equation (27) in above equation, we arrive at Now, using the Codazzi equation type condition on Q in the hypothesis, we get Using the second equation in equation (27), we compute the Lie derivative of g with respect to ζ to conclude that is, ζ is a Killing vector field and that the flow of ζ consists of isometries of the Riemannian manifold M. Thus, we have and using equation (27), we conclude Combining the above equation with equation (31), we have Taking the inner product with ζ in above equation, we conclude We claim that M being simply connected, α ≠ 0; for if α = 0, then by equation (27), we see that ζ is parallel and that η is closed, which implies η is exact; that is, η = df for a smooth function f on M. This implies ζ = gradf , and M being compact, there is a point q ∈ M such that ðgradf ÞðqÞ = 0, and we get ζðqÞ = 0, a contradiction to the fact that ζ is a unit vector field. Hence, α ≠ 0, and equation (36) implies UðαÞ = 0, U ∈ ΓðTMÞ; that is, α is a nonzero constant. Now, equation (28) gives QðζÞ = 2α 2 ζ, and taking the covariant derivative in this equation yields Using the condition in the hypothesis and equation (34) with α ≠ 0, in above equation, we get Operating F on above equation while using equation (1) and QðζÞ = 2α 2 ζ, we conclude This proves that M is an Einstein manifold. Finally, using equation (27), with α a nonzero constant, we compute Hence, by Theorem 1, we conclude that M is homothetic to a compact simply connected Einstein Sasakian manifold of positive scalar curvature. The converse is trivial.