On Characterizing a Three-Dimensional Sphere

: In this paper, we ﬁnd a characterization of the 3-sphere using 3-dimensional compact and simply connected trans-Sasakian manifolds of type ( α , β ) .


Introduction
The geometry of 3-dimensional Riemannian spaces has special importance owing to Thurston's conjecture (see [1]). In particular, spherical geometry, one of the eight Thurston geometries, is of primary relevance (cf. [2]). Remarkable examples of manifolds corresponding to this geometry are provided by the Poincaré dodecahedral space, threedimensional spheres, and also lens spaces. We will focus our study on three-dimensional spheres, these spaces being of special importance also from the perspective of their own elegant geometry.
We are going to specifically answer the question raised above by showing that a compact and simply connected TRSM (N, Ψ, ζ, η, g, α, β) with Ricci operator satisfying a Codazzi-type condition and β satisfying the differential Equation (1), and also with scalar curvature bounded above by a certain bound involving functions α and β, is isometric to a 3-sphere (see Theorem 1).
Next, we shall show that the sphere S 3 (c) of constant curvature c has a trans-Sasakian structure. It is clear that S 3 (c) is an embedded surface in the Euclidean space R 4 with unit normal ξ and shape operator B given by B = − √ cI. Using complex structure J on R 4 that is compatible with the Euclidean metric , and makes R 4 , J, , a Kähler manifold, we define an operator Ψ on S 3 (c) by where ζ = −Jξ is the unit vector field on S 3 (c), η is the 1-form dual to ζ with respect to the induced metric g on S 3 (c), and Ψ(E) is the projection of JE on S 3 (c). Then, it follows that the quadruplet (Ψ, ζ, η, g) satisfies (2) by virtue of the properties of the operator J, that is, where D is the Euclidean connection on R 4 , and D is the induced Riemannian connection on S 3 (c). Taking the covariant derivative in Equation (9) and using ζ = −Jξ, as well as Equation (10) and the fact that J is parallel, we get This proves that S 3 (c) has a trans-Sasakian structure (Ψ, ζ, η, g, α, β), where α = √ c and β = 0.

A Characterization of 3-Spheres
We are interested in characterizing 3-spheres using a TRSM (N, Ψ, ζ, η, g, α, β). In the following result, we see that the combination of the Fischer-Marsden differential equation and an upper bound on the scalar curvature involving the functions α, β helps us in reaching the goal.
We claim that α = 0. If we suppose α = 0, then (4) assumes the form and by virtue of the above equation, we derive immediately that η is closed. However, N being simply connected, we have η = dh for some smooth function h on N. Thus, with the assumption made, we get ζ = ∇h, and on compact N, there is a point p ∈ N such that ζ(p) = 0, which is contrary to the fact that ζ is a unit vector field. Hence, we have α = 0.
We claim now that (17) always implies β = 0. Suppose that α 2 − β 2 = s 4 holds in Equation (17). Then, using Equations (5) and (16), we get 2α(−2αβ) = 0, that is α 2 β = 0, and as α = 0 on connected M (being simply connected), we get β = 0. Thus, in view of (17), we derive that, indeed, we always have β = 0, and consequently, Equation (4) assumes the form which proves that ζ is a Killing vector field, and therefore the flow of ζ consists of isometries of N. We get £ ζ S = 0, which in view of Equation (18) gives Using Equation (11) in the above equation, we derive which in view of Equation (18) implies Next, using β = 0 in Equation (6), we get and inserting this equation in Equation (19) we obtain that is, where we have used Equations (8) and (18). Using now (3), (5), and β = 0 in the above equation, we arrive at which on taking the inner product with ζ gives 5αE(α) = 0, that is, E α 2 = 0, E ∈ X(N). This proves that α is a constant. Hence, because we have already shown that α = 0, we conclude that α is a nonzero constant. Now, the Equations (6) and (20) become S(ζ) = 2α 2 ζ and Ψ(S(E)) = 2α 2 Ψ(E), and by operating Ψ on the second equation while using the first equation, we get However, the above equation implies Now, using the following expression for the Riemannian curvature tensor field Rm of a 3-dimensional manifold (N, g): as well as Equations (21) and (22), we arrive at This proves that (N, g) is a space of constant curvature α 2 . As (N, g) is compact, it is complete, and as it is also simply connected, it is isometric to S 3 α 2 .

Remark 1.
Observe that if a Riemannian manifold (M, g) admits a non-trivial solution of the Fischer-Marsden Equation (1), then the scalar curvature s is a constant (cf. [18]). However, we would like to point out that in the statement of Theorem 1, the solution β of Equation (1) is not supposed to be a non-trivial solution (actually, in the proof of the theorem, it turns out to be exactly zero), and therefore we could not use the above argument to conclude the constancy of the scalar curvature.

Remark 2.
The assumption of Theorem 1 that β is a solution of Fischer-Marsden Equation (1) has the following justification. The two smooth functions involved in the definition of a trans-Sasakian manifold (N, Ψ, ζ, η, g, α, β), namely α and β, could be natural candidates for solutions of Equation (1). Moreover, our aim is at getting a characterization of a 3-dimensional sphere S 3 (c), knowing that S 3 (c) admits a trans-Sasakian structure (Ψ, ζ, η, g, α, β), where α = √ c and β = 0. However, α = √ c does not satisfy the Fischer-Marsden equation. Therefore, in view of our goal, the assumption that α satisfies Equation (1) is ruled out. This motivates the hypothesis of Theorem 1 that function β involved in the definition of a trans-Sasakian manifold satisfies the Fischer-Marsden equation. The proof of Theorem 1 shows that this assumption implies, in fact, that β is a trivial solution of (1), provided that the scalar curvature of N satisfies a certain inequality and the Ricci operator of N satisfies a Codazzi-type condition. Moreover, these assumptions also imply that α is a non-zero constant and N is a space of constant curvature, which ultimately leads to the conclusion that N is isometric to S 3 (α 2 ).