Trans-Sasakian static spaces

Static spaces (with perfect fluids) appeared in a natural way both in physics (cf. Hawking and Ellis, 1975) and mathematics (cf. A. Fischer and J. Marsden, Duke Math. J. 42 (3) (1975), 519-547). These arise as solutions of the perfect static fluid equation and play a key role in general relativity, providing patterns for some celestial objects. In this paper, we investigate compact and simply connected trans-Sasakian spaces having type (alpha, beta), whose defining functions alpha and beta satisfy the static perfect fluid equation. In particular, we derive some conditions that ensure these spaces are isometric to a 3-sphere. First result of this work shows that the function alpha satisfying static perfect fluid equation and the scalar curvature tau satisfying certain inequality are not only necessary, but also sufficient conditions for a 3-dimensional compact and simply connected trans-Sasakian manifold to be isometric to a 3-sphere. In the second result of this paper, we prove that the function beta satisfying the static perfect fluid equation, scalar curvature tau satisfying certain inequality and the Ricci operator satisfying a Coddazi-type equation are also requirements ensuring that a trans-Sasakian space (again compact and also simply connected) is isometric with a 3-sphere.