Trans-Sasakian static spaces

Abstract 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 ( α , β ) , whose defining functions α and β 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 α satisfying static perfect fluid equation and the scalar curvature τ 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 β satisfying the static perfect fluid equation, scalar curvature τ 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.