Trans-Sasakian space forms

In this paper, we derive an explicit expression for the Riemann--Christoffel curvature tensor of a trans-Sasakian manifold with pointwise constant $φ$-sectional curvature and introduce the notion of trans-Sasakian space forms. As consequences, we obtain explicit formulas for the Ricci tensor and the scalar curvature. We further prove that every $3$-dimensional trans-Sasakian manifold necessarily has pointwise constant $φ$-sectional curvature. In addition, we investigate statistical structures on trans-Sasakian manifolds and introduce the notion of trans-Sasakian statistical space forms. We then establish a classification of such statistical structures. Our results show that every trans-Sasakian statistical space form of dimension greater than $3$ is necessarily almost-trivial. In the $3$-dimensional case, we establish a rigidity theorem for trans-Sasakian statistical space forms and provide explicit examples showing the existence of non-trivial $3$-dimensional trans-Sasakian statistical space forms