Homogeneous structures of $3$-dimensional Sasakian space forms

TL;DR

This work classifies all homogeneous contact Riemannian structures on 3-dimensional Sasakian space forms, showing that their Ambrose–Singer connections are governed by Okumura’s one-parameter family when the holomorphic sectional curvature c satisfies c≥−3, and by an extended set that includes the Cartan–Schouten (−) connection for c<−3. By explicitly computing Levi–Civita connections, curvature, and Sasakian data on the 3D model groups, the authors connect Ambrose–Singer structures with Tanaka–Webster and Okumura-type connections, clarifying when the homogeneous structures arise from geometry of GA^{+}(1)×ℝ or from standard Sasakian models. The results provide explicit parametrizations for homogeneous contact Riemannian structures on S^{3}, Nil_{3}, and ilde{SL}_{2}(ℝ), and extend to Berger spheres via Abe’s framework, linking CR–geometry, natural reductivity, and coset representations. Overall, the paper bridges Ambrose–Singer and Okumura theories in dimension three, yielding a complete picture of homogeneous contact Riemannian structures on Sasakian space forms with concrete geometric realizations such as Nil_{3}, S^{3}, and GA^{+}(1)×ℝ models.

Abstract

We give explicit parametrizations for all the homogeneous contact Riemannian structures on $3$-dimensional Sasakian space forms.

Theorems & Definitions (51)

• Definition 1.1
• Proposition 1.1
• Definition 1.2
• Remark 1
• Theorem 1.1: AS
• Proposition 1.2
• Definition 1.3
• Theorem 1.2: TV, TV2
• Theorem 1.3: TV
• Definition 1.4