Geodesic
A Classification of Homogeneous Surfaces
Canadian journal of mathematics, Tome 33 (1981) no. 5, pp. 1097-1110

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Throughout this paper a surface is a 2-dimensional (not necessarily compact) complex manifold. A surface X is homogeneous if a complex Lie group G of holomorphic transformations acts holomorphically and transitively on it. Concisely, X is homogeneous if it can be identified with the left coset space G/H, where if is a closed complex Lie subgroup of G. We emphasize that the assumption that G is a complex Lie group is an essential part of the definition. For example, the 2-dimensional ball B 2 is certainly “homogeneous” in the sense that its automorphism group acts transitively. But it is impossible to realize B 2 as a homogeneous space in the above sense. The purpose of this paper is to give a detailed classification of the homogeneous surfaces. We give explicit descriptions of all possibilities. 
Huckleberry, A. T.; Livorni, E. L. A Classification of Homogeneous Surfaces. Canadian journal of mathematics, Tome 33 (1981) no. 5, pp. 1097-1110. doi: 10.4153/CJM-1981-084-x
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