Surfaces of constant curvature in 3-dimensional space forms

The study of surfaces immersed in a 3-dimensional ambient space plays a central role in the theory of submanifolds. In addition, Riemannian manifolds with constant sectional curvature can be considered as the most simple examples. Thus, one can think of surfaces with constant Gauss curvature in the Euclidean space R, hyperbolic space H or 3-sphere S as very natural objects of study. Through these notes we will study some classical results on complete surfaces with constant Gauss curvature in 3-dimensional space forms, using a modern approach. The notes are organized as follows. In Section 2 we establish some notation and recall some necessary preliminary concepts. In Section 3 we center our attention on the complete surfaces with positive constant Gauss curvature in S, R or H. We shall prove the Liebmann theorem and show that the only complete examples must be totally umbilical round spheres. In particular, there is no complete surface in S with constant Gauss curvature K(I) ∈ (0, 1). In Section 4 we prove the existence of a conformal representation for flat surfaces in the hyperbolic 3-space. We shall associate to each surface a pair of holomorphic 1-forms and see that the flat surface can be recovered in terms of this holomorphic pair. We prove that the two hyperbolic Gauss maps of a flat surface are holomorphic with respect to the conformal structure given by the second fundamental form, and classify the complete flat surfaces. Section 5 is devoted to the Hartman-Nirenberg theorem for flat surfaces in the Euclidean 3-space. We shall show that a complete flat surface in R must be a right cylinder on a planar curve which is defined for all values of its arc length parameter. In Section 6 we shall study the Bianchi-Spivak representation of a flat surface in S. We start showing the existence of global Tschebyscheff coordinates in a complete flat surface. We prove that the asymptotic curves of a flat surface have torsion ±1 when the curvature of the curves does not vanish. Moreover, these asymptotic curves are characterized in a simple way which generalizes the torsion ±1 concept. We recall the quaternionic model of S and show that any complete flat surface Σ in S can be recovered by multiplication of the two asymptotic curves passing across a point p ∈ Σ. We also prove the converse of this result in terms of curves with generalized torsion ±1. Finally, we note that every asymptotic curve of a flat torus is closed. Thus, flat tori can be classified.