2 - The Mean Curvature Sphere Congruence

Summary

A fundamental construction in conformal geometry of surfaces is the mean curvature sphere congruence, or central sphere congruence, the bundle of 2-spheres tangent to the surface and sharing with it mean curvature vector at each point (although the mean curvature vector is not conformally invariant, under a conformal change of the metric, it changes in the same way for the surface and the osculating 2-sphere). The concept has its origin in the nineteenth century, with the introduction of the mean curvature sphere of a surface at a point, by Germain. By the turn of the century, the family of the mean curvature spheres of a surface was known as the central sphere congruence, cf. Blaschke. Nowadays, after Bryant's paper, it goes as well by the name of conformal Gauss map. We introduce it in this chapter, starting by recalling some fundamental concepts in Riemannian Geometry.