Surfaces of Revolution in Homogeneous 3-Manifolds

The pseudosphere realizes a piece of the hyperbolic plane as a surface of revolution in Euclidean space, and then stops: it has a circular rim beyond which no isometric extension exists. That is not a defect of the hyperbolic plane. Put the same surface in hyperbolic 3-space and it continues without obstruction. Whether a rotationally symmetric surface can be built as a surface of revolution depends on both the surface and the space around it. In this project with Fabian Lander we work this out for every homogeneous 3-manifold, Riemannian and Lorentzian alike, and determine which surfaces embed in which spaces.

A surface of revolution in a general 3-manifold is a surface swept out by rotating a curve about a geodesic axis, and the homogeneous spaces that have such rotations are a short list: the constant curvature spaces, and the bundles over constant curvature surfaces such as $S^2 \times \mathbb{R}$, $\mathbb{H}^2 \times \mathbb{R}$, Nil and the Berger spheres, together with their Lorentzian counterparts. In each of these the question of which surfaces embed has a clean answer.

The pseudosphere is what happens when the Euclidean inequality fails at the rim, and the hyperbolic inequality does not. Comparing the inequalities across spaces gives the second result: a web of implications of the form “if a surface embeds in $X$, then it also embeds in $Y$,” organized by curvature and by how twisted the space is.

Status. The Riemannian case is done. Some work remains in the Lorentzian case, where horizons appear.