Surfaces of Revolution in Homogeneous Spaces

The pseudosphere realizes the hyperbolic plane as a surface of revolution in Euclidean space, but only up to a circular rim, beyond which no isometric extension exists. This failure reflects a limitation of the ambient geometry rather than of the surface itself: in hyperbolic space, the same surface extends without obstruction. Such behavior is common — rotationally symmetric surfaces arise throughout geometry, and whether a given one can be concretely realized inside an ambient 3-manifold depends sharply on the interplay between their geometries. In this talk we seek a classification: which surfaces embed in which ambient spaces, as the surfaces range over all rotationally symmetric (Riemannian and Lorentzian) surfaces, and the ambient spaces over all homogeneous 3-manifolds of either signature. This leads to a web of implications between geometries: if a surface embeds in geometry X, what does this force about its embeddability in geometry Y? Joint work with Fabian Lander.