8 Comparison geometry and non-negative curvature

Morgan–Tian devote their Chapter 2 to this material and it is used throughout everything below. An earlier version of this blueprint omitted it entirely, which understated the foundations by a full chapter.

Theorem 94 Cheeger–Gromoll splitting theorem

A complete manifold of non-negative Ricci curvature containing a line splits isometrically as a product with \(\mathbb {R}\). (Morgan–Tian 2.5.)

Consumed repeatedly in the classification of ancient solutions, where the limits that arise at infinity are shown to split.

Theorem 95 Soul theorem

A complete non-compact manifold of non-negative sectional curvature is diffeomorphic to the normal bundle of a compact totally geodesic submanifold. (Morgan–Tian 2.3, via Busemann functions, 2.1–2.2.)

Definition 96 \(\epsilon \)-neck

A region almost isometric, after rescaling, to a cylinder \(S^2 \times (-\epsilon ^{-1}, \epsilon ^{-1})\). (Morgan–Tian 2.6.)

The basic local model for where surgery happens. Purely a definition, but a fiddly one — the whole surgery construction is organized around which regions are necks and how deep.