Structures on manifolds

Wikibooks Topology has a page on the topic of Categories of Manifolds.

There are three main types of structures important on manifolds. The foundational geometric structures are piecewise linear, mostly studied in geometric topology, and smooth manifold structures on a given topological manifold, which are the concern of differential topology as far as classification goes. Building on a smooth structure, there are:

• various G-structures, which relate the tangent bundle to some subgroup G of the general linear group
• structures defined by holonomy conditions.

These can be related, and (for example for Calabi–Yau manifolds) their existence can be predicted using discrete invariants.