Structures on manifolds

The Wikibook 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.