Geometry and Topology

In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory.

Sharp distinctions between geometry and topology can be drawn, however, as discussed below.

It is also the title of a journal Geometry & Topology that covers these topics.

It is distinct from "geometric topology", which more narrowly involves applications of topology to geometry.

It includes:

It does not include such parts of algebraic topology as homotopy theory, but some areas of geometry and topology (such as surgery theory, particularly algebraic surgery theory) are heavily algebraic.

Geometry has local structure (or infinitesimal), while topology only has global structure. Alternatively, geometry has continuous moduli, while topology has discrete moduli.

By examples, an example of geometry is Riemannian geometry, while an example of topology is homotopy theory. The study of metric spaces is geometry, the study of topological spaces is topology.

The terms are not used completely consistently: symplectic manifolds are a boundary case, and coarse geometry is global, not local.

By definition, differentiable manifolds of a fixed dimension are all locally diffeomorphic to Euclidean space, so aside from dimension, there are no local invariants. Thus, differentiable structures on a manifold are topological in nature.

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