# Geometry and topology

*https://en.wikipedia.org/wiki/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.^{[
clarification needed]}

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

## Scope

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

It includes:

- Differential geometry and topology
- Geometric topology (including low-dimensional topology and surgery theory)

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.

## Distinction between geometry and topology

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.

### Local versus global structure

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.

By contrast, the
curvature of a
Riemannian manifold is a local (indeed, infinitesimal) invariant^{[
clarification needed]} (and is the only local invariant under
isometry).

### Moduli

If a structure has a discrete moduli (if it has no
deformations, or if a deformation of a structure is isomorphic to the original structure), the structure is said to be **rigid**, and its study (if it is a geometric or topological structure) is topology. If it has non-trivial deformations, the structure is said to be **flexible**, and its study is geometry.

The space of
homotopy classes of maps is discrete,^{
[a]} so studying maps up to homotopy is topology.
Similarly, differentiable structures on a manifold is usually a discrete space, and hence an example of topology, but
exotic **R**^{4}s have continuous moduli of differentiable structures.

Algebraic varieties have continuous moduli spaces, hence their study is algebraic geometry. These are finite-dimensional moduli spaces.

The space of Riemannian metrics on a given differentiable manifold is an infinite-dimensional space.

### Symplectic manifolds

Symplectic manifolds are a boundary case, and parts of their study are called symplectic topology and symplectic geometry.

By Darboux's theorem, a symplectic manifold has no local structure, which suggests that their study be called topology.

By contrast, the space of symplectic structures on a manifold form a continuous moduli, which suggests that their study be called geometry.

However, up to
isotopy, the space of symplectic structures is discrete (any family of symplectic structures are isotopic).^{
[1]}

## Notes

**^**Given point-set conditions, which are satisfied for manifolds; more generally homotopy classes form a totally disconnected but not necessarily discrete space; for example, the fundamental group of the Hawaiian earring.^{[ citation needed]}

## References

**^**Introduction to Lie Groups and Symplectic Geometry, by Robert Bryant, p. 103–104