In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, geometry and topology is an umbrella term

Umbrella term

An umbrella term is a word that provides a superset or grouping of concepts that all fall under a single common category. Umbrella term is also called a hypernym. For example, cryptology is an umbrella term that encompasses cryptography and cryptanalysis, among other fields...

 for geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 and topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, as the line between these two is often blurred, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem

Gauss–Bonnet theorem

The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...

 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.

Scope

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

It includes:

• Differential geometry and topology
  Differential geometry and topology
  Differential geometry is a mathematical discipline that uses the techniques of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis...
  
  
• Geometric topology
  Geometric topology
  In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
  
   (including low-dimensional topology
  Low-dimensional topology
  In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
  
   and surgery theory
  Surgery theory
  In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
  
  )

It does not include such parts of algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although 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

Pithily, 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

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, while an example of topology is homotopy theory. The study of metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

s is geometry, the study of topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s is topology.

The terms are not used completely consistently: symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

s are a boundary case, and coarse geometry is global, not local.

Local versus global structure

Differentiable manifold

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

s (of a given dimension) are all locally diffeomorphic (by definition), so there are no local invariants to a differentiable structure (beyond dimension). So differentiable structures on a manifold is an example of topology.

By contrast, the curvature

Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define it, now known as the curvature tensor...

 of a Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 is a local (indeed, infinitesimal) invariant (and is the only local invariant under isometry

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

).

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 have non-trivial deformations, the structure is said to be flexible, and its study is geometry.

The space of homotopy

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

 classes of maps is discrete, 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 R⁴s

Exotic R4

In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphic.The first examples were found by Robion Kirby and Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's...

 have continuous moduli of differentiable structures.

Algebraic varieties have continuous moduli space

Moduli space

In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...

s, hence their study is algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

. Note that these are finite-dimensional moduli spaces.

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

Symplectic manifolds

Symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

s are a boundary case, and parts of their study are called symplectic topology

Symplectic topology

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form...

 and symplectic geometry.

By Darboux's theorem

Darboux's theorem

Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry...

, 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).