- Geometry and topology
In

mathematics ,**geometry and topology**is anumbrella term forgeometry andtopology , as the line between these two is often blurred, most visibly in local to global theorems in Riemannian geometry, and results like theGauss–Bonnet theorem andChern-Weil theory .Sharp distinctions between geometry and topology can be drawn, as discussed below.

It is also the title of a journal

Geometry and Topology that covers these topics.**cope**It is distinct from "geometric topology", which more narrowly involves applications of topology to geometry.

It includes:

*Differential geometry and topology

*Geometric topology (includinglow-dimensional topology andsurgery theory )It does not include such parts of

algebraic topology ashomotopy theory , but some areas of geometry and topology (such as surgery theory, particularlyalgebraic 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 archetypal example of geometry is

Riemannian geometry , while an archetypal example of topology ishomotopy theory . The study ofmetric space s is geometry, the study oftopological space s is topology.The terms are not used completely consistently:

symplectic manifold s are a boundary case, andcoarse geometry is global, not local.**Local versus global structure**Differentiable manifold 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 of a

Riemannian manifold is a local (indeed, infinitesimal) invariant (and is the only local invariant underisometry ).**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 its 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 classes of maps is discrete [*Given point-set conditions, which are satisfied for manifolds; more generally homotopy classes form 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 exotictotally disconnected but not necessarily discrete space; for example, thefundamental group of theHawaiian earring .**R**^{4}s have continuous moduli of differentiable structures.Algebraic varieties have continuousmoduli space s, hence their study isalgebraic geometry . Note that these are finite-dimensional moduli spaces.The space of Riemannian metrics on a given differentiable manifold is an infinite-dimensional space.

**ymplectic manifolds**Symplectic manifold s are a boundary case, and parts of their study are calledsymplectic topology andsymplectic 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) [*[*] .*http://www.math.duke.edu/~bryant/ParkCityLectures.pdf Introduction to Lie Groups and Symplectic Geometry*] , by Robert Bryant, p. 103-104**References**

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