Differential topology

A sphere (from Ancient Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space. That given point is the center of the sphere, and the distance is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

220px|right|thumb|Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot. 220px|right|thumb|Tangent plane to a sphere

assignment of a vector to each point in a subset of Euclidean space

branch of mathematics

function defined by a relation of the form 𝑅(𝑥,𝑦)=0, where 𝑅 is a function of several variables and there is a unique 𝑦 that satisfies the relation for every 𝑥

vector space associated to a point in a smooth manifold, consisting of vectors tangent to it (in some embedding into Euclidean space)

product of the principal curvatures of a surface

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

assignment of a tensor continuously varying across a mathematical space

theorem that, if a function is continuously differentiable with nonzero Jacobian determinant at a given point, then it is locally invertible near that point

in homological algebra, a structure consisting of a sequence of modules and a sequence of homomorphisms between consecutive modules such that the image of each homomorphism is included in the kernel of the next

tangent spaces of a manifold considered together

right|thumb|A torus is an orientable surface alt=Animation of a flat disk walking on the surface of a Möbius strip, flipping with each revolution.|thumb|The Möbius strip is a non-orientable surface. Note how the disk flips with every loop. right|thumb|The Roman surface is non-orientable.

differentiable function whose derivative is everywhere injective

geometrical idea of transporting data along a curve or family of curves in a parallel and consistent manner

derivative of a tensor field along the flow defined by a vector field

in differential geometry, a smooth manifold equipped with a closed, nondegenerate differential 2-form

set of continuous functions from a topological space to the unit interval [0,1] such that for every point x, there is a neighborhood of x where a cofinite number of the functions are 0, and such that the sum of all the function values at x is 1

thumb|A cobordism (W; M, N). In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.

sets of coordinates which can be used to describe a physical system at any given point in time

geometric modification on manifolds

vector bundle of all cotangent spaces at every point in a manifold

alt=Hyperbolic symmetry comparison to Euclidean symmetry|thumb|23star Orbifold Example In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space.

theorem relating the Euler characteristic of a closed manifold to the number of zeros of a vector field on it

topological operation of turning a sphere inside-out without creasing

statement about integration of differential forms on manifolds

dual space to the tangent space in differential geometry

three dimensions self-intersecting surface

generalization of winding number

right inverse of a fiber bundle map

operator in differential topology

vector bundle, complementary to the tangent bundle, associated to an embedding

characteristic class defined for real vector bundles

mathematics glossary

one-dimensional vector bundle

thumb|160px|Immersed manifold straight line with self-intersections In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.

system with kinematic constraints that cannot be integrated in position-level constraints

smooth manifold that is homeomorphic but not diffeomorphic to a sphere

problem in geometric topology

Study in mathematical gauge theory

Mathematical theories

contnuous linear functional on the space of compactly supported differential forms

a differentiable manifold whose (co)tangent bundle is topologically trivial

fiber bundle constructed by a representation of a group and a principal bundle

In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an '''h-cobordism' (the h'' stands for homotopy equivalence) if the inclusion maps

infinitesimal version of a Lie groupoid: manifold M with vector bundle E, vector bundle map ρ: E→TM, and a Lie bracket on sections of E, so that [s,ft] = (ρ(s)f)t+f[s,t] for any function f: M→ℝ and sections s, t of E