Glossary Of Differential Geometry And Topology

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:
* Glossary of general topology
* Glossary of algebraic topology
* Glossary of Riemannian and metric geometry.

See also:
* List of differential geometry topics

Words in ''italics'' denote a self-reference to this glossary.

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A

* Atlas

B

* Bundle – see ''fiber bundle''.

* Basic element – A basic element ''$x$'' with respect to an element ''$y$'' is an element of a cochain complex $(C^*, d)$ (e.g., complex of ''differential forms'' on a manifold) that is closed: $dx = 0$ and the contraction of ''$x$'' by ''$y$'' is zero.

C

* Characteristic class
* Chart

* Cobordism

* Codimension – The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold.

* Connected sum

* Connection

* Cotangent bundle – the vector bundle of cotangent spaces on a manifold.

* Cotangent space
* Covering
* Cusp
* CW-complex

D

* Dehn twist
* Diffeomorphism – Given two differentiable manifolds ''$M$'' and ''$N$'', a bijective map $f$ from ''$M$'' to ''$N$'' is called a diffeomorphism – if both $f:M\to N$ and its inverse $f^:N\to M$ are smooth functions.
* Differential form
* Domain invariance

* Doubling – Given a manifold ''$M$'' with boundary, doubling is taking two copies of ''$M$'' and identifying their boundaries. As the result we get a manifold without boundary.

E

* Embedding
* Exotic structure – See exotic sphere and exotic $\R^4$.

F

* Fiber – In a fiber bundle, ''$\pi:E \to B$'' the preimage ''$\pi^(x)$'' of a point ''$x$'' in the base ''$B$'' is called the fiber over ''$x$'', often denoted ''$E_x$''.

* Fiber bundle

* Frame – A frame at a point of a differentiable manifold ''M'' is a basis of the tangent space at the point.

* Frame bundle – the principal bundle of frames on a smooth manifold.

* Flow

G

* Genus
* Germ
* Grassmannian bundle
* Grassmannian manifold

H

* Handle decomposition
* Hypersurface – A hypersurface is a submanifold of ''codimension'' one.

I

* Immersion

* Integration along fibers
* Irreducible manifold
* Isotopy

J

* Jet
* Jordan curve theorem

L

* Lens space – A lens space is a quotient of the 3-sphere (or (2''n'' + 1)-sphere) by a free isometric action of Z – _k.
* Local diffeomorphism

M

* Manifold – A topological manifold is a locally Euclidean Hausdorff space (usually also required to be second-countable). For a given regularity (e.g. piecewise-linear, $C^k$ or $C^\infty$ differentiable, real or complex analytic, Lipschitz, Hölder, quasi-conformal...), a manifold of that regularity is a topological manifold whose charts transitions have the prescribed regularity.
* Manifold with boundary
* Manifold with corners
* Mapping class group
* Morse function

N

* Neat submanifold – A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded.

O

* Orbifold
* Orientation of a vector bundle

P

* Pair of pants – An orientable compact surface with 3 boundary components. All compact orientable surfaces can be reconstructed by gluing pairs of pants along their boundary components.
* Parallelizable – A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial.
* Partition of unity
* PL-map

* Poincaré lemma

* Principal bundle – A principal bundle is a fiber bundle ''$P \to B$'' together with an action on ''$P$'' by a Lie group ''$G$'' that preserves the fibers of ''$P$'' and acts simply transitively on those fibers.

* Pullback

R

* Rham cohomology

S

* Section
* Seifert fiber space

* Submanifold – the image of a smooth embedding of a manifold.

* Submersion

* Surface – a two-dimensional manifold or submanifold.

* Systole – least length of a noncontractible loop.

T

* Tangent bundle – the vector bundle of tangent spaces on a differentiable manifold.

* Tangent field – a ''section'' of the tangent bundle. Also called a ''vector field''.

* Tangent space

* Thom space

* Torus

* Transversality – Two submanifolds ''$M$'' and ''$N$'' intersect transversally if at each point of intersection ''p'' their tangent spaces $T_p(M)$ and $T_p(N)$ generate the whole tangent space at ''p'' of the total manifold.
* Triangulation

* Trivialization
* Tubular neighborhood

V

* Vector bundle – a fiber bundle whose fibers are vector spaces and whose transition functions are linear maps.

* Vector field – a section of a vector bundle. More specifically, a vector field can mean a section of the tangent bundle.

W

* Whitney sum – A Whitney sum is an analog of the direct product for vector bundles. Given two vector bundles ''$\alpha$'' and ''$\beta$'' over the same base ''$B$'' their cartesian product is a vector bundle over $B\times B$. The diagonal map $B\to B\times B$ induces a vector bundle over ''$B$'' called the Whitney sum of these vector bundles and denoted by ''$\alpha \oplus \beta$''.
* Whitney topologies

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Wikipedia glossaries using unordered lists