# Glossary of differential geometry and topology

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Glossary of differential geometry and topology

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

*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".

C

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

D

Diffeomorphism. Given two differentiable manifolds "M" and "N", a bijective map $f$ from "M" to "N" is called a diffeomorphism if both $f:M o N$ and its inverse $f^\left\{-1\right\}:N o M$ are smooth functions.

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

F

Fiber. In a fiber bundle, &pi;: "E" &rarr; "B" the preimage &pi;−1("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

H

Hypersurface. A hypersurface is a submanifold of "codimension" one.

I

Immersion

L

Lens space. A lens space is a quotient of the 3-sphere (or (2"n" + 1)-sphere) by a free isometric action of Zk.

M

Manifold. A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second-countable.) A "Ck" manifold is a differentiable manifold whose chart overlap functions are "k" times continuously differentiable. A "C"&infin; or smooth manifold is a differentiable manifold whose chart overlap functions are infinitely continuously differentiable.

P

Parallelizable. A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial.

Principal bundle. A principal bundle is a fiber bundle "P" &rarr; "B" together with right action on "P" by a Lie group "G" that preserves the fibers of "P" and acts simply transitively on those fibers.

Pullback

Section

Submanifold. A submanifold is 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

Torus

Transversality. Two submanifolds "M" and "N" intersect transversally if at each point of intersection "p" their tangent spaces $T_p\left(M\right)$ and $T_p\left(N\right)$ generate the whole tangent space at "p" of the total manifold.

Trivialization

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 α and &beta; over the same base "B" their cartesian product is a vector bundle over "B" &times;"B". The diagonal map $B o B imes B$ induces a vector bundle over "B" called the Whitney sum of these vector bundles and denoted by α&oplus;&beta;.

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