The Glossary of differential geometry and topology reference article from the English Wikipedia on 24-Apr-2004
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Glossary of differential geometry and topology

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This is a glossary of some terms used in differential geometry and differential topology. It does not cover the more general glossary of general topology or the more specific glossary of Riemannian and metric geometry.

See also:

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

Table of contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

A

Atlas

B

Bundle, see fiber bundle.

C

Chart

Cobordism

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

Connected sum

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 from M to N is called a diffeomorphism if both and its inverse are smooth.

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, π: E → B the preimage π−1(x) of a point x in the base B is called the fiber over x, often denoted Ex.

Fiber bundle

Frame

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 (2n+1)-sphere) by a free isometric action of Zk.

M

Manifold. A topological manifold is a locally Eulidean 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∞ 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 → B together with right action on P by a Lie group G that preverses the fibers of P and acts simply transitively on those fibers.

Pullback

S

Section

Submanifold. A submanifold is the image of a smooth embedding of a manifold.

Submersion

Surface, a two-dimensional manifold or submanifold.

T

Tangent bundle, the vector bundle of tangent spaces on a differtiable 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 and 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 β over the same base B their cartesian product is a vector bundle over B×B. The diagonal map induces a vector bundle over B called the Whitney sum of these vector bundles and denoted by α⊕β.