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:

See also:

List of differential geometry topics

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

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

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 $f:M\to N$ and its inverse $f^{-1}:N\to M$ 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, $\pi :E\to B$ 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

Integration along fibers

L

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

M

Manifold – A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second-countable.) A $C^{k}$ manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A $C^{\infty }$ or smooth manifold is a differentiable manifold whose chart overlap functions are infinitely continuously differentiable.

N

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

O

Orientation of a vector bundle

P

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

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

S

Section

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.

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 $\alpha$ 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 $\alpha \oplus \beta$ .

Manifolds (Glossary)

Basic concepts

Main results (list)

Types of
manifolds

Vectors	Distribution Lie bracket Pushforward Tangent space bundle Torsion Vector field Vector flow
Covectors	Closed/Exact Covariant derivative Cotangent space bundle De Rham cohomology Differential form Vector-valued Exterior derivative Interior product Pullback Ricci curvature flow Riemann curvature tensor Tensor field density Volume form Wedge product
Bundles	Adjoint Affine Associated Cotangent Dual Fiber (Co) Fibration Jet Lie algebra (Stable) Normal Principal Spinor Subbundle Tangent Tensor Vector
Connections	Affine Cartan Ehresmann Form Generalized Koszul Levi-Civita Principal Vector Parallel transport

Related

Generalizations

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