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:

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 $\alpha \oplus \beta$ .