Glossary of differential geometry and topology

A

• Atlas

B

• Bundle – see fiber bundle.

• basic element – A basic element ${\displaystyle x}$ with respect to an element ${\displaystyle y}$ is an element of a cochain complex ${\displaystyle (C^{*},d)}$ (e.g., complex of differential forms on a manifold) that is closed: ${\displaystyle dx=0}$ and the contraction of ${\displaystyle x}$ by ${\displaystyle y}$ 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 ${\displaystyle M}$ and ${\displaystyle N}$, a bijective map ${\displaystyle f}$ from ${\displaystyle M}$ to ${\displaystyle N}$ is called a diffeomorphism – if both ${\displaystyle f:M\to N}$ and its inverse ${\displaystyle f^{-1}:N\to M}$ are smooth functions.

• Doubling – Given a manifold ${\displaystyle M}$ with boundary, doubling is taking two copies of ${\displaystyle M}$ and identifying their boundaries. As the result we get a manifold without boundary.

E

• Embedding

F

• Fiber – In a fiber bundle, ${\displaystyle \pi :E\to B}$ the preimage ${\displaystyle \pi ^{-1}(x)}$ of a point ${\displaystyle x}$ in the base ${\displaystyle B}$ is called the fiber over ${\displaystyle x}$, often denoted ${\displaystyle E_{x}}$.

• 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 ${\displaystyle C^{k}}$ manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A ${\displaystyle C^{\infty }}$ 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 ${\displaystyle P\to B}$ together with an action on ${\displaystyle P}$ by a Lie group ${\displaystyle G}$ that preserves the fibers of ${\displaystyle P}$ 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 ${\displaystyle M}$ and ${\displaystyle N}$ intersect transversally if at each point of intersection p their tangent spaces ${\displaystyle T_{p}(M)}$ and ${\displaystyle T_{p}(N)}$ 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 ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ over the same base ${\displaystyle B}$ their cartesian product is a vector bundle over ${\displaystyle B\times B}$. The diagonal map ${\displaystyle B\to B\times B}$ induces a vector bundle over ${\displaystyle B}$ called the Whitney sum of these vector bundles and denoted by ${\displaystyle \alpha \oplus \beta }$.