Glossary of differential geometry and topology explained

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:

• Glossary of general topology
• Glossary of algebraic topology
• Glossary of Riemannian and metric geometry.

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 x B . The diagonal map B\to B x B induces a vector bundle over B called the Whitney sum of these vector bundles and denoted by \alpha ⊕ \beta .