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

A

B

Bundle – see fiber bundle.Basic element – A basic element with respect to an element ' is an element of a cochain complex that is closed: and the contraction of ' by is zero.

C

Characteristic classChartCobordismCodimension – The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold.Connected sumConnectionCotangent bundle – the vector bundle of cotangent spaces on a manifold.Cotangent space

D

Dehn twistDiffeomorphism – Given two differentiable manifolds ' and ', a bijective map from ' to ' is called a diffeomorphism – if both and its inverse are smooth functions.Differential form
  • Domain invarianceDoubling – Given a manifold ' with boundary, doubling is taking two copies of ' and identifying their boundaries. As the result we get a manifold without boundary.

E

EmbeddingExotic structure – See exotic sphere and exotic .

F

Fiber – In a fiber bundle, ' the preimage ' of a point ' in the base ' is called the fiber over ', often denoted '.Fiber bundleFrame – 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.

G

Genus

H

Handle decompositionHypersurface – A hypersurface is a submanifold of codimension one.

I

ImmersionIntegration along fibersIrreducible manifold

J

L

Lens space – A lens space is a quotient of the 3-sphere by a free isometric action of Z – k.

M

Manifold – A topological manifold is a locally Euclidean Hausdorff space. For a given regularity, a manifold of that regularity is a topological manifold whose charts transitions have the prescribed regularity.Manifold with boundaryManifold with cornersMapping class group

N

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

O

Orbifold

P

Pair of pants – An orientable compact surface with 3 boundary components. All compact orientable surfaces can be reconstructed by gluing pairs of pants along their boundary components.Parallelizable – A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial.Partition of unityPL-mapPoincaré lemmaPrincipal bundle – A principal bundle is a fiber bundle ' together with an action on ' by a Lie group ' that preserves the fibers of ' and acts simply transitively on those fibers.

R

S

SectionSeifert fiber spaceSubmanifold – the image of a smooth embedding of a manifold.SubmersionSurface – 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 spaceThom spaceTorusTransversality – Two submanifolds ' and ' intersect transversally if at each point of intersection p their tangent spaces and generate the whole tangent space at p of the total manifold.

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 ' and ' over the same base ' their cartesian product is a vector bundle over. The diagonal map induces a vector bundle over ' called the Whitney sum of these vector bundles and denoted by '.Whitney topologies'''