Glossary of differential geometry and topology

Mathematics glossary

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 (e.g., complex of differential forms on a manifold) that is closed: and the contraction of by is zero.

C

  • Codimension – The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold.

D

  • Diffeomorphism – Given two differentiable manifolds and , a bijective map from to is called a diffeomorphism – if both and its inverse are smooth functions.
  • Doubling – 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

F

  • Fiber – In a fiber bundle, the preimage of a point in the base is called the fiber over , often denoted .
  • Frame bundle – the principal bundle of frames on a smooth manifold.

G

H

  • Hypersurface – A hypersurface is a submanifold of codimension one.

I

L

M

  • Manifold – A topological manifold is a locally Euclidean Hausdorff space. (In OODA WIKI, a manifold need not be paracompact or second-countable.) A manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A 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

P

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

S

  • Submanifold – the image of a smooth embedding of a manifold.
  • 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.
  • Transversality – 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.
  • 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 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 .