Glossary of differential geometry and topology

Short description: Mathematics glossary

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

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

E

• Embedding

F

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

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