Английская Википедия:Glossary of differential geometry and topology
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Шаблон:Use American English Шаблон:Short description Шаблон:Use mdy dates Шаблон:Unreferenced 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:
Words in italics denote a self-reference to this glossary.
A
B
- Bundle – see fiber bundle.
- basic element – A basic element <math>x</math> with respect to an element <math>y</math> is an element of a cochain complex <math>(C^*, d)</math> (e.g., complex of differential forms on a manifold) that is closed: <math>dx = 0</math> and the contraction of <math>x</math> by <math>y</math> is zero.
C
- Codimension – The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold.
- Cotangent bundle – the vector bundle of cotangent spaces on a manifold.
D
- Diffeomorphism – Given two differentiable manifolds <math>M</math> and <math>N</math>, a bijective map <math>f</math> from <math>M</math> to <math>N</math> is called a diffeomorphism – if both <math>f:M\to N</math> and its inverse <math>f^{-1}:N\to M</math> are smooth functions.
- Doubling – Given a manifold <math>M</math> with boundary, doubling is taking two copies of <math>M</math> and identifying their boundaries. As the result we get a manifold without boundary.
E
F
- Fiber – In a fiber bundle, <math>\pi:E \to B</math> the preimage <math>\pi^{-1}(x)</math> of a point <math>x</math> in the base <math>B</math> is called the fiber over <math>x</math>, often denoted <math>E_x</math>.
- 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.
G
H
- Hypersurface – A hypersurface is a submanifold of codimension one.
I
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>C^k</math> manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A <math>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
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 <math>P \to B</math> together with an action on <math>P</math> by a Lie group <math>G</math> that preserves the fibers of <math>P</math> 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 <math>M</math> and <math>N</math> intersect transversally if at each point of intersection p their tangent spaces <math>T_p(M)</math> and <math>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>\alpha</math> and <math>\beta</math> over the same base <math>B</math> their cartesian product is a vector bundle over <math>B\times B</math>. The diagonal map <math>B\to B\times B</math> induces a vector bundle over <math>B</math> called the Whitney sum of these vector bundles and denoted by <math>\alpha \oplus \beta</math>.
