Glossary of differential geometry and topology
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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
[edit]B
[edit]- 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
[edit]- 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
[edit]- Dehn twist
- Diffeomorphism – 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 invariance
- 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
[edit]- Embedding
- Exotic structure – See exotic sphere and exotic .
F
[edit]- Fiber – In a fiber bundle, the preimage of a point in the base is called the fiber over , often denoted .
- 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
[edit]H
[edit]- Handle decomposition
- Hypersurface – A hypersurface is a submanifold of codimension one.
I
[edit]J
[edit]L
[edit]- Lens space – A lens space is a quotient of the 3-sphere (or (2n + 1)-sphere) by a free isometric action of Z – k.
- Local diffeomorphism
M
[edit]- Manifold – A topological manifold is a locally Euclidean Hausdorff space (usually also required to be second-countable). For a given regularity (e.g. piecewise-linear, or differentiable, real or complex analytic, Lipschitz, Hölder, quasi-conformal...), a manifold of that regularity is a topological manifold whose charts transitions have the prescribed regularity.
- Manifold with boundary
- Manifold with corners
- Mapping class group
- Morse function
N
[edit]- Neat submanifold – A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded.
O
[edit]P
[edit]- 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 unity
- PL-map
- 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.
R
[edit]S
[edit]- 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
[edit]- 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.
- Triangulation
V
[edit]- 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
[edit]- 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