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Glossary of differential geometry and topology 
Summary
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.
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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
- 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.
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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.
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H edit
- Hypersurface – A hypersurface is a submanifold of codimension one.
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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.
M edit
- Manifold – A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, 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.
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- Neat submanifold – A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded.
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P edit
- 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.
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- Submanifold – the image of a smooth embedding of a manifold.
- Surface – a two-dimensional manifold or submanifold.
- Systole – least length of a noncontractible loop.
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- 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 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 .
