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This is a [[glossary]] of terms specific to [[differential geometry]] and [[differential topology]].
The following two glossaries are closely related:
*[[Glossary of general topology]]
*[[Glossary of Riemannian and metric geometry]].
 
See also:
*[[List of differential geometry topics]]
 
Words in ''italics'' denote a self-reference to this glossary.
 
{{compactTOC8|side=yes|top=yes|num=yes}}
__NOTOC__
 
==A==
 
'''[[Atlas (topology)|Atlas]]'''
 
==B==
 
'''Bundle''', see ''fiber bundle''.
 
==C==
 
'''[[Chart (topology)|Chart]]'''
 
'''[[Cobordism]]'''
 
'''[[Codimension]]'''. The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold.
 
'''[[Connected sum]]'''
 
'''[[Connection (mathematics)|Connection]]'''
 
'''[[Cotangent bundle]]''', the vector bundle of cotangent spaces on a manifold.
 
'''[[Cotangent space]]'''
 
==D==
 
'''[[Diffeomorphism]].''' Given two [[Manifold#Differentiable_manifolds|differentiable manifolds]]
''M'' and ''N'', a [[bijective map]] <math>f</math> from ''M'' to ''N'' is called a '''diffeomorphism''' if both <math>f:M\to N</math> and its inverse <math>f^{-1}:N\to M</math> are [[smooth function]]s.
 
'''Doubling,''' given a manifold ''M'' with boundary, doubling is taking two copies  of ''M'' and identifying their boundaries.
As the result we get a manifold without boundary.
 
==E==
 
'''[[Embedding]]'''
 
==F==
 
'''Fiber'''. In a fiber bundle, π: ''E'' → ''B'' the [[preimage]] π<sup>&minus;1</sup>(''x'') of a point ''x'' in the base ''B'' is called the fiber over ''x'', often denoted ''E''<sub>''x''</sub>.
 
'''[[Fiber bundle]]'''
 
'''Frame'''.  A '''frame''' at a point of a [[differentiable manifold]] ''M'' is a [[basis of a vector space|basis]] of the [[tangent space]] at the point. 
 
'''[[Frame bundle]]''', the principal bundle of frames on a smooth manifold.
 
'''[[Flow (mathematics)|Flow]]'''
 
==G==
 
'''[[Genus (mathematics)|Genus]]'''
 
==H==
 
'''Hypersurface'''. A hypersurface is a submanifold of ''codimension'' one.
 
==I==
 
'''[[Embedding|Immersion]]'''
 
==L==
 
'''[[Lens space]]'''. A lens space is a quotient of the [[3-sphere]] (or (2''n'' + 1)-sphere) by a free isometric [[group action|action]] of [[cyclic group|'''Z'''<sub>k</sub>]].
 
==M==
 
'''[[Manifold]]'''. A topological manifold is a locally Euclidean [[Hausdorff space]]. (In Wikipedia, a manifold need not be [[paracompact]] or [[second-countable space|second-countable]].) A ''C<sup>k</sup>'' manifold is a differentiable manifold whose chart overlap functions are ''k'' times continuously differentiable. A ''C''<sup></sup> 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.
 
==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 ''P'' → ''B'' together with an [[group action|action]] on ''P'' by a [[Lie group]] ''G'' that preserves the fibers of ''P'' and acts simply transitively on those  fibers.
 
'''[[Pullback]]'''
 
==S==
 
'''[[Section (fiber bundle)|Section]]'''
 
'''[[Submanifold]]''', the image of a smooth embedding of a manifold.
 
'''[[Submersion (mathematics)|Submersion]]'''
 
'''[[Surface]]''', a two-dimensional manifold or submanifold.
 
'''[[systolic geometry|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]]'''
 
'''[[Torus]]'''
 
'''[[Transversality (mathematics)|Transversality]]'''. Two submanifolds ''M'' and ''N'' 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 α and β over the same base ''B'' their [[cartesian product]] is a vector bundle over ''B'' &times;''B''. The diagonal map <math>B\to B\times B</math> induces a vector bundle over ''B'' called the Whitney sum of these vector bundles and denoted by α⊕β.
 
{{DEFAULTSORT:Glossary Of Differential Geometry And Topology}}
[[Category:Glossaries of mathematics|Geometry]]
[[Category:Differential geometry| ]]
[[Category:Differential topology| ]]

Latest revision as of 02:43, 14 December 2014

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