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.

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A

  • Atlas

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

  • Characteristic class
  • 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
  • Covering
  • Cusp
  • CW-complex

D

  • Dehn twist
  • 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.
  • Differential form
  • Domain invariance
  • 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

  • Embedding
  • Exotic structure – See exotic sphere and exotic <math display="inline">\R^4</math>.

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>.
  • 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
  • Germ
  • Grassmannian bundle
  • Grassmannian manifold

H

  • Handle decomposition
  • Hypersurface – A hypersurface is a submanifold of codimension one.

I

  • Immersion
  • Integration along fibers
  • Irreducible manifold
  • Isotopy

J

  • Jet
  • Jordan curve theorem

L

  • Lens space – A lens space is a quotient of the 3-sphere (or (2n + 1)-sphere) by a free isometric action of Z – <sub>k</sub>.
  • Local diffeomorphism

M

  • 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, <math display="inline">C^k</math> or <math display="inline">C^\infty</math> 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

  • Neat submanifold – A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded.

O

  • Orbifold
  • Orientation of a vector bundle

P

  • 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
  • Poincaré lemma
  • 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.
  • Pullback

R

  • Rham cohomology

S

  • Section
  • Seifert fiber space
  • 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>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.
  • Triangulation
  • Trivialization
  • Tubular neighborhood

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>.
  • Whitney topologies

References