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:

• List of differential geometry topics

Words in italics denote a self-reference to this glossary.

A

• Atlas

B

• Bundle – see fiber bundle.

• basic element – A basic element ${\displaystyle x}$ with respect to an element ${\displaystyle y}$ is an element of a
  cochain complex
  ${\displaystyle (C^{*},d)}$ (e.g., complex of differential forms on a manifold) that is closed: ${\displaystyle dx=0}$ and the contraction of ${\displaystyle x}$ by ${\displaystyle y}$ is zero.

C

• 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

D

• Diffeomorphism – Given two differentiable manifolds ${\displaystyle M}$ and ${\displaystyle N}$, a
  bijective map
  ${\displaystyle f}$ from ${\displaystyle M}$ to ${\displaystyle N}$ is called a diffeomorphism – if both ${\displaystyle f:M\to N}$ and its inverse ${\displaystyle f^{-1}:N\to M}$ are
  smooth functions
  .

• Doubling – Given a manifold ${\displaystyle M}$ with boundary, doubling is taking two copies of ${\displaystyle M}$ and identifying their boundaries. As the result we get a manifold without boundary.

E

• Embedding

F

• Fiber – In a fiber bundle, ${\displaystyle \pi :E\to B}$ the
  preimage
  ${\displaystyle \pi ^{-1}(x)}$ of a point ${\displaystyle x}$ in the base ${\displaystyle B}$ is called the fiber over ${\displaystyle x}$, often denoted ${\displaystyle E_{x}}$.

• Fiber bundle

• Frame – A frame at a point of a
  basis of the tangent space
  at the point.

• Frame bundle – the principal bundle of frames on a smooth manifold.

• Flow

G

• Genus

H

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

I

• Immersion

• Integration along fibers

L

• action of Z – _k
  .