Glossary of differential geometry and topology

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''.

A basic element ''$x$'' with respect to an element ''$y$'' is an element of a cochain complex $(C^*, d)$ (e.g., complex of differential forms on a manifold) that is closed: $dx = 0$ and the contraction of ''$x$'' by ''$y$'' 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 ''$M$'' and ''$N$'', a bijective map $f$ from ''$M$'' to ''$N$'' is called a diffeomorphism if both $f:M\to N$ and its inverse $f^:N\to M$ are smooth functions.

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, ''$\pi:E \to B$'' the preimage ''$\pi^(x)$'' of a point ''$x$'' in the base ''$B$'' is called the fiber over ''$x$'', often denoted ''$E_x$''.

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

H

Hypersurface. A hypersurface is a submanifold of ''codimension'' one.

I

Immersion

Integration along fibers

L

Lens space. A lens space is a quotient of the 3-sphere (or (2''n'' + 1)-sphere) by a free isometric action of Z_k.

M

Manifold. A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second-countable.) A ''$C^k$'' manifold is a differentiable manifold whose chart overlap functions are ''k'' times continuously differentiable. A ''$C^\infty$'' 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.

O

Orientation of a vector bundle

P

Parallelizable. A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial.

Poincaré lemma

Principal bundle. A principal bundle is a fiber bundle ''$P \to B$'' together with an action on ''$P$'' by a Lie group ''$G$'' that preserves the fibers of ''$P$'' and acts simply transitively on those fibers.

Pullback

S

Section

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 ''$M$'' and ''$N$'' intersect transversally if at each point of intersection ''p'' their tangent spaces $T_p(M)$ and $T_p(N)$ 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 ''$\alpha$'' and ''$\beta$'' over the same base ''$B$'' their cartesian product is a vector bundle over $B\times B$. The diagonal map $B\to B\times B$ induces a vector bundle over ''$B$'' called the Whitney sum of these vector bundles and denoted by ''$\alpha \oplus \beta$''.

{{DEFAULTSORT:Glossary Of Differential Geometry And Topology
Geometry
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