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.

Contents:

* Top
* 0–9
* A
* B
* C
* D
* E
* F
* G
* H
* I
* J
* K
* L
* M
* N
* O
* P
* Q
* R
* S
* T
* U
* V
* W
* X
* Y
* Z

## A

[edit]

* Atlas

## B

[edit]

* Bundle – see fiber bundle.

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

## C

[edit]

* 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

[edit]

* Dehn twist
* Diffeomorphism – Given two differentiable manifolds M {\displaystyle M} and  N {\displaystyle N} , a bijective map f {\displaystyle f} from  M {\displaystyle M} to  N {\displaystyle N} is called a diffeomorphism – if both  f : M -> N {\displaystyle f:M\to N} and its inverse  f − 1 : N -> M {\displaystyle f^{-1}:N\to M} are smooth functions.
* Differential form
* Domain invariance

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

## E

[edit]

* Embedding
* Exotic structure – See exotic sphere and exotic  R 4 {\textstyle \mathbb {R} ^{4}} .

## F

[edit]

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

[edit]

* Genus
* Germ
* Grassmannian bundle
* Grassmannian manifold

## H

[edit]

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

## I

[edit]

* Immersion

* Integration along fibers
* Irreducible manifold
* Isotopy

## J

[edit]

* Jet
* Jordan curve theorem

## 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.
* Local diffeomorphism

## M

[edit]

* 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,  C k {\textstyle C^{k}} or  C ∞ {\textstyle C^{\infty }} 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

[edit]

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

## O

[edit]

* Orbifold
* Orientation of a vector bundle

## P

[edit]

* 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  P -> B {\displaystyle P\to B} together with an action on  P {\displaystyle P} by a Lie group G {\displaystyle G} that preserves the fibers of  P {\displaystyle P} and acts simply transitively on those fibers.

* Pullback

## R

[edit]

* Rham cohomology

## S

[edit]

* 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

[edit]

* 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 {\displaystyle M} and  N {\displaystyle N} intersect transversally if at each point of intersection p their tangent spaces  T p ( M ) {\displaystyle T_{p}(M)} and  T p ( N ) {\displaystyle T_{p}(N)} generate the whole tangent space at p of the total manifold.
* Triangulation

* Trivialization
* Tubular neighborhood

## 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  α {\displaystyle \alpha } and  β {\displaystyle \beta } over the same base  B {\displaystyle B} their cartesian product is a vector bundle over  B × B {\displaystyle B\times B} . The diagonal map  B -> B × B {\displaystyle B\to B\times B} induces a vector bundle over  B {\displaystyle B} called the Whitney sum of these vector bundles and denoted by  α ⊕ β {\displaystyle \alpha \oplus \beta } .
* Whitney topologies

* v
* t
* e

Manifolds (Glossary, List, Category)

Basic concepts|

* Topological manifold
* Atlas
* Differentiable/Smooth manifold
* Differential structure
* Smooth atlas
* Submanifold
* Riemannian manifold
* Smooth map
* Submersion
* Pushforward
* Tangent space
* Differential form
* Vector field

Main results (list)|

* Atiyah–Singer index
* Darboux's
* De Rham's
* Frobenius
* Generalized Stokes
* Hopf–Rinow
* Noether's
* Sard's
* Whitney embedding

Maps|

* Curve
* Diffeomorphism
* Local
* Geodesic
* Exponential map
* in Lie theory
* Foliation
* Immersion
* Integral curve
* Lie derivative
* Section
* Submersion

Types of
manifolds|

* Closed
* Collapsing
* Complete
* (Almost) Complex
* (Almost) Contact
* Fibered
* Finsler
* (Almost) Flat
* G-structure
* Hadamard
* Hermitian
* Hyperbolic
* Kähler
* Kenmotsu
* Lie group
* Lie algebra
* Manifold with boundary
* Nilmanifold
* Oriented
* Parallelizable
* Poisson
* Prime
* Quaternionic
* Hypercomplex
* (Pseudo−, Sub−) Riemannian
* Rizza
* (Almost) Symplectic
* Tame

Tensors|

| Vectors|

* Distribution
* Lie bracket
* Pushforward
* Tangent space
* bundle
* Torsion
* Vector field
* Vector flow

|

Covectors|

* Closed/Exact
* Covariant derivative
* Cotangent space
* bundle
* De Rham cohomology
* Differential form
* Vector-valued
* Exterior derivative
* Interior product
* Pullback
* Ricci curvature
* flow
* Riemann curvature tensor
* Tensor field
* density
* Volume form
* Wedge product

Bundles|

* Adjoint
* Affine
* Associated
* Cotangent
* Dual
* Fiber
* (Co) Fibration
* Jet
* Lie algebra
* (Stable) Normal
* Principal
* Spinor
* Subbundle
* Tangent
* Tensor
* Vector

Connections|

* Affine
* Cartan
* Ehresmann
* Form
* Generalized
* Koszul
* Levi-Civita
* Principal
* Vector
* Parallel transport

Related|

* Classification of manifolds
* Gauge theory
* History
* Morse theory
* Moving frame
* Singularity theory

Generalizations|

* Banach manifold
* Diffeology
* Diffiety
* Fréchet manifold
* K-theory
* Orbifold
* Secondary calculus
* over commutative algebras
* Sheaf
* Stratifold
* Supermanifold
* Stratified space

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