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

* References

## A

* Atlas

## B

* Bundle – see fiber bundle.

* Basic element – 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

* 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[1]

## D

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

* 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
* Exotic structure – See exotic sphere and exotic ℝ4.

## F

* Fiber – In a fiber bundle, π:E->B the preimage π−1(x) of a point x in the base B is called the fiber over x, often denoted Ex.

* 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 – k.
* 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, Ck or C∞ 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 P->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

## 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 M and N intersect transversally if at each point of intersection p their tangent spaces Tp(M) and Tp(N) 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 α and β over the same base B their cartesian product is a vector bundle over B×B. The diagonal map B->B×B induces a vector bundle over B called the Whitney sum of these vector bundles and denoted by α⊕β.
* Whitney topologies

## References

1. ↑ Whitehead, J. H. C. (1949a). "Combinatorial homotopy. I.". Bulletin of the American Mathematical Society 55 (5): 213-245. doi:10.1090/S0002-9904-1949-09175-9. https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-55/issue-3.P1/Combinatorial-homotopy-I/bams/1183513543.pdf. (open access)

* 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 theorems (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|

* Calabi–Yau
* Closed
* Collapsing
* Complete
* (Almost) Complex
* (Almost) Contact
* Einstein
* Fibered
* Finsler
* (Almost, Ricci-) Flat
* G-structure
* Hadamard
* Hermitian
* Hyperbolic
* (Hyper) Kähler
* Kenmotsu
* Lie group
* Lie algebra
* Manifold with boundary
* Nilmanifold
* Oriented
* Parallelizable
* Poisson
* Prime
* Quaternionic
* Hypercomplex
* (Pseudo-, Sub-) Riemannian
* Rizza
* Sasakian
* Stein
* (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
* Complex
* Vector-valued
* One-form
* 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
* Diffeology
* Diffiety
* Fréchet
* Hilbert
* K-theory
* Non-Hausdorff
* Orbifold
* Secondary calculus
* over commutative algebras
* Sheaf
* Stratifold
* Supermanifold
* Stratified space

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