Glossary of differential geometry and topology
# 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:

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

* 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

* 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

* 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 $f^{-1}:N\to M$ 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 $\pi ^{-1}(x)$ 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 (2n + 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 $\alpha \oplus \beta$.
