Glossary of differential geometry and topology

A

Atlas

B

Bundle, see fiber bundle.

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 → N and its inverse f − 1 : N → 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, π: E → B the preimage π⁻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

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^∞ 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.

P

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

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

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

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 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.