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.

Basic element – A basic element with respect to an element is an element of a cochain complex (e.g., complex of differential forms on a manifold) that is closed: and the contraction of by 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 (open access)

D

Dehn twist Diffeomorphism – Given two differentiable manifolds and , a bijective map from to is called a diffeomorphism – if both and its inverse are smooth functions. Differential form Domain invariance

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

E

Embedding Exotic structure – See exotic sphere and exotic .

F

Fiber – In a fiber bundle, the preimage of a point in the base is called the fiber over , often denoted .

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, or 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 together with an action on by a Lie group that preserves the fibers of 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 and intersect transversally if at each point of intersection p their tangent spaces and 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 and over the same base their cartesian product is a vector bundle over . The diagonal map induces a vector bundle over called the Whitney sum of these vector bundles and denoted by . Whitney topologies

References