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.
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
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 and , a bijective map from to is called a diffeomorphism – if both and its inverse are smooth functions.
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
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
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 manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A 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 together with an action on by a Lie group that preserves the fibers of 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 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.
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 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 .