This articlerelies largely or entirely on asingle source. Relevant discussion may be found on thetalk page. Please helpimprove this article byintroducing citations to additional sources. Find sources: "Glossary of differential geometry and topology" – news ·newspapers ·books ·scholar ·JSTOR(October 2025)

This is aglossary of terms specific todifferential geometry anddifferential 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 initalics denote a self-reference to this glossary.

• Atlas

• Bundle – seefiber bundle.

• Basic element – Abasic element${\displaystyle x}$ with respect to an element${\displaystyle y}$ is an element of acochain complex${\displaystyle (C^{\ast },d)}$ (e.g., complex ofdifferential forms on a manifold) that is closed:${\displaystyle dx=0}$ and the contraction of${\displaystyle x}$ by${\displaystyle y}$ is zero.

• 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]

• Dehn twist
• Diffeomorphism – Given twodifferentiable manifolds${\displaystyle M}$ and${\displaystyle N}$, abijective map${\displaystyle f}$ from${\displaystyle M}$ to${\displaystyle N}$ is called adiffeomorphism – if both${\displaystyle f:M\to N}$ and its inverse${\displaystyle f^{-1}:N\to M}$ aresmooth functions.
• Differential form
• Domain invariance

• Doubling – Given a manifold${\displaystyle M}$ with boundary, doubling is taking two copies of${\displaystyle M}$ and identifying their boundaries. As the result we get a manifold without boundary.

• Embedding
• Exotic structure – Seeexotic sphere andexotic${\mathbb{R}}^4$.

• Fiber – In a fiber bundle,${\displaystyle \pi :E\to B}$ thepreimage${\displaystyle {\pi }^{-1}(x)}$ of a point${\displaystyle x}$ in the base${\displaystyle B}$ is called the fiber over${\displaystyle x}$, often denoted${\displaystyle E_x}$.

• Fiber bundle

• Frame – Aframe at a point of adifferentiable manifoldM is abasis of thetangent space at the point.

• Frame bundle – the principal bundle of frames on a smooth manifold.

• Flow

• Genus
• Germ
• Grassmannian bundle
• Grassmannian manifold

• Handle decomposition
• Hypersurface – A hypersurface is a submanifold ofcodimension one.

• Immersion

• Integration along fibers
• Irreducible manifold
• Isotopy

• Jet
• Jordan curve theorem

• Lens space – A lens space is a quotient of the3-sphere (or (2n + 1)-sphere) by a free isometricaction ofZ –_k.
• Local diffeomorphism

• Manifold – A topological manifold is a locally EuclideanHausdorff space (usually also required to besecond-countable). For a given regularity (e.g. piecewise-linear,$C^k$ or$C^{\mathrm{\infty }}$ 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

• Neat submanifold – A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded.

• Orbifold
• Orientation of a vector bundle

• 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 smoothglobal 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${\displaystyle P\to B}$ together with anaction on${\displaystyle P}$ by aLie group${\displaystyle G}$ that preserves the fibers of${\displaystyle P}$ and acts simply transitively on those fibers.

• Pullback

• Rham cohomology

• 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.

• Tangent bundle – the vector bundle of tangent spaces on a differentiable manifold.

• Tangent field – asection of the tangent bundle. Also called avector field.

• Tangent space

• Thom space

• Torus

• Transversality – Two submanifolds${\displaystyle M}$ and${\displaystyle N}$ intersect transversally if at each point of intersectionp their tangent spaces${\displaystyle T_p(M)}$ and${\displaystyle T_p(N)}$ generate the whole tangent space atp of the total manifold.
• Triangulation

• Trivialization
• Tubular neighborhood

• 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.

• Whitney sum – A Whitney sum is an analog of thedirect product for vector bundles. Given two vector bundles${\displaystyle \alpha }$ and${\displaystyle \beta }$ over the same base${\displaystyle B}$ theircartesian product is a vector bundle over${\displaystyle B\times B}$. The diagonal map${\displaystyle B\to B\times B}$ induces a vector bundle over${\displaystyle B}$ called the Whitney sum of these vector bundles and denoted by${\displaystyle \alpha \oplus \beta }$.
• Whitney topologies

• Glossaries of mathematics
• Differential geometry
• Differential topology

• Articles with short description
• Short description is different from Wikidata
• Use American English from March 2019
• All Wikipedia articles written in American English
• Use mdy dates from March 2019
• Articles needing additional references from October 2025
• All articles needing additional references
• Wikipedia glossaries using unordered lists