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 x {\displaystyle x} with respect to an element y {\displaystyle y} is an element of a cochain complex ( C ∗ , d ) {\displaystyle (C^{*},d)} (e.g., complex of differential forms on a manifold) that is closed: d x = 0 {\displaystyle dx=0} and the contraction of x {\displaystyle x} by y {\displaystyle y} 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
D Dehn twist Diffeomorphism – Given two differentiable manifolds M {\displaystyle M} and N {\displaystyle N} , a bijective map f {\displaystyle f} from M {\displaystyle M} to N {\displaystyle N} is called a diffeomorphism – if both f : M → N {\displaystyle f:M\to N} and its inverse f − 1 : N → M {\displaystyle f^{-1}:N\to M} are smooth functions. Differential form Domain invariance Doubling – Given a manifold M {\displaystyle M} with boundary, doubling is taking two copies of M {\displaystyle M} and identifying their boundaries. As the result we get a manifold without boundary.
E Embedding Exotic structure – See exotic sphere and exotic R 4 {\textstyle \mathbb {R} ^{4}} .
F Fiber – In a fiber bundle, π : E → B {\displaystyle \pi :E\to B} the preimage π − 1 ( x ) {\displaystyle \pi ^{-1}(x)} of a point x {\displaystyle x} in the base B {\displaystyle B} is called the fiber over x {\displaystyle x} , often denoted E x {\displaystyle 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 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, C k {\textstyle C^{k}} or C ∞ {\textstyle C^{\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
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 P → B {\displaystyle P\to B} together with an action on P {\displaystyle P} by a Lie group G {\displaystyle G} that preserves the fibers of P {\displaystyle P} 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 M {\displaystyle M} and N {\displaystyle N} intersect transversally if at each point of intersection p their tangent spaces T p ( M ) {\displaystyle T_{p}(M)} and T p ( N ) {\displaystyle T_{p}(N)} 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 α {\displaystyle \alpha } and β {\displaystyle \beta } over the same base B {\displaystyle B} their cartesian product is a vector bundle over B × B {\displaystyle B\times B} . The diagonal map B → B × B {\displaystyle B\to B\times B} induces a vector bundle over B {\displaystyle B} called the Whitney sum of these vector bundles and denoted by α ⊕ β {\displaystyle \alpha \oplus \beta } . Whitney topologies
References