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Branch of mathematics
mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology
Differential_topology
Branch of mathematics
are given over the space. Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with
Differential_geometry
Branch of mathematics
of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. The
Topology
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
r-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds. Contact geometry studies 1-forms that
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
surfaces (Topology) De Rham's theorem (differential topology) Dehn-Nielsen-Baer theorem (geometric topology) Donaldson's theorem (differential topology) Ehresmann's
List_of_theorems
Branch of differential geometry and differential topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds
Symplectic_geometry
specific to differential geometry and differential topology. The following three glossaries are closely related: Glossary of general topology Glossary of
Glossary of differential geometry and topology
Glossary_of_differential_geometry_and_topology
Topological structure of 4D spacetime
topology of Minkowski space". Topology. 6 (2): 161–170. doi:10.1016/0040-9383(67)90033-X. Penrose, Roger (1972), Techniques of Differential Topology in
Spacetime_topology
have a number of nice properties and are an important concept in differential topology. Three special cases of constant rank maps occur. A constant rank
Rank_(differential_topology)
Mathematical notion of infinitesimal difference
mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology. The term differential is used nonrigorously in calculus
Differential_(mathematics)
Concept of vector calculus
and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0); and an exact form is a differential form
Closed and exact differential forms
Closed_and_exact_differential_forms
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic
Integrability conditions for differential systems
Integrability_conditions_for_differential_systems
Analyzes the topology of a manifold by studying differentiable functions on that manifold
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions
Morse_theory
Smooth manifold
structure in differential geometry Rizza manifold – Manifold in differential geometry Symplectic manifold – Type of manifold in differential geometry Van
Almost_complex_manifold
American mathematician (born 1931)
February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems
John_Milnor
Branch of topology
branches of topology, including differential topology, geometric topology, and algebraic topology. The fundamental concepts in point-set topology are continuity
General_topology
Theorem in differential topology
The hairy ball theorem of algebraic topology (formally, the Sphere Vector Field Theory, sometimes called the hedgehog theorem) states that there is no
Hairy_ball_theorem
Two-dimensional manifold
may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a two-dimensional space; this
Surface_(topology)
Frobenius theorem (differential topology) Distribution (differential geometry) integral curve foliation integrability conditions for differential systems Fiber
List of differential geometry topics
List_of_differential_geometry_topics
motivated by systems of linear partial differential equations, it is a branch of algebraic geometry and algebraic topology that uses methods from sheaf theory
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Relates the geometric vector bundles to algebraic projective modules
In the mathematical fields of differential geometry, topology and algebraic geometry, the Serre–Swan theorem, also called Swan's theorem, relates the
Serre–Swan_theorem
American mathematician (1930–2026)
of topology, including Topology (an undergraduate-level text), Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology
James_Munkres
Results in differential topology
In differential topology in mathematics, the Wall theorems are four results, which connect the smooth structure, auto-diffeomorphisms and the intersection
Wall_theorems
areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric
Glossary_of_general_topology
Inclusion of one mathematical structure in another, preserving properties of interest
necessarily a discrete subspace of its domain X . {\displaystyle X.} In differential topology: Let M {\displaystyle M} and N {\displaystyle N} be smooth manifolds
Embedding
Branch of differential geometry
well as analysis, and spurred the development of algebraic and differential topology. Riemannian geometry was first put forward in generality by Bernhard
Riemannian_geometry
Catalan economist and politician from Spain
theory, his monograph gave a thorough exposition of research using differential topology. His textbook Microeconomic Theory, co-authored with Michael Whinston
Andreu_Mas-Colell
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
theorem) is an important theorem in differential topology that relates zeros of continuous vector field on compact differential manifold to Euler characteristic
Poincaré–Hopf_theorem
Theorem in differential topology
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: The strong Whitney embedding
Whitney_embedding_theorem
Hrushovski and Sokolovic. The Kolchin topology on K m is defined by taking sets of solutions of systems of differential equations over K in m variables as
Differentially_closed_field
Mathematical relation consisting of a multi-variable function equal to zero
simultaneous solutions are called affine algebraic sets. The solutions of differential equations generally appear expressed by an implicit function. In economics
Implicit_function
On when a definite intersection form of a smooth 4-manifold is diagonalizable
In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a closed, oriented
Donaldson's_theorem
Soviet mathematician (1908–1988)
in a number of fields of mathematics, including algebraic topology, differential topology and optimal control. Pontryagin was born in Moscow and lost
Lev_Pontryagin
Mathematical structure
that allows for differential calculus on the manifold. If M is already a topological manifold, it is required that the new topology be identical to the
Differential_structure
Description of how spaces intersect in mathematics
general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecting
Transversality
American mathematician (1934–2025)
American mathematician, who specialized in algebraic topology, differential topology and differential geometry. He served as president of the American Mathematical
William Browder (mathematician)
William_Browder_(mathematician)
Differential map between manifolds whose differential is everywhere surjective
differentiable manifolds whose differential pushforward is everywhere surjective. It is a basic concept in differential topology, dual to that of an immersion
Submersion_(mathematics)
In mathematics, especially differential topology, the Gromoll–Meyer sphere is a special seven-dimensional exotic sphere with several unique properties
Gromoll–Meyer_sphere
Abelian group, in mathematics
In mathematics, especially differential topology and cobordism theory, a Kervaire–Milnor group is an abelian group defined as the h-cobordism classes
Kervaire–Milnor_group
Russian-American mathematician
Stanford University. His research interests are differential topology, symplectic topology, and contact topology. He was awarded many prizes for his work, including
Yakov_Eliashberg
Mathematics award
7-dimensional sphere can have several differential structures; this led to the creation of the field of differential topology" (see exotic sphere). 1966 Moscow
Fields_Medal
Topics referred to by the same term
interpreted as infinitesimals Differential equation, an equation relating derivatives of a function Differential topology Differential (pushforward) The total
Differential
Topological operation of turning a sphere inside-out without creasing
In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means
Sphere_eversion
Motion of particles in a fluid
the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector
Flow_(mathematics)
Fixed-point theorem for smooth manifolds
system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears
Atiyah–Bott fixed-point theorem
Atiyah–Bott_fixed-point_theorem
Study of smooth real-valued functions on manifold and their singularities
In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions
Cerf_theory
How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification
Vector_fields_on_spheres
Area of differential geometry and topology
In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various
Spin_geometry
square-integrable differential forms. The notion of square-integrability makes sense because the metric on M gives rise to a norm on differential forms and a
L²_cohomology
Topological space that locally resembles Euclidean space
(1990) Topology of 4-Manifolds. Princeton University Press. ISBN 0-691-08577-3. Guillemin, Victor and Pollack, Alan (1974) Differential Topology. Prentice-Hall
Manifold
Operator in differential topology
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector
Lie_bracket_of_vector_fields
Cohomology with real coefficients computed using differential forms
Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about
De_Rham_cohomology
Gives the signature of a smooth compact oriented manifold in terms of Pontryagin numbers
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's
Hirzebruch_signature_theorem
Set of functions from a topological space to [0,1] which sum to 1 for any input
ISBN 978-3-031-23816-1. Evans, Lawrence (2010-03-02), "Sobolev spaces", Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical
Partition_of_unity
on the whole phase-portrait. In the particular case of an autonomous differential equation defined on a compact set X, a complete Lyapunov function V from
Conley's fundamental theorem of dynamical systems
Conley's_fundamental_theorem_of_dynamical_systems
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)
Hopf_fibration
Swiss mathematician
1990) was a Swiss mathematician, known for his contributions to differential topology. Georges de Rham was born on 10 September 1903 in Roche, a small
Georges_de_Rham
In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy
Regular_homotopy
Fundamental construction of differential calculus
exterior derivative and the Lie derivative form a Lie superalgebra. In differential topology, a vector field may be defined as a derivation on the ring of smooth
Generalizations of the derivative
Generalizations_of_the_derivative
Mathematical invariants used to classify plane curves
double point. Plane curve Knot invariant Whitney–Graustein theorem Differential topology Arnold, V. I. (1994). Topological Invariants of Plane Curves and
Arnold_invariants
Polynomial with all terms of degree two
theory (orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology (intersection forms of manifolds
Quadratic_form
Topological degree is the only homotopy invariant of continuous maps to spheres
The Hopf theorem (named after Heinz Hopf) is a statement in differential topology, saying that the topological degree is the only homotopy invariant of
Hopf_theorem
American mathematician (born 1937)
arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology. Born in New York City, Mazur attended the Bronx
Barry_Mazur
Type of optimization problem
trajectory stays within a submanifold. See integrability conditions for differential systems for how to decide whether a system of Pfaffian constraints. In
Nonholonomic_system
On the preimage of points in a manifold under the action of a smooth map
In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the
Preimage_theorem
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to
Exotic_sphere
Algebraic structure in homological algebra
– particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic
Differential_graded_algebra
Typically linear operator defined in terms of differentiation of functions
play a major role in setting up and solving partial differential equations. In differential topology, the exterior derivative and Lie derivative operators
Differential_operator
American mathematician (1892–1977)
variations in the large, a subject where he introduced the technique of differential topology now known as Morse theory. The Morse–Palais lemma, one of the key
Marston_Morse
On when a smooth map between smooth manifolds is a locally trivial fibration
In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that if a smooth mapping f : M →
Ehresmann's_lemma
Concept in differential topology
In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold-with-boundary which is
Mazur_manifold
Vector bundle of rank 1
is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of rank 1. Line
Line_bundle
vol. 218, Springer, ISBN 9781441999825 Hirsch, Morris W. (1976). Differential topology. New York Heidelberg Berlin: Springer-Verlag. ISBN 978-1-4684-9449-5
Collar_neighbourhood
Way to join two given mathematical manifolds together
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds
Connected_sum
Set of points equidistant from a center
is a 2-dimensional surface which is embedded in 3-dimensional space. In topology, the n-sphere is an example of a compact topological manifold without boundary
Sphere
Right conoid ruled surface
In geometry, the Whitney umbrella or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella, is
Whitney_umbrella
Subbundle of the tangent bundle
Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology. Even though they share the same name, distributions presented in
Distribution (differential geometry)
Distribution_(differential_geometry)
Theory in differential topology
In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed
Morse_homology
Differentiable function whose derivative is everywhere injective
to differential geometry (Volume 1), Publish or Perish, ISBN 0-914098-70-5 Spring, David (2005), "The golden age of immersion theory in topology: 1959–1973:
Immersion_(mathematics)
solutions of differential equations. It was later given a more explicitly topological interpretation by Vladimir Arnold, who related it to the topology of the
Maslov_index
American mathematician (1913–1973)
taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization of knot theory
Ralph_Fox
Product of the principal curvatures of a surface
In differential geometry, the Gaussian curvature or Gauss curvature (symbol Κ, named after Carl Friedrich Gauss) of a smooth surface in three-dimensional
Gaussian_curvature
Theorem in differential topology
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m > 1 {\displaystyle m>1} , any smooth m {\displaystyle
Whitney_immersion_theorem
Concepts in mathematics
field. These appear in a number of different contexts, including differential topology, Riemannian geometry and Lie group theory. Let V be a smooth vector
Vector_flow
Continuous surjection satisfying a local triviality condition
general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles. Mappings between total spaces
Fiber_bundle
Type of functional equation (mathematics)
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions
Differential_equation
Degree of differentiability of a function or map
different degrees of regularity for partial differential equations. They are used in differential topology to define different classes of differentiable
Smoothness
Topological construct
In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres. Consider
Clutching_construction
Type of manifold in differential geometry
matrix – Mathematical concept Symplectic topology – Branch of differential geometry and differential topologyPages displaying short descriptions of redirect
Symplectic_manifold
In differential topology in mathematics, Donaldson invariants are invariants of the smooth structure of four-dimensional smooth manifolds (short 4-manifolds)
Donaldson_invariant
Topological space in mathematics
In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain sense "longer". It behaves
Long_line_(topology)
In topology in mathematics, Freedman's classification (or Freedman's theorem) is a central result about four-dimensional topological manifolds (short 4-manifolds)
Freedman_classification
Statement about integration on manifolds
JSTOR 2690275. Katz, Victor J. (1999). "5. Differential Forms". In James, I. M. (ed.). History of Topology. Amsterdam: Elsevier. pp. 111–122. ISBN 9780444823755
Generalized_Stokes_theorem
sphere in the tangent space. The unit tangent bundle carries a variety of differential geometric structures. The metric on M induces a contact structure on
Unit_tangent_bundle
Topological space associated to a vector bundle
construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over
Thom_space
(partial differential equations) Grönwall's lemma Lax–Milgram lemma Pugh's closing lemma Weyl's lemma (Laplace equation) (partial differential equations)
List_of_lemmas
Study in mathematical gauge theory
(1983), "An Application of Gauge Theory to Four Dimensional Topology", Journal of Differential Geometry, 18 (2): 279–315, doi:10.4310/jdg/1214437665, MR 0710056
Donaldson_theory
Assignment of vector fields to manifolds
space of possible velocities for a particle moving on the manifold. In differential geometry, one can attach to every point x {\displaystyle x} of a differentiable
Tangent_space
Mathematics concept
to be ker ( d π e ) {\displaystyle \ker(d\pi _{e})} . That is, the differential d π e : T e E → T b B {\displaystyle d\pi _{e}\colon T_{e}E\to T_{b}B}
Vertical and horizontal bundles
Vertical_and_horizontal_bundles
Tool in homological algebra
acyclicity criterion Differential graded module "Graph complex". Bott, Raoul; Tu, Loring W. (1982), Differential Forms in Algebraic Topology, Berlin, New York:
Chain_complex
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