Get it on Google Play
New! Download Unionpedia on your Android™ device!
Faster access than browser!
And Ads-free!


In mathematics, a manifold is a topological space that resembles Euclidean space near each point. [1]

275 relations: Abelian variety, Absolute value, Affine connection, Algebraic geometry, Algebraic topology, Algebraic variety, Ambient space, Analysis Situs (paper), Analytic continuation, Analytic function, Analytic manifold, Analytical mechanics, Andrey Markov, Jr., Angle, Arc length, Area, Atiyah–Singer index theorem, Atlas, Atlas (topology), Ball (mathematics), Banach manifold, Banach space, Bernhard Riemann, Betti number, Bijection, Boundary (topology), Boy's surface, Calculus, Carl Friedrich Gauss, Carl Gustav Jacob Jacobi, Cartesian coordinate system, Cartesian product, Category theory, Characteristic class, Circle, Classical mechanics, Closed manifold, Cohomology, Compact space, Complement (set theory), Complete set of invariants, Complex manifold, Complex number, Connected space, Conservation law, Contact geometry, Continuous function, Countable set, Covering space, CR manifold, ..., Cross-cap, Cubic plane curve, Curvature, Curvature of Riemannian manifolds, Curve, CW complex, Cylinder (geometry), Derivative, Diffeomorphism, Differentiable function, Differentiable manifold, Differential form, Differential geometry, Differential structure, Differential topology, Digital manifold, Digital topology, Dimension, Directional statistics, Disjoint union, Disk (mathematics), Distance, Divergence, Dot product, Earth, Elliptic geometry, Elliptic integral, Embedding, Equivalence class, Euclid, Euclidean space, Euler characteristic, Filtration (mathematics), Finite group, Foundations of mathematics, Fréchet manifold, Fréchet space, Functional (mathematics), Functional analysis, Gauss–Bonnet theorem, General linear group, General relativity, General topology, Generalized coordinates, Generalized Poincaré conjecture, Genus (mathematics), Geodesic, Geometric topology, Geometrization conjecture, Geometry, Geometry and topology, German language, Giovanni Girolamo Saccheri, Gradient, Graph of a function, Grigori Perelman, Group (mathematics), Group action, Hamiltonian mechanics, Handlebody, Harmonic analysis, Harmonic function, Hassler Whitney, Hausdorff space, Hearing the shape of a drum, Heat kernel, Henri Poincaré, Hermann Weyl, Hilbert manifold, Hilbert space, Holomorphic function, Homeomorphism, Homology (mathematics), Homology manifold, Homotopy, Hyperbola, Hyperbolic geometry, Hypersphere, Immersion (mathematics), Implicit function, Implicit function theorem, Inner product space, Interval (mathematics), Invariant (mathematics), Inverse function, János Bolyai, John Milnor, Joseph-Louis Lagrange, Klein bottle, Knot theory, Lagrangian mechanics, Laplace operator, Lemniscate, Length, Leonhard Euler, Lie group, Line (geometry), Linear algebra, List of manifolds, Local homeomorphism, Locally connected space, Locally constant function, Locus (mathematics), Long line (topology), Manifold, Map, Map (mathematics), Map projection, Maps of manifolds, Mathematical analysis, Mathematical induction, Mathematical physics, Mathematics, Mathematics of general relativity, Matrix (mathematics), Max Dehn, Möbius strip, Metric (mathematics), Michael Freedman, Michael Spivak, Morphism of algebraic varieties, Morse theory, Nash embedding theorem, Nautical chart, Neighbourhood (mathematics), Niels Henrik Abel, Nikolai Lobachevsky, Non-Euclidean geometry, Non-Hausdorff manifold, Normal (geometry), Normed vector space, Orbifold, Orientability, Orthogonal group, Parabola, Parallel postulate, Partial differential equation, Phase space, Piecewise linear manifold, Plane (geometry), Poincaré conjecture, Poisson bracket, Polar coordinate system, Polytope, Poul Heegaard, Power series, Product topology, Projection (mathematics), Projective plane, Pseudo-Riemannian manifold, Pseudogroup, Quotient space (topology), Ramification (mathematics), Real number, Real projective plane, Real projective space, Rectifiable set, René Thom, Riemann surface, Riemannian manifold, Riemannian submersion, Ringed space, Scheme (mathematics), Second-countable space, Semialgebraic set, Sergei Novikov (mathematician), Sheaf (mathematics), Siméon Denis Poisson, Simon Donaldson, Simplicial complex, Simply connected space, Simultaneous equations, Singular homology, Singular point of an algebraic variety, Singularity (mathematics), Smoothness, Sophus Lie, Spacetime, Sphere, Spherical harmonics, Square, Stable normal bundle, Stephen Smale, Subanalytic set, Submanifold, Submersion (mathematics), Surface, Surgery theory, Symmetry group, Symplectic manifold, Symplectomorphism, Synonym, Table of Lie groups, Tangent, Tangent space, Tangent vector, Theorema Egregium, Topological manifold, Topological map, Topological property, Topological space, Topological vector space, Topologically stratified space, Topology, Torsion tensor, Torus, Two-dimensional space, Uniformization theorem, Unit circle, Unit disk, Unit sphere, Vector field, Vertex (geometry), Volume, Well-behaved, Whitney conditions, Whitney embedding theorem, Whitney immersion theorem, William Kingdon Clifford, William Rowan Hamilton, William Thurston, Yang–Mills theory, 3-manifold, 4-manifold, 5-manifold. Expand index (225 more) »

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.

New!!: Manifold and Abelian variety · See more »

Absolute value

In mathematics, the absolute value (or modulus) of a real number is the non-negative value of without regard to its sign.

New!!: Manifold and Absolute value · See more »

Affine connection

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.

New!!: Manifold and Affine connection · See more »

Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

New!!: Manifold and Algebraic geometry · See more »

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

New!!: Manifold and Algebraic topology · See more »

Algebraic variety

In mathematics, algebraic varieties (also called varieties) are one of the central objects of study in algebraic geometry.

New!!: Manifold and Algebraic variety · See more »

Ambient space

An ambient space or ambient configuration space is the space surrounding an object.

New!!: Manifold and Ambient space · See more »

Analysis Situs (paper)

"Analysis Situs" is a seminal mathematics paper that Henri Poincaré published in 1895.

New!!: Manifold and Analysis Situs (paper) · See more »

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.

New!!: Manifold and Analytic continuation · See more »

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

New!!: Manifold and Analytic function · See more »

Analytic manifold

In mathematics, an analytic manifold is a topological manifold with analytic transition maps.

New!!: Manifold and Analytic manifold · See more »

Analytical mechanics

In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics.

New!!: Manifold and Analytical mechanics · See more »

Andrey Markov, Jr.

Andrey Andreyevich Markov Jr. (Андре́й Андре́евич Ма́рков; St. Petersburg, September 22, 1903 – Moscow, October 11, 1979) was a Soviet mathematician, the son of the Russian mathematician Andrey Andreyevich Markov Sr, and one of the key founders of the Russian school of constructive mathematics and logic.

New!!: Manifold and Andrey Markov, Jr. · See more »


In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

New!!: Manifold and Angle · See more »

Arc length

Determining the length of an irregular arc segment is also called rectification of a curve.

New!!: Manifold and Arc length · See more »


Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.

New!!: Manifold and Area · See more »

Atiyah–Singer index theorem

In differential geometry, the Atiyah–Singer index theorem, proved by, states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data).

New!!: Manifold and Atiyah–Singer index theorem · See more »


An atlas is a collection of maps; it is typically a map of Earth or a region of Earth, but there are atlases of the other planets (and their satellites) in the Solar System.

New!!: Manifold and Atlas · See more »

Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

New!!: Manifold and Atlas (topology) · See more »

Ball (mathematics)

In mathematics, a ball is the space inside a sphere.

New!!: Manifold and Ball (mathematics) · See more »

Banach manifold

In mathematics, a Banach manifold is a manifold modeled on Banach spaces.

New!!: Manifold and Banach manifold · See more »

Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

New!!: Manifold and Banach space · See more »

Bernhard Riemann

Georg Friedrich Bernhard Riemann (September 17, 1826 – July 20, 1866) was an influential German mathematician who made lasting and revolutionary contributions to analysis, number theory, and differential geometry.

New!!: Manifold and Bernhard Riemann · See more »

Betti number

In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.

New!!: Manifold and Betti number · See more »


In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elements of two sets, where every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set.

New!!: Manifold and Bijection · See more »

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.

New!!: Manifold and Boundary (topology) · See more »

Boy's surface

In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 (he discovered it on assignment from David Hilbert to prove that the projective plane could not be immersed in 3-space).

New!!: Manifold and Boy's surface · See more »


Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.

New!!: Manifold and Calculus · See more »

Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß,; Carolus Fridericus Gauss) (30 April 177723 February 1855) was a German mathematician who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, mechanics, electrostatics, astronomy, matrix theory, and optics.

New!!: Manifold and Carl Friedrich Gauss · See more »

Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

New!!: Manifold and Carl Gustav Jacob Jacobi · See more »

Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.

New!!: Manifold and Cartesian coordinate system · See more »

Cartesian product

In mathematics, a Cartesian product is a mathematical operation which returns a set (or product set or simply product) from multiple sets.

New!!: Manifold and Cartesian product · See more »

Category theory

Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms).

New!!: Manifold and Category theory · See more »

Characteristic class

In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not.

New!!: Manifold and Characteristic class · See more »


A circle is a simple shape in Euclidean geometry.

New!!: Manifold and Circle · See more »

Classical mechanics

In physics, classical mechanics and quantum mechanics are the two major sub-fields of mechanics.

New!!: Manifold and Classical mechanics · See more »

Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary.

New!!: Manifold and Closed manifold · See more »


In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex.

New!!: Manifold and Cohomology · See more »

Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

New!!: Manifold and Compact space · See more »

Complement (set theory)

In set theory, a complement of a set A refers to things not in (that is, things outside of) A. The relative complement of A with respect to a set B is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of all elements in U but not in A.

New!!: Manifold and Complement (set theory) · See more »

Complete set of invariants

In mathematics, a complete set of invariants for a classification problem is a collection of maps (where X is the collection of objects being classified, up to some equivalence relation, and the Y_i are some sets), such that x \sim x' if and only if f_i(x).

New!!: Manifold and Complete set of invariants · See more »

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

New!!: Manifold and Complex manifold · See more »

Complex number

A complex number is a number that can be expressed in the form, where and are real numbers and is the imaginary unit, that satisfies the equation.

New!!: Manifold and Complex number · See more »

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

New!!: Manifold and Connected space · See more »

Conservation law

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time.

New!!: Manifold and Conservation law · See more »

Contact geometry

In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'.

New!!: Manifold and Contact geometry · See more »

Continuous function

In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output.

New!!: Manifold and Continuous function · See more »

Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

New!!: Manifold and Countable set · See more »

Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

New!!: Manifold and Covering space · See more »

CR manifold

In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.

New!!: Manifold and CR manifold · See more »


In mathematics, a cross-cap is a two-dimensional surface in 3-space that is one-sided and the continuous image of a Möbius strip that intersects itself in an interval.

New!!: Manifold and Cross-cap · See more »

Cubic plane curve

In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation applied to homogeneous coordinates x:y:z for the projective plane; or the inhomogeneous version for the affine space determined by setting z.

New!!: Manifold and Cubic plane curve · See more »


In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

New!!: Manifold and Curvature · See more »

Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point.

New!!: Manifold and Curvature of Riemannian manifolds · See more »


In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but which is not required to be straight.

New!!: Manifold and Curve · See more »

CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory.

New!!: Manifold and CW complex · See more »

Cylinder (geometry)

A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler") is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder.

New!!: Manifold and Cylinder (geometry) · See more »


The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable).

New!!: Manifold and Derivative · See more »


In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

New!!: Manifold and Diffeomorphism · See more »

Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

New!!: Manifold and Differentiable function · See more »

Differentiable manifold

In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

New!!: Manifold and Differentiable manifold · See more »

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.

New!!: Manifold and Differential form · See more »

Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

New!!: Manifold and Differential geometry · See more »

Differential structure

In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold.

New!!: Manifold and Differential structure · See more »

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.

New!!: Manifold and Differential topology · See more »

Digital manifold

In mathematics, a digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space.

New!!: Manifold and Digital manifold · See more »

Digital topology

Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images that correspond to topological properties (e.g., connectedness) or topological features (e.g., boundaries) of objects.

New!!: Manifold and Digital topology · See more »


In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

New!!: Manifold and Dimension · See more »

Directional statistics

Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Rn), axes (lines through the origin in Rn) or rotations in Rn.

New!!: Manifold and Directional statistics · See more »

Disjoint union

In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in.

New!!: Manifold and Disjoint union · See more »

Disk (mathematics)

In geometry, a disk (also spelled disc).

New!!: Manifold and Disk (mathematics) · See more »


Distance is a numerical description of how far apart objects are.

New!!: Manifold and Distance · See more »


In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar.

New!!: Manifold and Divergence · See more »

Dot product

In mathematics, the dot product or scalar product (sometimes inner product in the context of Euclidean space, or rarely projection product for emphasizing the geometric significance), is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.

New!!: Manifold and Dot product · See more »


Earth (also the world, in Greek: Gaia, or in Latin: Terra), is the third planet from the Sun, the densest planet in the Solar System, the largest of the Solar System's four terrestrial planets, and the only astronomical object known to accommodate life.

New!!: Manifold and Earth · See more »

Elliptic geometry

Elliptic geometry, a special case of Riemannian geometry, is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p, as all lines in elliptic geometry intersect.

New!!: Manifold and Elliptic geometry · See more »

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.

New!!: Manifold and Elliptic integral · See more »


In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

New!!: Manifold and Embedding · See more »

Equivalence class

In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of elements that are related to one another, forming what are called equivalence classes.

New!!: Manifold and Equivalence class · See more »


Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "Father of Geometry".

New!!: Manifold and Euclid · See more »

Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

New!!: Manifold and Euclidean space · See more »

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.

New!!: Manifold and Euler characteristic · See more »

Filtration (mathematics)

In mathematics, a filtration \mathcal is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that.

New!!: Manifold and Filtration (mathematics) · See more »

Finite group

In abstract algebra, a finite group is a mathematical group with a finite number of elements.

New!!: Manifold and Finite group · See more »

Foundations of mathematics

Foundations of mathematics is the study of the logical and philosophical basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.

New!!: Manifold and Foundations of mathematics · See more »

Fréchet manifold

In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.

New!!: Manifold and Fréchet manifold · See more »

Fréchet space

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.

New!!: Manifold and Fréchet space · See more »

Functional (mathematics)

In mathematics, and particularly in functional analysis and the calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers.

New!!: Manifold and Functional (mathematics) · See more »

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense.

New!!: Manifold and Functional analysis · See more »

Gauss–Bonnet theorem

The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).

New!!: Manifold and Gauss–Bonnet theorem · See more »

General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

New!!: Manifold and General linear group · See more »

General relativity

General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

New!!: Manifold and General relativity · See more »

General topology

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.

New!!: Manifold and General topology · See more »

Generalized coordinates

In analytical mechanics, specifically the study of the rigid body dynamics of multibody systems, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration.

New!!: Manifold and Generalized coordinates · See more »

Generalized Poincaré conjecture

In the mathematical area of topology, the term Generalized Poincaré conjecture refers to a statement that a manifold which is a homotopy sphere 'is' a sphere.

New!!: Manifold and Generalized Poincaré conjecture · See more »

Genus (mathematics)

In mathematics, genus (plural genera) has a few different, but closely related, meanings.

New!!: Manifold and Genus (mathematics) · See more »


In mathematics, particularly differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

New!!: Manifold and Geodesic · See more »

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

New!!: Manifold and Geometric topology · See more »

Geometrization conjecture

In mathematics, Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them.

New!!: Manifold and Geometrization conjecture · See more »


Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

New!!: Manifold and Geometry · See more »

Geometry and topology

In mathematics, geometry and topology is an umbrella term for geometry and topology, as the line between these two is often blurred, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory.

New!!: Manifold and Geometry and topology · See more »

German language

German (Deutsch) is a West Germanic language that derives most of its vocabulary from the Germanic branch of the Indo-European language family.

New!!: Manifold and German language · See more »

Giovanni Girolamo Saccheri

Giovanni Girolamo Saccheri (5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician.

New!!: Manifold and Giovanni Girolamo Saccheri · See more »


In mathematics, the gradient is a generalization of the usual concept of derivative of a function in one dimension to a function in several dimensions.

New!!: Manifold and Gradient · See more »

Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs.

New!!: Manifold and Graph of a function · See more »

Grigori Perelman

Grigori Yakovlevich Perelman (a; Григо́рий Я́ковлевич Перельма́н; born 13 June 1966) is a Russian mathematician who made landmark contributions to Riemannian geometry and geometric topology before apparently withdrawing from mathematics.

New!!: Manifold and Grigori Perelman · See more »

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element.

New!!: Manifold and Group (mathematics) · See more »

Group action

In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object.

New!!: Manifold and Group action · See more »

Hamiltonian mechanics

Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.

New!!: Manifold and Hamiltonian mechanics · See more »


In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces.

New!!: Manifold and Handlebody · See more »

Harmonic analysis

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).

New!!: Manifold and Harmonic analysis · See more »

Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R (where U is an open subset of Rn) which satisfies Laplace's equation, i.e. everywhere on U. This is usually written as or.

New!!: Manifold and Harmonic function · See more »

Hassler Whitney

Hassler Whitney (23 March 190710 May 1989) was an American mathematician.

New!!: Manifold and Hassler Whitney · See more »

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

New!!: Manifold and Hausdorff space · See more »

Hearing the shape of a drum

To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones, via the use of mathematical theory.

New!!: Manifold and Hearing the shape of a drum · See more »

Heat kernel

In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions.

New!!: Manifold and Heat kernel · See more »

Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and a philosopher of science.

New!!: Manifold and Henri Poincaré · See more »

Hermann Weyl

Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.

New!!: Manifold and Hermann Weyl · See more »

Hilbert manifold

In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces.

New!!: Manifold and Hilbert manifold · See more »

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

New!!: Manifold and Hilbert space · See more »

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis.

New!!: Manifold and Holomorphic function · See more »


In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

New!!: Manifold and Homeomorphism · See more »

Homology (mathematics)

In mathematics (especially algebraic topology and abstract algebra), homology (in part from Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.

New!!: Manifold and Homology (mathematics) · See more »

Homology manifold

In mathematics, a homology manifold (or generalized manifold) is a locally compact topological space X that looks locally like a topological manifold from the point of view of homology theory.

New!!: Manifold and Homology manifold · See more »


In topology, two continuous functions from one topological space to another are called homotopic (Greek ὁμός (homós).

New!!: Manifold and Homotopy · See more »


In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set.

New!!: Manifold and Hyperbola · See more »

Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

New!!: Manifold and Hyperbolic geometry · See more »


In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center.

New!!: Manifold and Hypersphere · See more »

Immersion (mathematics)

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.

New!!: Manifold and Immersion (mathematics) · See more »

Implicit function

In mathematics, an implicit equation is a relation of the form R(x1,..., xn).

New!!: Manifold and Implicit function · See more »

Implicit function theorem

In multivariable calculus, the implicit function theorem, also known, especially in Italy, as Dini's theorem, is a tool that allows relations to be converted to functions of several real variables.

New!!: Manifold and Implicit function theorem · See more »

Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

New!!: Manifold and Inner product space · See more »

Interval (mathematics)

In mathematics, an (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

New!!: Manifold and Interval (mathematics) · See more »

Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.

New!!: Manifold and Invariant (mathematics) · See more »

Inverse function

In mathematics, an inverse function is a function that "reverses" another function.

New!!: Manifold and Inverse function · See more »

János Bolyai

János Bolyai (15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, one of the founders of non-Euclidean geometry — a geometry that differs from Euclidean geometry in its definition of parallel lines.

New!!: Manifold and János Bolyai · See more »

John Milnor

John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems.

New!!: Manifold and John Milnor · See more »

Joseph-Louis Lagrange

Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia or Giuseppe Ludovico De la Grange Tournier (also reported as Giuseppe Luigi Lagrange or Lagrangia) (25 January 1736 – 10 April 1813) was an Italian Enlightenment Era mathematician and astronomer.

New!!: Manifold and Joseph-Louis Lagrange · See more »

Klein bottle

In mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.

New!!: Manifold and Klein bottle · See more »

Knot theory

In topology, knot theory is the study of mathematical knots.

New!!: Manifold and Knot theory · See more »

Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

New!!: Manifold and Lagrangian mechanics · See more »

Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

New!!: Manifold and Laplace operator · See more »


In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves.

New!!: Manifold and Lemniscate · See more »


In geometric measurements, length is the most extended dimension of an object.

New!!: Manifold and Length · See more »

Leonhard Euler

Leonhard Euler (17071783) was a pioneering Swiss mathematician and physicist.

New!!: Manifold and Leonhard Euler · See more »

Lie group

In mathematics, a Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

New!!: Manifold and Lie group · See more »

Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

New!!: Manifold and Line (geometry) · See more »

Linear algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces.

New!!: Manifold and Linear algebra · See more »

List of manifolds

This is a list of particular manifolds, by Wikipedia page.

New!!: Manifold and List of manifolds · See more »

Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure.

New!!: Manifold and Local homeomorphism · See more »

Locally connected space

In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.

New!!: Manifold and Locally connected space · See more »

Locally constant function

In mathematics, a function f from a topological space A to a set B is called locally constant, if for every a in A there exists a neighborhood U of a, such that f is constant on U. Every constant function is locally constant.

New!!: Manifold and Locally constant function · See more »

Locus (mathematics)

In geometry, a locus (plural: loci) is a set of points whose location satisfies or is determined by one or more specified conditions.

New!!: Manifold and Locus (mathematics) · See more »

Long line (topology)

In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer".

New!!: Manifold and Long line (topology) · See more »


In mathematics, a manifold is a topological space that resembles Euclidean space near each point.

New!!: Manifold and Manifold · See more »


A map is a symbolic depiction highlighting relationships between elements of some space, such as objects, regions, and themes.

New!!: Manifold and Map · See more »

Map (mathematics)

In mathematics, the term mapping, usually shortened to map, refers to either.

New!!: Manifold and Map (mathematics) · See more »

Map projection

Commonly, a map projection is a systematic transformation of the latitudes and longitudes of locations on the surface of a sphere or an ellipsoid into locations on a plane.

New!!: Manifold and Map projection · See more »

Maps of manifolds

In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed.

New!!: Manifold and Maps of manifolds · See more »

Mathematical analysis

Mathematical analysis is a branch of mathematics that studies continuous change and includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions.

New!!: Manifold and Mathematical analysis · See more »

Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers.

New!!: Manifold and Mathematical induction · See more »

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics.

New!!: Manifold and Mathematical physics · See more »


Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

New!!: Manifold and Mathematics · See more »

Mathematics of general relativity

The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying and formulating Albert Einstein's theory of general relativity.

New!!: Manifold and Mathematics of general relativity · See more »

Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array—of numbers, symbols, or expressions, arranged in rows and columns—that is interpreted and manipulated in certain prescribed ways.

New!!: Manifold and Matrix (mathematics) · See more »

Max Dehn

Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German-born American mathematician and student of David Hilbert.

New!!: Manifold and Max Dehn · See more »

Möbius strip

The Möbius strip or Möbius band ((non-rhotic) or), also Mobius or Moebius, is a surface with only one side and only one boundary.

New!!: Manifold and Möbius strip · See more »

Metric (mathematics)

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.

New!!: Manifold and Metric (mathematics) · See more »

Michael Freedman

Michael Hartley Freedman (born 21 April 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara.

New!!: Manifold and Michael Freedman · See more »

Michael Spivak

Michael David Spivak (born May 25, 1940)Biographical sketch in, Vol.

New!!: Manifold and Michael Spivak · See more »

Morphism of algebraic varieties

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.

New!!: Manifold and Morphism of algebraic varieties · See more »

Morse theory

In differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold.

New!!: Manifold and Morse theory · See more »

Nash embedding theorem

The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space.

New!!: Manifold and Nash embedding theorem · See more »

Nautical chart

A nautical chart is a graphic representation of a maritime area and adjacent coastal regions.

New!!: Manifold and Nautical chart · See more »

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

New!!: Manifold and Neighbourhood (mathematics) · See more »

Niels Henrik Abel

Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.

New!!: Manifold and Niels Henrik Abel · See more »

Nikolai Lobachevsky

Nikolai Ivanovich Lobachevsky (a; &ndash) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry.

New!!: Manifold and Nikolai Lobachevsky · See more »

Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

New!!: Manifold and Non-Euclidean geometry · See more »

Non-Hausdorff manifold

In mathematics, it is a usual axiom of a manifold to be a Hausdorff space, and this is assumed throughout geometry and topology: "manifold" means "(second countable) Hausdorff manifold".

New!!: Manifold and Non-Hausdorff manifold · See more »

Normal (geometry)

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.

New!!: Manifold and Normal (geometry) · See more »

Normed vector space

In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn.

New!!: Manifold and Normed vector space · See more »


T^3/S_3 – the quotient of the 3-torus by the symmetric group on 3 letters. --> In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.

New!!: Manifold and Orbifold · See more »


In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.

New!!: Manifold and Orientability · See more »

Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

New!!: Manifold and Orthogonal group · See more »


A parabola (plural parabolas or parabolae, adjective parabolic, from παραβολή) is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane.

New!!: Manifold and Parabola · See more »

Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry.

New!!: Manifold and Parallel postulate · See more »

Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

New!!: Manifold and Partial differential equation · See more »

Phase space

In mathematics and physics, a phase space of a dynamical system is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.

New!!: Manifold and Phase space · See more »

Piecewise linear manifold

In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it.

New!!: Manifold and Piecewise linear manifold · See more »

Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface.

New!!: Manifold and Plane (geometry) · See more »

Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

New!!: Manifold and Poincaré conjecture · See more »

Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system.

New!!: Manifold and Poisson bracket · See more »

Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

New!!: Manifold and Polar coordinate system · See more »


In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope.

New!!: Manifold and Polytope · See more »

Poul Heegaard

Poul Heegaard (November 2, 1871, Copenhagen - February 7, 1948, Oslo) was a Danish mathematician active in the field of topology.

New!!: Manifold and Poul Heegaard · See more »

Power series

In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term, c is a constant, and x varies around c (for this reason one sometimes speaks of the series as being centered at c).

New!!: Manifold and Power series · See more »

Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

New!!: Manifold and Product topology · See more »

Projection (mathematics)

In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition (or, in other words, which is idempotent).

New!!: Manifold and Projection (mathematics) · See more »

Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane.

New!!: Manifold and Projective plane · See more »

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite.

New!!: Manifold and Pseudo-Riemannian manifold · See more »


In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example).

New!!: Manifold and Pseudogroup · See more »

Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

New!!: Manifold and Quotient space (topology) · See more »

Ramification (mathematics)

In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign.

New!!: Manifold and Ramification (mathematics) · See more »

Real number

In mathematics, a real number is a value that represents a quantity along a continuous line.

New!!: Manifold and Real number · See more »

Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface.

New!!: Manifold and Real projective plane · See more »

Real projective space

In mathematics, real projective space, or RPn, is the topological space of lines passing through the origin 0 in Rn+1.

New!!: Manifold and Real projective space · See more »

Rectifiable set

In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense.

New!!: Manifold and Rectifiable set · See more »

René Thom

René Frédéric Thom (September 2, 1923 – October 25, 2002) was a French mathematician.

New!!: Manifold and René Thom · See more »

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold.

New!!: Manifold and Riemann surface · See more »

Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

New!!: Manifold and Riemannian manifold · See more »

Riemannian submersion

In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.

New!!: Manifold and Riemannian submersion · See more »

Ringed space

In mathematics, a ringed space can be equivalently thought of either Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry.

New!!: Manifold and Ringed space · See more »

Scheme (mathematics)

In mathematics, schemes connect the fields of algebraic geometry, commutative algebra and number theory.

New!!: Manifold and Scheme (mathematics) · See more »

Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability.

New!!: Manifold and Second-countable space · See more »

Semialgebraic set

In mathematics, a semialgebraic set is a subset S of Rn for some real closed field R (for example R could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n).

New!!: Manifold and Semialgebraic set · See more »

Sergei Novikov (mathematician)

Sergei Petrovich Novikov (also Serguei) (Russian: Серге́й Петро́вич Но́виков) (born 20 March 1938) is a Soviet and Russian mathematician, noted for work in both algebraic topology and soliton theory.

New!!: Manifold and Sergei Novikov (mathematician) · See more »

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

New!!: Manifold and Sheaf (mathematics) · See more »

Siméon Denis Poisson

Siméon Denis Poisson (21 June 1781 – 25 April 1840), was a French mathematician, geometer, and physicist.

New!!: Manifold and Siméon Denis Poisson · See more »

Simon Donaldson

Sir Simon Kirwan Donaldson FRS (born 20 August 1957), is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds.

New!!: Manifold and Simon Donaldson · See more »

Simplicial complex

In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).

New!!: Manifold and Simplicial complex · See more »

Simply connected space

In topology, a topological space is called simply-connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other such path while preserving the two endpoints in question (see below for an informal discussion).

New!!: Manifold and Simply connected space · See more »

Simultaneous equations

In mathematics, a set of simultaneous equations, also known as a system of equations, is a finite set of equations for which common solutions are sought.

New!!: Manifold and Simultaneous equations · See more »

Singular homology

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X).

New!!: Manifold and Singular homology · See more »

Singular point of an algebraic variety

In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined.

New!!: Manifold and Singular point of an algebraic variety · See more »

Singularity (mathematics)

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

New!!: Manifold and Singularity (mathematics) · See more »


In mathematical analysis, smoothness has to do with how many derivatives of a function exist and are continuous.

New!!: Manifold and Smoothness · See more »

Sophus Lie

Marius Sophus Lie (17 December 1842 – 18 February 1899) was a Norwegian mathematician.

New!!: Manifold and Sophus Lie · See more »


In physics, spacetime (also space–time, space time or space–time continuum) is any mathematical model that combines space and time into a single interwoven continuum.

New!!: Manifold and Spacetime · See more »


A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions).

New!!: Manifold and Sphere · See more »

Spherical harmonics

In mathematics, spherical harmonics are a series of special functions defined on the surface of a sphere used to solve some kinds of differential equations.

New!!: Manifold and Spherical harmonics · See more »


In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or right angles).

New!!: Manifold and Square · See more »

Stable normal bundle

In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data.

New!!: Manifold and Stable normal bundle · See more »

Stephen Smale

Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan.

New!!: Manifold and Stephen Smale · See more »

Subanalytic set

In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points (for example in Euclidean space) defined in a way broader than for semianalytic sets (roughly speaking, those satisfying conditions requiring certain real power series to be positive there).

New!!: Manifold and Subanalytic set · See more »


In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.

New!!: Manifold and Submanifold · See more »

Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.

New!!: Manifold and Submersion (mathematics) · See more »


In mathematics, specifically, in topology, a surface is a two-dimensional, topological manifold.

New!!: Manifold and Surface · See more »

Surgery theory

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by.

New!!: Manifold and Surgery theory · See more »

Symmetry group

In abstract algebra, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.

New!!: Manifold and Symmetry group · See more »

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.

New!!: Manifold and Symplectic manifold · See more »


In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds.

New!!: Manifold and Symplectomorphism · See more »


A synonym is a word or phrase that means exactly or nearly the same as another word or phrase in the same language.

New!!: Manifold and Synonym · See more »

Table of Lie groups

This article gives a table of some common Lie groups and their associated Lie algebras.

New!!: Manifold and Table of Lie groups · See more »


In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

New!!: Manifold and Tangent · See more »

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.

New!!: Manifold and Tangent space · See more »

Tangent vector

In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point.

New!!: Manifold and Tangent vector · See more »

Theorema Egregium

Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.

New!!: Manifold and Theorema Egregium · See more »

Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.

New!!: Manifold and Topological manifold · See more »

Topological map

In cartography and geology, a topological map is a type of diagram that has been simplified so that only vital information remains and unnecessary detail has been removed.

New!!: Manifold and Topological map · See more »

Topological property

In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.

New!!: Manifold and Topological property · See more »

Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods.

New!!: Manifold and Topological space · See more »

Topological vector space

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

New!!: Manifold and Topological vector space · See more »

Topologically stratified space

In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way.

New!!: Manifold and Topologically stratified space · See more »


In mathematics, topology (from the Greek τόπος, place, and λόγος, study), is the study of topological spaces.

New!!: Manifold and Topology · See more »

Torsion tensor

In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve.

New!!: Manifold and Torsion tensor · See more »


In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

New!!: Manifold and Torus · See more »

Two-dimensional space

In physics and mathematics, two-dimensional space or bi-dimensional space is a geometric model of the planar projection of the physical universe.

New!!: Manifold and Two-dimensional space · See more »

Uniformization theorem

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere.

New!!: Manifold and Uniformization theorem · See more »

Unit circle

In mathematics, a unit circle is a circle with a radius of one.

New!!: Manifold and Unit circle · See more »

Unit disk

In mathematics, the open unit disk (or disc) around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1: The closed unit disk around P is the set of points whose distance from P is less than or equal to one: Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself.

New!!: Manifold and Unit disk · See more »

Unit sphere

In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point.

New!!: Manifold and Unit sphere · See more »

Vector field

In vector calculus, a vector field is an assignment of a vector to each point in a subset of space.

New!!: Manifold and Vector field · See more »

Vertex (geometry)

In geometry, a vertex (plural vertices) is a special kind of point that describes the corners or intersections of geometric shapes.

New!!: Manifold and Vertex (geometry) · See more »


Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

New!!: Manifold and Volume · See more »


Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object—a function, a set, a space of one sort or another—is "well-behaved".

New!!: Manifold and Well-behaved · See more »

Whitney conditions

In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965.

New!!: Manifold and Whitney conditions · See more »

Whitney embedding theorem

In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney.

New!!: Manifold and Whitney embedding theorem · See more »

Whitney immersion theorem

In differential topology, the Whitney immersion theorem states that for m>1, any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space, and a (not necessarily one-to-one) immersion in (2m-1)-space.

New!!: Manifold and Whitney immersion theorem · See more »

William Kingdon Clifford

William Kingdon Clifford FRS (4 May 1845 – 3 March 1879) was an English mathematician and philosopher.

New!!: Manifold and William Kingdon Clifford · See more »

William Rowan Hamilton

Sir William Rowan Hamilton (midnight, 3–4 August 1805 – 2 September 1865) was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra.

New!!: Manifold and William Rowan Hamilton · See more »

William Thurston

William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician.

New!!: Manifold and William Thurston · See more »

Yang–Mills theory

Yang–Mills theory is a gauge theory based on the SU(''N'') group, or more generally any compact, semi-simple Lie group.

New!!: Manifold and Yang–Mills theory · See more »


In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.

New!!: Manifold and 3-manifold · See more »


In mathematics, 4-manifold is a 4-dimensional topological manifold.

New!!: Manifold and 4-manifold · See more »


In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure.

New!!: Manifold and 5-manifold · See more »

Redirects here:

0-manifold, Abstract Manifold, Abstract manifold, Boundary of a manifold, Coordinate chart, Interior of a manifold, ManiFold, Manifold (Mathematics), Manifold (geometry), Manifold (mathematics), Manifold (topology), Manifold theory, Manifold with boundary, Manifold with corners, Manifold/old2, Manifold/rewrite, Manifolds, Manifolds with boundary, Maximal Atlas, Real manifold, Regular domain, Two-dimensional manifold.


[1] https://en.wikipedia.org/wiki/Manifold

Hey! We are on Facebook now! »