286 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, Arctic Circle, 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, Calculus on Manifolds (book), 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 geometry, 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, 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, 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), Intrinsic and extrinsic properties, Invariant (mathematics), Inverse function, James Munkres, 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, Morris Hirsch, 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, North America, Orbifold, Orientability, Orthogonal group, Parabola, Parallel postulate, Partial differential equation, Pathological (mathematics), 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, Robion Kirby, Scheme (mathematics), Second-countable space, Semialgebraic set, Sergei Novikov (mathematician), Sheaf (mathematics), Siméon Denis Poisson, Simon Donaldson, Simplicial complex, Simply connected space, Singular homology, Singular point of an algebraic variety, Singularity (mathematics), Smoothness, Sophus Lie, South America, Spacetime, Sphere, Spherical harmonics, Square, Stable normal bundle, Stephen Smale, Subanalytic set, Submanifold, Submersion (mathematics), Surface (topology), Surgery theory, Symmetry group, Symplectic manifold, Symplectomorphism, Synonym, System of equations, Table of Lie groups, Tangent, Tangent space, Tangent vector, Theorema Egregium, Timeline of manifolds, 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), Victor Guillemin, Volume, Whitney conditions, Whitney embedding theorem, Whitney immersion theorem, William Kingdon Clifford, William Rowan Hamilton, William S. Massey, William Thurston, Yang–Mills theory, 3-manifold, 4-manifold, 5-manifold. Expand index (236 more) » « Shrink index
In mathematics, particularly in algebraic geometry, complex analysis and algebraic 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.
In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.
In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
Algebraic varieties are the central objects of study in algebraic geometry.
An ambient space or ambient configuration space is the space surrounding an object.
"Analysis Situs" is a seminal mathematics paper that Henri Poincaré published in 1895.
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.
In mathematics, an analytic function is a function that is locally given by a convergent power series.
In mathematics, an analytic manifold is a topological manifold with analytic transition maps.
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics.
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.
In plane 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.
Determining the length of an irregular arc segment is also called rectification of a curve.
The Arctic Circle is the most northerly of the five major circles of latitude as shown on maps of Earth.
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.
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).
An atlas is a collection of maps; it is typically a bundle of maps of Earth or a region of Earth.
In mathematics, particularly topology, one describes a manifold using an atlas.
In mathematics, a ball is the space bounded by a sphere.
In mathematics, a Banach manifold is a manifold modeled on Banach spaces.
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
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.
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).
Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief (146 pp.) monograph on the theory of vector-valued functions of several real variables (f: Rn→Rm) and differentiable manifolds in Euclidean space.
Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.
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.
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 to the point from two fixed perpendicular directed lines, measured in the same unit of length.
In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
In mathematics, a characteristic class is a way of associating to each principal bundle X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" — and whether it possesses sections.
A circle is a simple closed shape.
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary.
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.
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).
In set theory, the complement of a set refers to elements not in.
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).
In mathematics, complex geometry is the study of complex manifolds and functions of several complex variables.
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.
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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.
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.
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'.
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
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.
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.
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.
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equation.
In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.
In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory.
A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.
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.
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.
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.
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.
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.
In mathematics, a digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space.
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.
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.
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.
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.
In geometry, a disk (also spelled disc).
Distance is a numerical measurement of how far apart objects are.
In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point.
In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.
Earth is the third planet from the Sun and the only astronomical object known to harbor life.
Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.
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.
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.
Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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.
In mathematics, a filtration \mathcal is an indexed set S_i 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 If the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic object S_i gaining in complexity with time.
In abstract algebra, a finite group is a mathematical group with a finite number of elements.
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.
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.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
In mathematics, the term functional (as a noun) has at least two meanings.
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 functions defined on these spaces and respecting these structures in a suitable sense.
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).
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.
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.
In the mathematical area of topology, the Generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere 'is' a sphere.
In mathematics, genus (plural genera) has a few different, but closely related, meanings.
In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
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.
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.
In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory.
German (Deutsch) is a West Germanic language that is mainly spoken in Central Europe.
Giovanni Girolamo Saccheri (5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician.
In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.
Grigori Yakovlevich Perelman (a; born 13 June 1966) is a Russian mathematician.
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.
In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces.
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).
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 that satisfies Laplace's equation, i.e. everywhere on U. This is usually written as or.
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician.
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.
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.
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.
Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.
Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces.
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.
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.
In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
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.
In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
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.
In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.
In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center.
In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.
In mathematics, an implicit equation is a relation of the form R(x_1,\ldots, x_n).
In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables.
In linear algebra, an inner product space is a vector space with an additional structure called an inner product.
In mathematics, a (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.
An intrinsic property is a property of a system or of a material itself or within.
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.
In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.
James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including Topology (an undergraduate-level text), Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology.
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.
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems.
Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.
In topology, a branch of 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.
In topology, knot theory is the study of mathematical knots.
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves.
In geometric measurements, length is the most extended dimension of an object.
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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.
Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.
This is a list of particular manifolds, by Wikipedia page.
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
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.
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.
In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.
In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer".
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.
A map projection is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane.
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.
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
Mathematical induction is a mathematical proof technique.
Mathematical physics refers to the development of mathematical methods for application to problems in physics.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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.
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German-born American mathematician and student of David Hilbert.
The Möbius strip or Möbius band, also spelled Mobius or Moebius, is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary.
In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.
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.
Michael David Spivak (born May 25, 1940)Biographical sketch in, Vol.
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.
Morris William Hirsch (born June 28, 1933) is an American mathematician, formerly at the University of California, Berkeley.
"Morse function" redirects here.
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.
A nautical chart is a graphic representation of a maritime area and adjacent coastal regions.
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.
Nikolai Ivanovich Lobachevsky (a; –) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry and also his fundamental study on Dirichlet integrals known as Lobachevsky integral formula.
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space.
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.
In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.
North America is a continent entirely within the Northern Hemisphere and almost all within the Western Hemisphere; it is also considered by some to be a northern subcontinent of the Americas.
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.
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.
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.
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.
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.
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved.
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.
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.
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.
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.
In elementary geometry, a polytope is a geometric object with "flat" sides.
Poul Heegaard (November 2, 1871, Copenhagen - February 7, 1948, Oslo) was a Danish mathematician active in the field of topology.
In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.
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.
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).
In mathematics, a projective plane is a geometric structure that extends the concept of a plane.
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, but need only be a non-degenerate bilinear form, which is a weaker condition.
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).
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.
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.
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface.
In mathematics, real projective space, or RPn or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in Rn+1.
In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense.
René Frédéric Thom (2 September 1923 – 25 October 2002) was a French mathematician.
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex 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 differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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.
In mathematics, a ringed space can be equivalently thought of as either Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry.
Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology.
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.
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).
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.
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
Baron Siméon Denis Poisson FRS FRSE (21 June 1781 – 25 April 1840) was a French mathematician, engineer, and physicist, who made several scientific advances.
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 and Donaldson–Thomas theory.
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.
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).
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.
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.
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
Marius Sophus Lie (17 December 1842 – 18 February 1899) was a Norwegian mathematician.
South America is a continent in the Western Hemisphere, mostly in the Southern Hemisphere, with a relatively small portion in the Northern Hemisphere.
In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.
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 the circular objects in two dimensions, where a "circle" circumscribes its "disk").
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere.
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted.
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.
Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan.
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).
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.
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.
In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.
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.
In group theory, 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.
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds.
A synonym is a word or phrase that means exactly or nearly the same as another word or phrase in the same language.
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought.
This article gives a table of some common Lie groups and their associated Lie algebras.
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.
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 that gives the displacement of the one point from the other.
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point.
Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.
This is a timeline of manifolds, one of the major geometric concepts of mathematics.
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.
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.
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.
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, satisfying a set of axioms relating points and neighbourhoods.
In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.
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.
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve.
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.
Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.
In mathematics, a unit circle is a circle with a radius of one.
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.
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.
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.
Victor William Guillemin (born 1937 in Boston) is a mathematician working in the field of symplectic geometry, who has also made contributions to the fields of microlocal analysis, spectral theory, and mathematical physics.
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.
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.
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney.
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.
William Kingdon Clifford FRS (4 May 1845 – 3 March 1879) was an English mathematician and philosopher.
Sir William Rowan Hamilton MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician who made important contributions to classical mechanics, optics, and algebra.
William Schumacher Massey (August 23, 1920 - June 17, 2017) was an American mathematician, known for his work in algebraic topology.
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.
Yang–Mills theory is a gauge theory based on the SU(''N'') group, or more generally any compact, reductive Lie algebra.
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.
In mathematics, a 4-manifold is a 4-dimensional topological manifold.
In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure.
0-manifold, Abstract Manifold, Abstract manifold, Boundary of a manifold, 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, Two-dimensional manifold.