Faster access than browser!
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) »
Abelian variety
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.
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, so it 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
Algebraic varieties are 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 »
Angle
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.
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 »
Arctic Circle
The Arctic Circle is the most northerly of the five major circles of latitude as shown on maps of Earth.
New!!: Manifold and Arctic Circle · See more »
Area
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 »
Atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or a region of Earth.
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 bounded by 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 (17 September 1826 – 20 July 1866) was a German mathematician who made 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 »
Bijection
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.
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
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.
New!!: Manifold and Calculus · See more »
Calculus on Manifolds (book)
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.
New!!: Manifold and Calculus on Manifolds (book) · See more »
Carl Friedrich Gauss
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.
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 to the point from two fixed perpendicular directed lines, measured in the same unit of length.
New!!: Manifold and Cartesian coordinate system · See more »
Cartesian product
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.
New!!: Manifold and Cartesian product · See more »
Category theory
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).
New!!: Manifold and Category theory · See more »
Characteristic class
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.
New!!: Manifold and Characteristic class · See more »
Circle
A circle is a simple closed shape.
New!!: Manifold and Circle · See more »
Classical mechanics
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
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 »
Cohomology
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.
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, the complement of a set refers to elements not in.
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 geometry
In mathematics, complex geometry is the study of complex manifolds and functions of several complex variables.
New!!: Manifold and Complex geometry · 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 a solution of 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 satisfying a condition called 'complete non-integrability'.
New!!: Manifold and Contact geometry · See more »
Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily 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 »
Cross-cap
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 for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equation.
New!!: Manifold and Cubic plane curve · See more »
Curvature
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 greater than 2 is too complicated to be described by a single number at a given point.
New!!: Manifold and Curvature of Riemannian manifolds · See more »
Curve
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.
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
A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.
New!!: Manifold and Cylinder · See more »
Derivative
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).
New!!: Manifold and Derivative · See more »
Diffeomorphism
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 (also differential 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 »
Dimension
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
Distance is a numerical measurement of how far apart objects are.
New!!: Manifold and Distance · See more »
Divergence
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.
New!!: Manifold and Divergence · See more »
Dot product
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.
New!!: Manifold and Dot product · See more »
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life.
New!!: Manifold and Earth · See more »
Elliptic geometry
Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.
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 »
Embedding
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 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.
New!!: Manifold and Equivalence class · See more »
Euclid
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".
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 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.
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 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.
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, the term functional (as a noun) has at least two meanings.
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 functions defined on 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 (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.
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 Generalized Poincaré conjecture is 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 »
Geodesic
In 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
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 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.
New!!: Manifold and Geometry and topology · See more »
German language
German (Deutsch) is a West Germanic language that is mainly spoken in Central Europe.
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 »
Graph of a function
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.
New!!: Manifold and Graph of a function · See more »
Grigori Perelman
Grigori Yakovlevich Perelman (a; born 13 June 1966) is a Russian mathematician.
New!!: Manifold and Grigori Perelman · See more »
Group (mathematics)
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.
New!!: Manifold and Group (mathematics) · See more »
Group action
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.
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 »
Handlebody
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 that 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 (March 23, 1907 – May 10, 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 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, 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.
New!!: Manifold and Holomorphic function · See more »
Homeomorphism
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, 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.
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 »
Homotopy
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.
New!!: Manifold and Homotopy · See more »
Hyperbola
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 »
Hypersphere
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(x_1,\ldots, x_n).
New!!: Manifold and Implicit function · See more »
Implicit function theorem
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.
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, 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.
New!!: Manifold and Interval (mathematics) · See more »
Intrinsic and extrinsic properties
An intrinsic property is a property of a system or of a material itself or within.
New!!: Manifold and Intrinsic and extrinsic properties · 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 (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.
New!!: Manifold and Inverse function · See more »
James Munkres
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.
New!!: Manifold and James Munkres · 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 (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.
New!!: Manifold and Joseph-Louis Lagrange · See more »
Klein bottle
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.
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 »
Lemniscate
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves.
New!!: Manifold and Lemniscate · See more »
Length
In geometric measurements, length is the most extended dimension of an object.
New!!: Manifold and Length · See more »
Leonhard Euler
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.
New!!: Manifold and Leonhard Euler · See more »
Lie group
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.
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 linear equations such as linear functions such as and their representations through matrices and vector 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 a function between topological spaces that, intuitively, preserves local (though not necessarily global) 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) (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.
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 »
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
New!!: Manifold and Manifold · See more »
Map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
New!!: Manifold and Map · See more »
Map (mathematics)
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.
New!!: Manifold and Map (mathematics) · See more »
Map projection
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.
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 the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
New!!: Manifold and Mathematical analysis · See more »
Mathematical induction
Mathematical induction is a mathematical proof technique.
New!!: Manifold and Mathematical induction · See more »
Mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics.
New!!: Manifold and Mathematical physics · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, 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.
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, also spelled Mobius or Moebius, is a surface with only one side (when embedded in three-dimensional Euclidean space) 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 »
Morris Hirsch
Morris William Hirsch (born June 28, 1933) is an American mathematician, formerly at the University of California, Berkeley.
New!!: Manifold and Morris Hirsch · See more »
Morse theory
"Morse function" redirects here.
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; –) 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.
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 geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space.
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, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.
New!!: Manifold and Normed vector space · See more »
North America
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.
New!!: Manifold and North America · See more »
Orbifold
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 »
Orientability
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 »
Parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.
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 »
Pathological (mathematics)
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved.
New!!: Manifold and Pathological (mathematics) · See more »
Phase space
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.
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 that extends infinitely far.
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 »
Polytope
In elementary geometry, a polytope is a geometric object with "flat" sides.
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 and c is a constant.
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, but need only be a non-degenerate bilinear form, which is a weaker condition.
New!!: Manifold and Pseudo-Riemannian manifold · See more »
Pseudogroup
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 of a continuous quantity that can represent a distance along a 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; in other words, a one-sided surface.
New!!: Manifold and Real projective plane · See more »
Real projective space
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.
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 (2 September 1923 – 25 October 2002) was a French mathematician.
New!!: Manifold and René Thom · See more »
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface 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 differentiable 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 as 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 »
Robion Kirby
Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology.
New!!: Manifold and Robion Kirby · See more »
Scheme (mathematics)
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.
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 whose topology has a countable base.
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
Baron Siméon Denis Poisson FRS FRSE (21 June 1781 – 25 April 1840) was a French mathematician, engineer, and physicist, who made several scientific advances.
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 and Donaldson–Thomas theory.
New!!: Manifold and Simon Donaldson · See more »
Simplicial complex
In mathematics, a simplicial complex is a set composed of 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, 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.
New!!: Manifold and Simply connected space · 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 »
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that 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 »
South America
South America is a continent in the Western Hemisphere, mostly in the Southern Hemisphere, with a relatively small portion in the Northern Hemisphere.
New!!: Manifold and South America · See more »
Spacetime
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.
New!!: Manifold and Spacetime · See more »
Sphere
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").
New!!: Manifold and Sphere · See more »
Spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere.
New!!: Manifold and Spherical harmonics · See more »
Square
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.
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 »
Submanifold
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 »
Surface (topology)
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.
New!!: Manifold and Surface (topology) · 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 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.
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 »
Symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds.
New!!: Manifold and Symplectomorphism · See more »
Synonym
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 »
System of equations
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.
New!!: Manifold and System of equations · 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 »
Tangent
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 that gives the displacement of the one point from 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 major result of differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.
New!!: Manifold and Theorema Egregium · See more »
Timeline of manifolds
This is a timeline of manifolds, one of the major geometric concepts of mathematics.
New!!: Manifold and Timeline of manifolds · 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, satisfying 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 »
Topology
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.
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 »
Torus
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
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).
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 Riemann surfaces: 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 and physics, 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 or vertexes) is a point where two or more curves, lines, or edges meet.
New!!: Manifold and Vertex (geometry) · See more »
Victor Guillemin
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.
New!!: Manifold and Victor Guillemin · See more »
Volume
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.
New!!: Manifold and Volume · 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 MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician who made important contributions to classical mechanics, optics, and algebra.
New!!: Manifold and William Rowan Hamilton · See more »
William S. Massey
William Schumacher Massey (August 23, 1920 - June 17, 2017) was an American mathematician, known for his work in algebraic topology.
New!!: Manifold and William S. Massey · See more »
William Thurston
William Paul Thurston (October 30, 1946August 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, reductive Lie algebra.
New!!: Manifold and Yang–Mills theory · See more »
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.
New!!: Manifold and 3-manifold · See more »
4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold.
New!!: Manifold and 4-manifold · See more »
5-manifold
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, 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.
References
[1] https://en.wikipedia.org/wiki/Manifold
