Differential geometry of curves

Differential geometry of curves

Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....

• List of curve topics
• Frenet–Serret formulas
• Curves in differential geometry
• Line element
  Line element
  A line element ds in mathematics can most generally be thought of as the change in a position vector in an affine space expressing the change of the arc length. An easy way of visualizing this relationship is by parametrizing the given curve by Frenet–Serret formulas...
  
  
• Curvature
  Curvature
  In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
  
  
• Radius of curvature
  Radius of curvature
  The distance from the center of a circle or sphere to its surface is its radius. For other curved lines or surfaces, the radius of curvature at a given point is the radius of a circle that mathematically best fits the curve at that point....
  
  
• Osculating circle
  Osculating circle
  In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p...
  
  
• Curve
  Curve
  In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
  
  
• Fenchel's theorem

Differential geometry of surfaces

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....

• Theorema egregium
  Theorema Egregium
  Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...
  
  
• Gauss–Bonnet theorem
  Gauss–Bonnet theorem
  The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
  
  
• First fundamental form
  First fundamental form
  In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3. It permits the calculation of curvature and metric properties of a surface such as length and...
  
  
• Second fundamental form
• Gauss–Codazzi–Mainardi equations
• Dupin indicatrix
  Dupin indicatrix
  The Dupin indicatrix is a method for characterising the local shape of a surface. Draw a plane parallel to the tangent plane and a small distance away from it. Consider the intersection of the surface with this plane. The shape of the intersection is related to the Gaussian curvature. The Dupin...
  
  
• Asymptotic curve
  Asymptotic curve
  In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface . It is sometimes called an asymptotic line, although it need not be a line....
  
  
• Curvature
  Curvature
  In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
  
  
  • Principal curvatures
  • Mean curvature
    Mean curvature
    In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space....
    
    
  • Gauss curvature
  • Elliptic point
• Types of surfaces
  • Minimal surface
    Minimal surface
    In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
    
    
  • Ruled surface
    Ruled surface
    In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
    
    
  • Conical surface
    Conical surface
    In geometry, a conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex...
    
    
  • Developable surface
    Developable surface
    In mathematics, a developable surface is a surface with zero Gaussian curvature. That is, it is a "surface" that can be flattened onto a plane without distortion . Conversely, it is a surface which can be made by transforming a plane...
    
    

Calculus on manifolds

• Manifold
  Manifold
  In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
  
  
  • Differentiable manifold
    Differentiable manifold
    A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
    
    
  • Smooth manifold
  • Banach manifold
    Banach manifold
    In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space...
    
    
  • Fréchet manifold
    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....
    
    
• Tensor analysis
  • Tangent vector
    Tangent vector
    A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
    
    
  • Tangent space
    Tangent space
    In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
    
    
  • Tangent bundle
    Tangent bundle
    In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
    
    
  • Cotangent space
    Cotangent space
    In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
    
    
  • Cotangent bundle
    Cotangent bundle
    In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
    
    
  • Tensor
    Tensor
    Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
    
    
  • Tensor bundle
    Tensor bundle
    In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed....
    
    
  • Vector field
    Vector field
    In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
    
    
  • Tensor field
    Tensor field
    In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
    
    
  • Differential form
    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. Differential forms provide a better definition for integrands in calculus...
    
    
  • Exterior derivative
    Exterior derivative
    In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
    
    
  • Lie derivative
    Lie derivative
    In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
    
    
  • pullback (differential geometry)
  • pushforward (differential)
• jet (mathematics)
  Jet (mathematics)
  In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...
  
  
  • Contact (mathematics)
    Contact (mathematics)
    In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...
    
    
  • jet bundle
    Jet bundle
    In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...
    
    
• Frobenius theorem (differential topology)
  Frobenius theorem (differential topology)
  In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...
  
  
• Integral curve
  Integral curve
  In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
  
  

Differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

• Diffeomorphism
  Diffeomorphism
  In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
  
  
  • Large diffeomorphism
    Large diffeomorphism
    In mathematics and theoretical physics, a large diffeomorphism is a diffeomorphism that cannot be continuously connected to the identity diffeomorphism ....
    
    
• Orientability
  Orientability
  In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
  
  
• characteristic class
  Characteristic class
  In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
  
  
  • Chern class
    Chern class
    In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...
    
    
  • Pontrjagin class
  • spin structure
    Spin structure
    In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
    
    
• differentiable map
  • submersion
    Submersion
    Submersion may refer to:*Being underwater or going underwater: see scuba diving or submarine or :wikt:submerge.*Submersion , in the mathematical sense.*Submersion , an episode of the television series Stargate Atlantis....
    
    
  • immersion
  • Embedding
    Embedding
    In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
    
    
    • Whitney embedding theorem
• Critical value
  Critical value
  -Differential topology:In differential topology, a critical value of a differentiable function between differentiable manifolds is the image ƒ in N of a critical point x in M.The basic result on critical values is Sard's lemma...
  
  
  • Sard's theorem
  • Saddle point
    Saddle point
    In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
    
    
  • Morse theory
    Morse theory
    In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
    
    
• Lie derivative
  Lie derivative
  In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
  
  
• Hairy ball theorem
  Hairy ball theorem
  The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on an even-dimensional n-sphere. An ordinary sphere is a 2-sphere, so that this theorem will hold for an ordinary sphere...
  
  
• Poincaré–Hopf theorem
  Poincaré–Hopf theorem
  In mathematics, the Poincaré–Hopf theorem is an important theorem that is still used today in differential topology...
  
  
• Stokes' theorem
  Stokes' theorem
  In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
  
  
• De Rham cohomology
  De Rham cohomology
  In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
  
  
• Smale's paradox
  Smale's paradox
  In differential topology, Smale's paradox states that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process often called sphere eversion...
  
  
• Frobenius theorem (differential topology)
  Frobenius theorem (differential topology)
  In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...
  
  
  • Distribution (differential geometry)
    Distribution (differential geometry)
    In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties...
    
    
  • integral curve
    Integral curve
    In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
    
    
  • foliation
    Foliation
    In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
    
    
  • integrability conditions for differential systems
    Integrability conditions for differential systems
    In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a...
    
    

Fiber bundles

• Fiber bundle
  Fiber bundle
  In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
  
  
• Principal bundle
  Principal bundle
  In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
  
  
  • Frame bundle
    Frame bundle
    In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered bases, or frames, for Ex...
    
    
  • Hopf bundle
    Hopf bundle
    In the mathematical field of topology, the Hopf fibration describes a 3-sphere in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle...
    
    
• Associated bundle
  Associated bundle
  In mathematics, the theory of fiber bundles with a structure group G allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with a group action of G...
  
  
• Vector bundle
  Vector bundle
  In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
  
  
  • Tangent bundle
    Tangent bundle
    In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
    
    
  • Cotangent bundle
    Cotangent bundle
    In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
    
    
  • Line bundle
    Line bundle
    In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
    
    
• Jet bundle
  Jet bundle
  In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...
  
  

Fundamental structures

• Sheaf (mathematics)
  Sheaf (mathematics)
  In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
  
  
• Pseudogroup
  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...
  
  
• G-structure
  G-structure
  In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....
  
  
• synthetic differential geometry
  Synthetic differential geometry
  In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...
  
  

Fundamental notions

• Metric tensor
  Metric tensor
  In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
  
  
• Riemannian manifold
  Riemannian manifold
  In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
  
  
  • Pseudo-Riemannian manifold
    Pseudo-Riemannian manifold
    In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
    
    
• Levi-Civita connection
  Levi-Civita connection
  In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
  
  

Non-Euclidean geometry

• Non-Euclidean geometry
  Non-Euclidean geometry
  Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
  
  
• Elliptic geometry
  Elliptic geometry
  Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...
  
  
  • Spherical geometry
    Spherical geometry
    Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....
    
    
  • Sphere-world
    Sphere-world
    The idea of a sphere-world was constructed by Henri Poincaré who, while pursuing his argument for conventionalism , offered a thought experiment about a sphere with strange properties....
    
    
  • Angle excess
    Angle excess
    Angle excess, also known as spherical excess is the amount by which the sum of the angles of a polygon on a sphere exceeds the sum of the angles of a polygon with the same number of sides in a plane. For instance, a plane triangle has an angle sum of 180°; an octant is a spherical triangle with...
    
    
• hyperbolic geometry
  Hyperbolic geometry
  In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
  
  
  • hyperbolic space
    Hyperbolic space
    In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
    
    
  • hyperboloid model
    Hyperboloid model
    In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model , is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in -dimensional Minkowski space and m-planes are...
    
    
  • Poincaré disc model
  • Poincaré half-plane model
    Poincaré half-plane model
    In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry....
    
    
  • Poincaré metric
    Poincaré metric
    In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces.There are three equivalent...
    
    
  • Angle of parallelism
    Angle of parallelism
    In hyperbolic geometry, the angle of parallelism φ, also known as Π, is the angle at one vertex of a right hyperbolic triangle that has two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism φ...
    
    

Geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

• Prime geodesic
  Prime geodesic
  In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once...
  
  
• Geodesic flow
• Exponential map
  Exponential map
  In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
  
  
• Injectivity radius
• Geodesic deviation equation
  Geodesic deviation equation
  In general relativity, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring geodesics or, equivalently, the tidal force experienced by a rigid body moving along a geodesic...
  
  
  • Jacobi field
    Jacobi field
    In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space...
    
    

Symmetric spaces (and related topics)

• Riemannian symmetric space
  Riemannian symmetric space
  In differential geometry, representation theory and harmonic analysis, a symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. There are two ways to formulate the inversion symmetry, via Riemannian geometry or via Lie theory...
  
  
  • Margulis lemma
    Margulis lemma
    In mathematics, the Margulis lemma is a result about discrete subgroups of isometries of a symmetric space , or more generally a space of non-positive curvature....
    
    
• Space form
  Space form
  In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three obvious examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.-Reduction to generalized crystallography:It is a...
  
  
  • Constant curvature
    Constant curvature
    In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points...
    
    
  • taut submanifold
• Uniformization theorem
  Uniformization theorem
  In mathematics, the uniformization theorem says that any simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature...
  
  
  • Myers theorem
  • Gromov's compactness theorem
    Gromov's compactness theorem
    Gromov's compactness theorem can refer to either of two mathematical theorems:* Gromov's compactness theorem in Riemannian geometry* Gromov's compactness theorem in symplectic topology...
    
    

Riemannian submanifold

Riemannian submanifold

A Riemannian submanifold N of a Riemannian manifold M is a submanifold of M equipped with the Riemannian metric inherited from M. The image of an isometric immersion is a Riemannian submanifold....

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• Gauss–Codazzi equations
  Gauss–Codazzi equations
  In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds...
  
  
• Darboux frame
  Darboux frame
  In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space...
  
  
• Hypersurface
  Hypersurface
  In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
  
  
• Induced metric
  Induced metric
  In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold which is calculated from the metric tensor on a larger manifold into which the submanifold is embedded...
  
  
• Nash embedding theorem
  Nash embedding theorem
  The Nash embedding theorems , named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path...
  
  
• minimal surface
  Minimal surface
  In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
  
  
  • Helicoid
    Helicoid
    The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...
    
    
  • Catenoid
    Catenoid
    A catenoid is a three-dimensional surface made by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste...
    
    
  • Costa's minimal surface
    Costa's minimal surface
    In mathematics, Costa's minimal surface is an embedded minimal surface and was discovered in 1982 by the Brazilian mathematician Celso Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a compact surface...
    
    
• Hsiang–Lawson's conjecture

Curvature of Riemannian manifolds

Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define it, now known as the curvature tensor...

• Theorema Egregium
  Theorema Egregium
  Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces...
  
  
• Gauss–Bonnet theorem
  Gauss–Bonnet theorem
  The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
  
  
  • Chern–Gauss–Bonnet theorem
  • Chern–Weil homomorphism
• Gauss map
  Gauss map
  In differential geometry, the Gauss map maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X → S2 such that N is a unit vector orthogonal to X at p, namely the normal vector to X at p.The Gauss map can be defined...
  
  
• Second fundamental form
• Curvature form
  Curvature form
  In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...
  
  
• Curvature tensor
  Curvature tensor
  In differential geometry, the term curvature tensor may refer to:* the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;* the curvature of an affine connection or covariant derivative ;...
  
  
• Geodesic curvature
  Geodesic curvature
  In Riemannian geometry, the geodesic curvature k_g of a curve lying on a submanifold of the ambient space measures how far the curve is from being a geodesic...
  
  
• Scalar curvature
  Scalar curvature
  In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...
  
  
• Sectional curvature
  Sectional curvature
  In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...
  
  
• Ricci curvature
  Ricci curvature
  In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...
  
  , Ricci flat
• Ricci decomposition
  Ricci decomposition
  In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties...
  
  
  • Schouten tensor
  • Weyl curvature
• Ricci flow
  Ricci flow
  In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....
  
  
• Einstein manifold
  Einstein manifold
  In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric...
  
  
• holonomy
  Holonomy
  In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...
  
  

Theorems in Riemannian geometry

• Gauss–Bonnet theorem
  Gauss–Bonnet theorem
  The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
  
  
• Hopf–Rinow theorem
  Hopf–Rinow theorem
  In mathematics, the Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow .-Statement of the theorem:...
  
  
• Cartan–Hadamard theorem
  Cartan–Hadamard theorem
  The Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point...
  
  
• Myers theorem
• Rauch comparison theorem
  Rauch comparison theorem
  In Riemannian geometry, the Rauch comparison theorem is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for large curvature, geodesics tend to converge, while for small curvature,...
  
  
• Morse index theorem
• Synge theorem
• Weinstein theorem
• Toponogov theorem
• Sphere theorem
  Sphere theorem
  In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows...
  
  
• Hodge theorem
• Uniformization theorem
  Uniformization theorem
  In mathematics, the uniformization theorem says that any simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature...
  
  
• Yamabe problem
  Yamabe problem
  The Yamabe problem in differential geometry concerns the existence of Riemannian metrics with constant scalar curvature, and takes its name from the mathematician Hidehiko Yamabe. Although claimed to have a solution in 1960, a critical error...
  
  

Laplace–Beltrami operator

• Hodge star operator
• Weitzenböck identity
  Weitzenbock identity
  In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity expresses a relationship between two second-order elliptic operators on a manifold with the same leading symbol...
  
  
• Laplacian operators in differential geometry
  Laplacian operators in differential geometry
  In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them.- Connection Laplacian :...
  
  

Formulas and other tools

• List of coordinate charts
• List of formulas in Riemannian geometry
• Christoffel symbols
  Christoffel symbols
  In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
  
  

Related structures

• Intrinsic metric
  Intrinsic metric
  In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to that distance...
  
  
• Pseudo-Riemannian manifold
  Pseudo-Riemannian manifold
  In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
  
  
• Sub-Riemannian manifold
  Sub-Riemannian manifold
  In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces....
  
  
• Finsler geometry
• General relativity
  General relativity
  General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
  
  
• G2 manifold
  G2 manifold
  In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group G2. The group G_2 is one of the five exceptional simple Lie groups...
  
  
• Information geometry
  Information geometry
  Information geometry is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. It derives its name from the fact that the Fisher information is used as the Riemannian metric when considering the geometry of probability distribution families...
  
  
  • Fisher information metric
    Fisher information metric
    In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space....
    
    

Connections

• covariant derivative
  Covariant derivative
  In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
  
  
  • exterior covariant derivative
    Exterior covariant derivative
    In mathematics, the exterior covariant derivative, sometimes also covariant exterior derivative, is a very useful notion for calculus on manifolds, which makes it possible to simplify formulas which use a principal connection....
    
    
• Levi-Civita connection
  Levi-Civita connection
  In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
  
  
• parallel transport
  Parallel transport
  In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
  
  
  • Development (differential geometry)
    Development (differential geometry)
    In classical differential geometry, development refers to the simple idea of rolling one smooth surface over another in Euclidean space. For example, the tangent plane to a surface at a point can be rolled around the surface to obtain the tangent-plane at other points.The tangential contact...
    
    
• connection form
  Connection form
  In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms....
  
  
• Cartan connection
  Cartan connection
  In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the...
  
  
  • affine connection
    Affine connection
    In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
    
    
  • conformal connection
    Conformal connection
    In conformal differential geometry, a conformal connection is a Cartan connection on an n-dimensional manifold M arising as a deformation of the Klein geometry given by the celestial n-sphere, viewed as the homogeneous space...
    
    
  • projective connection
    Projective connection
    In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection. Much like affine connections,...
    
    
  • method of moving frames
  • Cartan's equivalence method
    Cartan's equivalence method
    In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism...
    
    
  • Vierbein, tetrad
    Tetrad
    Tetrad may refer to:* Tetrad , Bivalents or Tetrad of homologous chromosomes consisting of four synapsed chromatids that become visible during the Pachytene stage of meiotic prophase...
    
    
  • Cartan connection applications
    Cartan connection applications
    The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms...
    
    
  • Einstein–Cartan theory
    Einstein–Cartan theory
    In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory or the Cartan–Sciama–Kibble theory is a classical theory of gravitation similar to general relativity but relaxing the assumption that the metric be torsion-free. Introducing torsion allows...
    
    
• connection (vector bundle)
  Connection (vector bundle)
  In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
  
  
• connection (principal bundle)
  Connection (principal bundle)
  In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
  
  
• Ehresmann connection
  Ehresmann connection
  In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...
  
  
• curvature
  Curvature
  In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
  
  
  • curvature form
    Curvature form
    In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...
    
    
  • holonomy
    Holonomy
    In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...
    
    , local holonomy
  • Chern–Weil homomorphism
  • Curvature vector
  • Curvature form
    Curvature form
    In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...
    
    
  • Curvature tensor
    Curvature tensor
    In differential geometry, the term curvature tensor may refer to:* the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;* the curvature of an affine connection or covariant derivative ;...
    
    
  • Cocurvature
    Cocurvature
    In mathematics in the branch of differential geometry, the cocurvature of a connection on a manifold is the obstruction to the integrability of the vertical bundle.-Definition:...
    
    
• torsion (differential geometry)

Complex manifolds

• Riemann surface
  Riemann surface
  In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
  
  
• Complex projective space
  Complex projective space
  In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
  
  
• Kähler manifold
  Kähler manifold
  In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
  
  
• Dolbeault operator
• CR manifold
  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....
  
  
• Stein manifold
  Stein manifold
  In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...
  
  
• Almost complex structure
• Hermitian manifold
  Hermitian manifold
  In mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. Specifically, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each tangent space...
  
  
• Newlander–Nirenberg theorem
• Generalized complex manifold
• Calabi–Yau manifold
• Hyperkähler manifold
  Hyperkähler manifold
  In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...
  
  
• K3 surface
  K3 surface
  In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
  
  
• hypercomplex manifold
  Hypercomplex manifold
  In differential geometry, a hypercomplex manifold is a manifold with the tangent bundleequipped with an action by the algebra of quaternionsin such a way that the quaternions I, J, Kdefine integrable almost complex structures.- Examples :...
  
  
• Quaternion-Kähler manifold
  Quaternion-Kähler manifold
  In differential geometry, a quaternion-Kähler manifold is a Riemannian manifold whose Riemannian holonomy group is a subgroup of Sp·Sp....
  
  

Symplectic geometry

• Symplectic topology
  Symplectic topology
  Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form...
  
  
• Symplectic space
  Symplectic space
  A symplectic space is either a symplectic manifold or a symplectic vector space. The latter is a special case of the former....
  
  
• Symplectic manifold
  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. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
  
  
• Symplectic structure
• Symplectomorphism
  Symplectomorphism
  In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.-Formal definition:A diffeomorphism between two symplectic manifolds f: \rightarrow is called symplectomorphism, iff^*\omega'=\omega,...
  
  
• Contact structure
• Contact geometry
  Contact geometry
  In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'...
  
  
• Hamiltonian system
  Hamiltonian system
  In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
  
  
• Sasakian manifold
  Sasakian manifold
  In differential geometry, a Sasakian manifold is a contact manifold equipped with a special kind of Riemannian metric g, called a Sasakian metric.-Definition:...
  
  
• Poisson manifold
  Poisson manifold
  In mathematics, a Poisson manifold is a differentiable manifold M such that the algebra C^\infty\, of smooth functions over M is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra...
  
  

Conformal geometry

Conformal geometry

In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...

• Möbius transformation
• Conformal map
  Conformal map
  In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...
  
  
• conformal connection
  Conformal connection
  In conformal differential geometry, a conformal connection is a Cartan connection on an n-dimensional manifold M arising as a deformation of the Klein geometry given by the celestial n-sphere, viewed as the homogeneous space...
  
  
• tractor bundle
  Tractor bundle
  In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation of the conformal group ....
  
  
• Weyl curvature
• Weyl–Schouten theorem

• ambient construction
  Ambient construction
  In conformal geometry, the ambient construction refers to a construction of Charles Fefferman and Robin Graham for which a conformal manifold of dimension n is realized as the boundary of a certain Poincaré manifold, or alternatively as the celestial sphere of a certain pseudo-Riemannian...
  
  
• Willmore energy
  Willmore energy
  In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the mean curvature squared...
  
  
• Willmore flow

Index theory

• Atiyah–Singer index theorem
  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 is equal to the topological index...
  
  
• de Rham cohomology
  De Rham cohomology
  In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
  
  
• Dolbeault cohomology
  Dolbeault cohomology
  In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold...
  
  
• elliptic complex
  Elliptic complex
  In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for...
  
  
• Hodge theory
  Hodge theory
  In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
  
  
• pseudodifferential operator

Homogeneous spaces

• Klein geometry
  Klein geometry
  In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry.For background and motivation...
  
  , Erlangen programme
• symmetric space
  Symmetric space
  A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...
  
  
• space form
  Space form
  In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three obvious examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.-Reduction to generalized crystallography:It is a...
  
  
• Maurer–Cartan form
• Examples
  • hyperbolic space
    Hyperbolic space
    In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
    
    
  • Gauss–Bolyai–Lobachevsky space
  • Grassmannian
    Grassmannian
    In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
    
    
  • Complex projective space
    Complex projective space
    In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
    
    
  • Real projective space
    Real projective space
    In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
    
    
  • Euclidean space
    Euclidean space
    In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
    
    
  • Stiefel manifold
    Stiefel manifold
    In mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel...
    
    
  • Upper half-plane
  • Sphere
    Sphere
    A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
    
    

Systolic geometry

• Loewner's torus inequality
  Loewner's torus inequality
  In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus.-Statement:...
  
  
• Pu's inequality
• Gromov's inequality for complex projective space
• Wirtinger inequality (2-forms)
  Wirtinger inequality (2-forms)
  In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that the exterior \scriptstyle\nuth power of the standard symplectic form ω, when evaluated on a simple -vector ζ of unit volume, is bounded above by \scriptstyle\nu!...
  
  
• Gromov's systolic inequality for essential manifolds
  Gromov's systolic inequality for essential manifolds
  In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold...
  
  
• Essential manifold
  Essential manifold
  In mathematics, in algebraic topology and differential geometry, the notion of an essential manifold seems to have been first introduced explicitly in Mikhail Gromov's classic text in 1983 .-Definition:...
  
  
• Filling radius
  Filling radius
  In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the...
  
  
• Filling area conjecture
  Filling area conjecture
  In mathematics, in Riemannian geometry, Mikhail Gromov's filling area conjecture asserts that among all possible fillings of the Riemannian circle of length 2π by a surface with the strongly isometric property, the round hemisphere has the least area...
  
  
• Bolza surface
  Bolza surface
  In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve , is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely 48. An affine model for the Bolza surface can be obtained as the locus of the...
  
  
• First Hurwitz triplet
  First Hurwitz triplet
  In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 . The explanation for this phenomenon is arithmetic...
  
  
• Hermite constant
• Systoles of surfaces
  Systoles of surfaces
  In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 . Given a closed surface, its systole, denoted sys, is defined to the least length of a loop that cannot be contracted to a point on the surface. The systolic area of a metric is defined to...
  
  
• Systolic freedom
  Systolic freedom
  In differential geometry, systolic freedom refers to the fact that closed Riemannian manifolds may have arbitrarily small volume regardless of their systolic invariants....
  
  
• Systolic category
  Systolic category
  Systolic category is a numerical invariant of a closed manifold M, introduced by Mikhail Katz and Yuli Rudyak in 2006, by analogy with the Lusternik–Schnirelmann category. The invariant is defined in terms of the systoles of M and its covers, as the largest number of systoles in a product...