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Euler characteristic of an orbifold

In differential geometry, the Euler characteristic of an orbifold, or orbifold Euler characteristic, is a generalization of the topological Euler characteristic that includes contributions coming from

Lie sphere geometry

Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main

Minkowski problem

In differential geometry, the Minkowski problem, named after Hermann Minkowski, asks for the construction of a strictly convex compact surface S whose Gaussian curvature is specified. More precisely,

Presymplectic form

In Geometric Mechanics a presymplectic form is a closed differential 2-form of constant rank on a manifold. However, some authors use different definitions. Recently, Hajduk and Walczak defined a pres

Ricci decomposition

In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into

Abstract differential geometry

The adjective abstract has often been applied to differential geometry before, but the abstract differential geometry (ADG) of this article is a form of differential geometry without the calculus noti

Third fundamental form

In differential geometry, the third fundamental form is a surface metric denoted by . Unlike the second fundamental form, it is independent of the surface normal.

Mabuchi functional

In mathematics, and especially complex geometry, the Mabuchi functional or K-energy functional is a functional on the space of Kähler potentials of a compact Kähler manifold whose critical points are

Maximal surface

In the mathematical field of differential geometry, a maximal surface is a certain kind of submanifold of a Lorentzian manifold. Precisely, given a Lorentzian manifold (M, g), a maximal surface is a s

Yau's conjecture

In differential geometry, Yau's conjecture from 1982, is a mathematical conjecture which states that a closed Riemannian 3-manifold has an infinite number of smooth closed immersed minimal surfaces. I

Finsler manifold

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski functional F(x, −) is provided on each tangent space TxM,

Mean curvature

In mathematics, the mean curvature of a surface is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambie

Banach bundle

In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.

G-fibration

In algebraic topology, a G-fibration or principal fibration is a generalization of a principal G-bundle, just as a fibration is a generalization of a fiber bundle. By definition, given a topological m

Homological mirror symmetry

Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists s

Yamabe invariant

In mathematics, in the field of differential geometry, the Yamabe invariant, also referred to as the sigma constant, is a real number invariant associated to a smooth manifold that is preserved under

Minimal surface

In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because the

Milnor–Wood inequality

In mathematics, more specifically in differential geometry and geometric topology, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure. It is named

Neovius surface

In differential geometry, the Neovius surface is a triply periodic minimal surface originally discovered by Finnish mathematician Edvard Rudolf Neovius (the uncle of Rolf Nevanlinna). The surface has

Rizza manifold

In differential geometry a Rizza manifold, named after Giovanni Battista Rizza, is an almost complex manifold also supporting a Finsler structure: this kind of manifold is also referred as almost Herm

Projective vector field

A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold (p.ex. spacetime) whose flow preserves the geodesic structure of without necessarily preserving the affine

Weitzenböck identity

In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two sec

Maurer–Cartan form

In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G. It was much used by Él

Supergeometry

Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories i

Eguchi–Hanson space

In mathematics and theoretical physics, the Eguchi–Hanson space is a non-compact, self-dual, asymptotically locally Euclidean (ALE) metric on the cotangent bundle of the 2-sphere T*S2. The holonomy gr

Laplace 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.

Localization formula for equivariant cohomology

In differential geometry, the localization formula states: for an equivariantly closed equivariant differential form on an orbifold M with a torus action and for a sufficient small in the Lie algebra

Yang–Mills equations

In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or princi

Glossary of differential geometry and topology

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:
* Glossary of general topology
* Glossary of algebraic to

Arithmetic Fuchsian group

Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an ar

Hitchin's equations

In mathematics, and in particular differential geometry and gauge theory, Hitchin's equations are a system of partial differential equations for a connection and Higgs field on a vector bundle or prin

Generalized flag variety

In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex n

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 embedde

Equivariant index theorem

In differential geometry, the equivariant index theorem, of which there are several variants, computes the (graded) trace of an element of a compact Lie group acting in given setting in terms of the i

Principal geodesic analysis

In geometric data analysis and statistical shape analysis, principal geodesic analysis is a generalization of principal component analysis to a non-Euclidean, non-linear setting of manifolds suitable

Deformed Hermitian Yang–Mills equation

In mathematics and theoretical physics, and especially gauge theory, the deformed Hermitian Yang–Mills (dHYM) equation is a differential equation describing the equations of motion for a D-brane in th

Spin geometry

In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fun

Chern–Weil homomorphism

In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of

Hermitian Yang–Mills connection

In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomo

3-torus

The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, In contrast, the usual torus is the Cartesian product of on

Santaló's formula

In differential geometry, Santaló's formula describes how to integrate a function on the unit sphere bundle of a Riemannian manifold by first integrating along every geodesic separately and then over

Equivariant differential form

In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map from the Lie algebra to the space of differential forms on M that are equivar

Spray (mathematics)

In differential geometry, a spray is a vector field H on the tangent bundle TM that encodes a quasilinear second order system of ordinary differential equations on the base manifold M. Usually a spray

Acceleration (differential geometry)

In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of

Plücker embedding

In mathematics, the Plücker map embeds the Grassmannian , whose elements are k-dimensional subspaces of an n-dimensional vector space V, in a projective space, thereby realizing it as an algebraic var

Courant algebroid

In a field of mathematics known as differential geometry, a Courant geometry was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids

Geodesics on an ellipsoid

The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a

Hermitian manifold

In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a

Liouville's equation

In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor f of a metric f2(dx2 + dy2) on a surface

Transversality theorem

In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse intersection p

Björling problem

In differential geometry, the Björling problem is the problem of finding a minimal surface passing through a given curve with prescribed normal (or tangent planes). The problem was posed and solved by

Hermitian symmetric space

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural

Flat map

In differential geometry, flat map is a mapping that converts vectors into corresponding 1-forms, given a non-degenerate (0,2)-tensor.

Lie derivative

In differential geometry, the Lie derivative (/liː/ LEE), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-fo

Connection (fibred manifold)

In differential geometry, a fibered manifold is surjective submersion of smooth manifolds Y → X. Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manif

Natural bundle

In mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle for some . It turns out that its transition functions depend functionally on local changes of coordinates in the b

Envelope (mathematics)

In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically,

Gromov's compactness theorem (geometry)

In the mathematical field of metric geometry, Mikhael Gromov proved a fundamental compactness theorem for sequences of metric spaces. In the special case of Riemannian manifolds, the key assumption of

Natural pseudodistance

In size theory, the natural pseudodistance between two size pairs , is the value , where varies in the set of all homeomorphisms from the manifold to the manifold and is the supremum norm. If and are

Secondary vector bundle structure

In mathematics, particularly differential topology, the secondary vector bundle structurerefers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smo

Web (differential geometry)

In mathematics, a web permits an intrinsic characterization in terms of Riemannian geometry of the additive separation of variables in the Hamilton–Jacobi equation.

Dynamic fluid film equations

Fluid films, such as soap films, are commonly encountered in everyday experience. A soap film can be formed by dipping a closed contour wire into a soapy solution as in the figure on the right. Altern

G2-structure

In differential geometry, a -structure is an important type of G-structure that can be defined on a smooth manifold. If M is a smooth manifold of dimension seven, then a G2-structure is a reduction of

K-stability

In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stab

Covariance and contravariance of vectors

In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a chan

Inverse mean curvature flow

In the mathematical fields of differential geometry and geometric analysis, inverse mean curvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian manifold. It has

Sharp map

In differential geometry, the sharp map is the mapping that converts 1-forms into corresponding vectors, given a non-degenerate (0,2)-tensor.

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: The Gaussian radius of curvatur

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generaliza

Center of curvature

In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the

Invariant differential operator

In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on , func

Huisken's monotonicity formula

In differential geometry, Huisken's monotonicity formula states that, if an n-dimensional surface in (n + 1)-dimensional Euclidean space undergoes the mean curvature flow, then its convolution with an

Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneere

Polar action

In mathematics, a polar action is a proper and isometric action of a Lie group G on a complete Riemannian manifold M for which there exists a complete submanifold Σ that meets all the orbits and meets

Wirtinger inequality (2-forms)

In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that on a Kähler manifold M, the exterior kth power of the symplectic form (Kähler form) ω, when evaluated o

Tensor density

In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to

Tensor product bundle

In differential geometry, the tensor product of vector bundles E, F (over same space ) is a vector bundle, denoted by E ⊗ F, whose fiber over a point is the tensor product of vector spaces Ex ⊗ Fx. Ex

Complex hyperbolic space

In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex hyperbolic space is a Kähler mani

Geometric analysis

Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential ge

Moving frame

In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in

Gromov's inequality for complex projective space

In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality , valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attainedb

Contact (mathematics)

In mathematics, two functions have a contact of order k if, at a point P, they have the same value and k equal derivatives. This is an equivalence relation, whose equivalence classes are generally cal

Instanton

An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a , non-zero action, either in quantum

Glossary of Riemannian and metric geometry

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either con

Winding number

In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around

Liouville field theory

In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville th

Affine geometry of curves

In the mathematical field of differential geometry, the affine geometry of curves is the study of curves in an affine space, and specifically the properties of such curves which are invariant under th

Pullback (differential geometry)

Suppose that φ : M → N is a smooth map between smooth manifolds M and N. Then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to

Bitangent

In geometry, a bitangent to a curve C is a line L that touches C in two distinct points P and Q and that has the same direction as C at these points. That is, L is a tangent line at P and at Q.

Time dependent vector field

In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which moves as time passes. Fo

Thurston norm

In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of hom

Bundle gerbe

In mathematics, a bundle gerbe is a geometrical model of certain 1-gerbes with connection, or equivalently of a 2-class in Deligne cohomology.

Frenet–Serret formulas

In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space R3, or the geometric prope

Diffeology

In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are. The concept was first introduc

Weyl integration formula

In mathematics, the Weyl integration formula, introduced by Hermann Weyl, is an integration formula for a compact connected Lie group G in terms of a maximal torus T. Precisely, it says there exists a

General covariance

In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable coordi

L² cohomology

In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrab

Parallelization (mathematics)

In mathematics, a parallelization of a manifold of dimension n is a set of n global smooth linearly independent vector fields.

Immersion (mathematics)

In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, f : M → N is an immersion if is an i

Bernstein's problem

In differential geometry, Bernstein's problem is as follows: if the graph of a function on Rn−1 is a minimal surface in Rn, does this imply that the function is linear? This is true in dimensions n at

Geodesic

In geometry, a geodesic (/ˌdʒiː.əˈdɛsɪk, -oʊ-, -ˈdiːsɪk, -zɪk/) is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifol

Iwasawa manifold

In mathematics, in the field of differential geometry, an Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup. An Iwasawa manifold is a

Quaternionic manifold

In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the

(G,X)-manifold

In geometry, if X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a (G, X)-structure on a topological space is a way to formalise it being locally is

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.

Reilly formula

In the mathematical field of Riemannian geometry, the Reilly formula is an important identity, discovered by Robert Reilly in 1977. It says that, given a smooth Riemannian manifold-with-boundary (M, g

Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, i

Haken manifold

In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one consid

G-structure on a manifold

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures include

Affine bundle

In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.

Affine connection

In 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

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,

Torsion of a curve

In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a s

Upper half-plane

In mathematics, the upper half-plane, is the set of points (x, y) in the Cartesian plane with y > 0.

Analytic torsion

In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dim

Isothermal coordinates

In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in i

Round function

In topology and in calculus, a round function is a scalar function , over a manifold , whose critical points form one or several connected components, each homeomorphic to the circle , also called cri

Kulkarni–Nomizu product

In the mathematical field of differential geometry, the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu) is defined for two (0, 2)-tensors and gives as a result a (0, 4

Lichnerowicz formula

The Lichnerowicz formula (also known as the Lichnerowicz–Weitzenböck formula) is a fundamental equation in the analysis of spinors on pseudo-Riemannian manifolds. In dimension 4, it forms a piece of S

Courant bracket

In a field of mathematics known as differential geometry, the Courant bracket is a generalization of the Lie bracket from an operation on the tangent bundle to an operation on the direct sum of the ta

Associated bundle

In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to ,

Kähler identities

No description available.

Pseudogroup

In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating h

Shape analysis (digital geometry)

This article describes shape analysis to analyze and process geometric shapes.

Affine curvature

Special affine curvature, also known as the equiaffine curvature or affine curvature, is a particular type of curvature that is defined on a plane curve that remains unchanged under a special affine t

Gauge covariant derivative

The gauge covariant derivative is a variation of the covariant derivative used in general relativity, quantum field theory and fluid dynamics. If a theory has gauge transformations, it means that some

Involute

In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut str

Coordinate-induced basis

In mathematics, a coordinate-induced basis is a basis for the tangent space or cotangent space of a manifold that is induced by a certain coordinate system. Given the coordinate system , the coordinat

Frame (linear algebra)

In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology of signal processing, a frame provides

Hilbert's lemma

Hilbert's lemma was proposed at the end of the 19th century by mathematician David Hilbert. The lemma describes a property of the principal curvatures of surfaces. It may be used to prove Liebmann's t

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 po

Monopole (mathematics)

In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle.

Pushforward (differential)

In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ, , at a po

Myers–Steenrod theorem

Two theorems in the mathematical field of Riemannian geometry bear the name Myers–Steenrod theorem, both from a 1939 paper by Myers and Steenrod. The first states that every distance-preserving map (t

Spherical Bernstein's problem

The spherical Bernstein's problem is a possible generalization of the original Bernstein's problem in the field of global differential geometry, first proposed by Shiing-Shen Chern in 1969, and then l

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifol

Darboux vector

In differential geometry, especially the theory of space curves, the Darboux vector is the angular velocity vector of the Frenet frame of a space curve. It is named after Gaston Darboux who discovered

Petrov classification

In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each

Differentiable stack

A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas,

Sine-Gordon equation

The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduce

Reflection lines

Engineers use reflection lines to judge a surface's quality. Reflection lines reveal surface flaws, particularly discontinuities in normals indicating that the surface is not . Reflection lines may be

Normal bundle

In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).

Curvature form

In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.

Whitehead manifold

In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to J. H. C. Whitehead discovered this puzzling object while he was trying to prove the Poincaré

Connection (composite bundle)

Composite bundles play a prominent role in gauge theory with symmetry breaking, e.g., gauge gravitation theory, non-autonomous mechanics where is the time axis, e.g., mechanics with time-dependent par

Conformal Killing vector field

In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric (also called a conformal Killing vector, CKV, or conformal colineation), is a vecto

Connection (principal bundle)

In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fiber

Bäcklund transform

In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions. They are an im

Frölicher–Nijenhuis bracket

In mathematics, the Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the study of conne

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 fram

Osculating plane

In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order

Minakshisundaram–Pleijel zeta function

The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced by Subbaramiah Minakshisundaram and Åke Plei

Diffiety

In mathematics, a diffiety (/dəˈfaɪəˌtiː/) is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equation

Catalan's minimal surface

In differential geometry, Catalan's minimal surface is a minimal surface originally studied by Eugène Charles Catalan in 1855. It has the special property of being the minimal surface that contains a

Kronheimer–Mrowka basic class

In mathematics, the Kronheimer–Mrowka basic classes are elements of the second cohomology H2(X) of a simple smooth 4-manifold X that determine its Donaldson polynomials. They were introduced by P. B.

Normal plane (geometry)

A normal plane is any plane containing the normal vector of a surface at a particular point. The normal plane also refers to the plane that is perpendicular to the tangent vector of a space curve; (th

Evolute

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the res

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.That is, systolic invariants or

Metric signature

In mathematics, the signature (v, p, r) of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (c

Bundle metric

In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric

Kenmotsu manifold

In the mathematical field of differential geometry, a Kenmotsu manifold is an almost-contact manifold endowed with a certain kind of Riemannian metric.

Tensor field

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic

Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distributio

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 (such as the sphere

Ribbon (mathematics)

In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector. More formally, a ribbon denoted by includes a curve given by a three-dimen

Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the

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 (see associated bundle)

Curvature of Space and Time, with an Introduction to Geometric Analysis

Curvature of Space and Time, with an Introduction to Geometric Analysis is an undergraduate-level textbook for mathematics and physics students on differential geometry, focusing on applications to ge

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 f

Interior product

In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)

Lebrun manifold

In mathematics, a LeBrun manifold is a connected sum of copies of the complex projective plane, equipped with an explicit self-dual metric. Here, self-dual means that the Weyl tensor is its own Hodge

Motion (geometry)

In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion

Geometry processing

Geometry processing, or mesh processing, is an area of research that uses concepts from applied mathematics, computer science and engineering to design efficient algorithms for the acquisition, recons

Hamiltonian field theory

In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It a

Shape of the universe

The shape of the universe, in physical cosmology, is the and of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the u

Levi-Civita parallelogramoid

In the mathematical field of differential geometry, the Levi-Civita parallelogramoid is a quadrilateral in a curved space whose construction generalizes that of a parallelogram in the Euclidean plane.

Hopf conjecture

In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.

Riemannian connection on a surface

In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the e

Lie bracket of vector fields

In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two ve

Nearly Kähler manifold

In mathematics, a nearly Kähler manifold is an almost Hermitian manifold , with almost complex structure ,such that the (2,1)-tensor is skew-symmetric. So, for every vector field on . In particular, a

Nonlinear partial differential equation

In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation t

Haefliger structure

In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André Haefliger in 1970. Any foliation on a manifold induces a special kind

Translation surface (differential geometry)

In differential geometry a translation surface is a surface that is generated by translations:
* For two space curves with a common point , the curve is shifted such that point is moving on . By this

Complex manifold

In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in , such that the transition maps are holomorphic. The term complex manif

Equiareal map

In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of figures.

Nonholonomic system

A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differe

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to expre

Chentsov's theorem

In information geometry, Chentsov's theorem states that the Fisher information metric is, up to rescaling, the unique Riemannian metric on a statistical manifold that is invariant under sufficient sta

Closed manifold

In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components.

Lie group–Lie algebra correspondence

In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorph

Lanczos tensor

The Lanczos tensor or Lanczos potential is a rank 3 tensor in general relativity that generates the Weyl tensor. It was first introduced by Cornelius Lanczos in 1949. The theoretical importance of the

Symplectic space

No description available.

Sphere

A sphere (from Ancient Greek σφαῖρα (sphaîra) 'globe, ball') is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the

General covariant transformations

In physics, general covariant transformations are symmetries of gravitation theory on a world manifold . They are gauge transformations whose parameter functions are vector fields on . From the physic

Heat kernel signature

A heat kernel signature (HKS) is a feature descriptor for use in deformable shape analysis and belongs to the group of spectral shape analysis methods. For each point in the shape, HKS defines its fea

Multivector

In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra Λ(V) of a vector space V. This algebra is graded, associative and alternating, and consis

Siegel upper half-space

In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is p

Double vector bundle

In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent of a vector bundle and the double tangent bundle .

Tangential and normal components

In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and anoth

Hilbert manifold

In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbe

Spherical image

In differential geometry, the spherical image of a unit-speed curve is given by taking the curve's tangent vectors as points, all of which must lie on the unit sphere. The movement of the spherical im

Essential manifold

In geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov.

Bochner's formula

In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.

Classification of manifolds

In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain.

Principal bundle

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian prod

Pseudotensor

In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally chang

Jet (mathematics)

In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. Although this is the defini

Poisson manifold

In differential geometry, a Poisson structure on a smooth manifold is a Lie bracket (called a Poisson bracket in this special case) on the algebra of smooth functions on , subject to the Leibniz rule

Spin connection

In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as th

Dual curve

In projective geometry, a dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each

Fundamental theorem of curves

In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size or scale) completely dete

Gauss map

In differential geometry, the Gauss map (named after Carl F. Gauss) 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

Parabolic geometry (differential geometry)

In differential geometry and the study of Lie groups, a parabolic geometry is a homogeneous space G/P which is the quotient of a semisimple Lie group G by a parabolic subgroup P. More generally, the c

Differentiable curve

Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have

Gibbons–Hawking ansatz

In mathematics, the Gibbons–Hawking ansatz is a method of constructing gravitational instantons introduced by Gary Gibbons and Stephen Hawking . It gives examples of hyperkähler manifolds in dimension

Almost-contact manifold

In the mathematical field of differential geometry, an almost-contact structure is a certain kind of geometric structure on a smooth manifold. Such structures were introduced by Shigeo Sasaki in 1960.

Geodesic map

In mathematics—specifically, in differential geometry—a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". More precisely, given two (pseudo-)Riema

Hyperkähler quotient

In mathematics, the hyperkähler quotient of a hyperkähler manifold acted on by a Lie group G is the quotient of a fiber of a hyperkähler moment map over a G-fixed point by the action of G. It was intr

Monopole moduli space

In mathematics, the monopole moduli space is a space parametrizing monopoles (solutions of the Bogomolny equations). Atiyah and Hitchin studied the moduli space for 2 monopoles in detail and used it t

Variational bicomplex

In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is th

Developable surface

In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can

Affine Grassmannian (manifold)

In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all k-dimensional affine subspaces of Rn (described on this page), while in the other the

Mostow rigidity theorem

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimen

Geodesic manifold

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, you can follow a "straight" line indefinitely along any

Calculus of moving surfaces

The calculus of moving surfaces (CMS) is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative whose original definition was put for

Tangent space

In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the cont

Nadirashvili surface

In differential geometry, a Nadirashvili surface is an immersed complete bounded minimal surface in R3 with negative curvature. The first example of such a surface was constructed by in . This simulta

Isotropic manifold

In mathematics, an isotropic manifold is a manifold in which the geometry does not depend on directions. Formally, we say that a Riemannian manifold is isotropic if for any point and unit vectors , th

Exterior covariant derivative

In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector

Riemann's minimal surface

In differential geometry, Riemann's minimal surface is a one-parameter family of minimal surfaces described by Bernhard Riemann in a posthumous paper published in 1867. Surfaces in the family are sing

Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a

Synthetic differential geometry

In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulatio

Symmetric space

In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be stu

Bochner identity

In mathematics — specifically, differential geometry — the Bochner identity is an identity concerning harmonic maps between Riemannian manifolds. The identity is named after the American mathematician

Closed geodesic

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the

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.

Exponential map (Riemannian geometry)

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determin

Real projective space

In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension n, and is a special case of a Grassman

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 mea

Atiyah–Hitchin–Singer theorem

In differential geometry and gauge theory, the Atiyah–Hitchin–Singer theorem, introduced by Michael Atiyah, Nigel Hitchin, and Isadore Singer , states that the space of SU(2) anti self dual Yang–Mills

Density on a manifold

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a

G2 manifold

In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group is one of the five exceptional simple Lie groups. It can be described

Volume entropy

The volume entropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls in its universal cover. This concept is closely r

Atiyah conjecture

In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.

Arithmetic group

In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other

Weierstrass–Enneper parameterization

In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry. Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as

Differential invariant

In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space. Differential invariants are fundame

Nonmetricity tensor

In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It is therefore a tensor field of order three. It vanishes for the case of Riemannian

Spectral geometry

Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Be

Symmetry (physics)

In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of pa

Affine sphere

In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point. The term affine sphere is used because they play

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 con

Curved space

Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some

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

Associate family

In differential geometry, the associate family (or Bonnet family) of a minimal surface is a one-parameter family of minimal surfaces which share the same Weierstrass data. That is, if the surface has

Integration along fibers

In differential geometry, the integration along fibers of a k-form yields a -form where m is the dimension of the fiber, via "integration".

Holonomic basis

In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold M is a set of basis vector fields {e1, ..., en} defined at every point P of a region of the

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

Affine focal set

In mathematics, and especially affine differential geometry, the affine focal set of a smooth submanifold M embedded in a smooth manifold N is the caustic generated by the affine normal lines. It can

Weakly symmetric space

In mathematics, a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg in the 1950s as a generalisation of symmetric space, due to Élie Cartan. Geometrically the s

Cayley's ruled cubic surface

In differential geometry, Cayley's ruled cubic surface is the ruled cubic surface It contains a nodal line of self-intersection and two cuspital points at infinity. In projective coordinates it is .

List of differential geometry topics

This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics.

Grassmannian

In mathematics, the Grassmannian Gr(k, V) is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V. For example, the Grassmannian Gr(1, V) is the space of l

Hopf fibration

In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles an

Schouten–Nijenhuis bracket

In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie brack

Equivalent latitude

In differential geometry, the equivalent latitude is a Lagrangian coordinate .It is often used in atmospheric science,particularly in the study of stratospheric dynamics.Each isoline in a map of equiv

Yau's conjecture on the first eigenvalue

In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks: Is it true that the first eigenvalue for the Laplace–Beltra

Parallel curve

A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of parallel (straight) lines. It can also be defined as a curve whose points ar

Lie groupoid

In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map a

Frankel conjecture

In the mathematical fields of differential geometry and algebraic geometry, the Frankel conjecture was a problem posed by Theodore Frankel in 1961. It was resolved in 1979 by Shigefumi Mori, and by Yu

Lie algebroid

In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of

Monodromy

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name

Paneitz operator

In the mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension n. It is named after , who discovered it

Tame manifold

In geometry, a tame manifold is a manifold with a well-behaved compactification. More precisely, a manifold is called tame if it is homeomorphic to a compact manifold with a closed subset of the bound

Arthur Besse

Arthur Besse is a pseudonym chosen by a group of French differential geometers, led by Marcel Berger, following the model of Nicolas Bourbaki. A number of monographs have appeared under the name.

Stiefel manifold

In mathematics, the Stiefel manifold is the set of all orthonormal k-frames in That is, it is the set of ordered orthonormal k-tuples of vectors in It is named after Swiss mathematician Eduard Stiefel

Tangential angle

In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis. (Some authors define the

Generalized complex structure

In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic

Inflection point

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature

Stable principal bundle

In mathematics, and especially differential geometry and algebraic geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles.

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 aro

Chern's conjecture for hypersurfaces in spheres

Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates from the Chern's unanswered question: Consi

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

Margulis lemma

In differential geometry, the Margulis lemma (named after Grigory Margulis) is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold (e.g. the hyperbolic n-spa

Directional derivative

In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the

Discrete differential geometry

Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. I

Negative pedal curve

In geometry, a negative pedal curve is a plane curve that can be constructed from another plane curve C and a fixed point P on that curve. For each point X ≠ P on the curve C, the negative pedal curve

Alexandrov space

In geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are locally compa

Lyusternik–Fet theorem

In mathematics, the Lyusternik–Fet theorem states that on every compact Riemannian manifold there exists a closed geodesic. It is named after Lazar Lyusternik and Abram Ilyich Fet.

Pedal curve

In mathematics, a pedal curve of a given curve results from the orthogonal projection of a fixed point on the tangent lines of this curve. More precisely, for a plane curve C and a given fixed pedal p

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 c

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. Historically, c

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. Such geodesics are called prime

Hitchin system

In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies o

Theorema Egregium

Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says

Henneberg surface

In differential geometry, the Henneberg surface is a non-orientable minimal surface named after Lebrecht Henneberg. It has parametric equation and can be expressed as an order-15 algebraic surface. It

Schwarz minimal surface

In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz. In the 1880s Schwarz and his student E. R. Neovius described periodic mini

Clifford analysis

Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. E

Gauss curvature flow

In the mathematical fields of differential geometry and geometric analysis, the Gauss curvature flow is a geometric flow for oriented hypersurfaces of Riemannian manifolds. In the case of curves in a

Double tangent bundle

In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πT

Tortuosity

Tortuosity is widely used as a critical parameter to predict transport properties of porous media, such as rocks and soils. But unlike other standard microstructural properties, the concept of tortuos

Volume form

In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold of dimension , a volume form is an -form. It i

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. For example, if M and N are

Kähler–Einstein metric

In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admi

Anti-de Sitter space

In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are

Triply periodic minimal surface

In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystall

Induced metric

In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, th

Teichmüller space

In mathematics, the Teichmüller space of a (real) topological (or differential) surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the

Projective differential geometry

In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifold

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 t

Pseudosphere

In geometry, a pseudosphere is a surface with constant negative Gaussian curvature. A pseudosphere of radius R is a surface in having curvature −1/R2 in each point. Its name comes from the analogy wit

Quaternion-Kähler symmetric space

In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symm

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 (a more i

Bel–Robinson tensor

In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by: Alternatively, where is the Weyl tensor. It was introduced by Lluís Bel

Affine manifold

In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset

Bogomolny equations

In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation where is the curvature of a connection on a principal -bundle over a 3-manifold , is a sectio

Weyl transformation

In theoretical physics, the Weyl transformation, named after Hermann Weyl, is a local rescaling of the metric tensor: which produces another metric in the same conformal class. A theory or an expressi

Gauge group (mathematics)

A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle with a structure Lie group , a gauge group is defin

Connection (affine bundle)

Let Y → X be an affine bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J1Y of the jet bundle J1Y → Y of Y is an affine bu

Second fundamental form

In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by (read

Crofton formula

In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line int

Caustic (mathematics)

In differential geometry, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the concept of caustics in geometric optics. The ray's source may be a point (

Chern's conjecture (affine geometry)

Chern's conjecture for affinely flat manifolds was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2018, it remains an unsolved mathematical problem. Chern's conjecture st

Fibered manifold

In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion that is, a surjective differentiable mapping such that at each point the tangent ma

Ricci calculus

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is

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. The torsion of a curve, as it appears in the Frenet–Serret formulas, fo

Spectral shape analysis

Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami

Myers's theorem

Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asse

Taut submanifold

In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every the distance function is a perfect Morse function. If N is not compact, one n

Filling area conjecture

In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introduci

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.

Quillen metric

In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen for certain elliptic o

Tameness theorem

In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a

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. There are analogs for gener

Matsushima's formula

In mathematics, Matsushima's formula, introduced by Matsushima, is a formula for the Betti numbers of a quotient of a symmetric space G/H by a discrete group, in terms of unitary representations of th

Calibrated geometry

In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibratio

Last geometric statement of Jacobi

In differential geometry the last geometric statement of Jacobi is a conjecture named after Carl Gustav Jacob Jacobi. According to this conjecture: Every caustic from any point on an ellipsoid other t

Total absolute curvature

In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is

Radius of curvature

In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For sur

Chern–Simons form

In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteris

Covariant transformation

In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. The transformation that describes the new basis vector

Higher gauge theory

In mathematical physics higher gauge theory is the general study of counterparts of gauge theory that involve higher-degree differential forms instead of the traditional connection forms of gauge theo

K-noid

In differential geometry, a k-noid is a minimal surface with k catenoid openings. In particular, the 3-noid is often called trinoid. The first k-noid minimal surfaces were described by Jorge and Meeks

ADHM construction

In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel

Clairaut's relation (differential geometry)

In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ

Novikov–Shubin invariant

In mathematics, a Novikov–Shubin invariant, introduced by Sergei Novikov and Mikhail Shubin, is an invariant of a compact Riemannian manifold related to the spectrum of the Laplace operator acting on

Affine differential geometry

Affine differential geometry is a type of differential geometry which studies invariants of volume-preserving affine transformations. The name affine differential geometry follows from Klein's Erlange

Calabi–Yau manifold

In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical phys

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. Leibniz defined it as the line through a pair of i

Mean curvature flow

In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euc

Twist (mathematics)

In differential geometry, the twist of a ribbon is its rate of axial rotation. Let a ribbon be composted of space curve , where is the arc length of , and the a unit normal vector, perpendicular at ea

Coframe

In mathematics, a coframe or coframe field on a smooth manifold is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of , one has a

Warped geometry

In mathematics and physics, in particular differential geometry and general relativity, a warped geometry is a Riemannian or Lorentzian manifold whose metric tensor can be written in form The geometry

Lattice (discrete subgroup)

In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case

String group

In topology, a branch of mathematics, a string group is an infinite-dimensional group introduced by as a -connected cover of a spin group. A string manifold is a manifold with a lifting of its frame b

Gauge theory (mathematics)

In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in

Musical isomorphism

In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle and the cotangent bundle of a pseudo-Riemanni

Bogomolov–Miyaoka–Yau inequality

In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality between Chern numbers of compact complex surfaces of general type. Its major interest is the way it restricts the possible topolo

Scherk surface

In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly period

De Sitter space

In mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of

Tetrad formalism

The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a lo

Representation up to homotopy

A Representation up to homotopy has several meanings. One of the earliest appeared in the `physical' context of constrained Hamiltonian systems. The essential idea is lifting a non-representation on a

Group analysis of differential equations

Group analysis of differential equations is a branch of mathematics that studies the symmetry properties of differential equations with respect to various transformations of independent and dependent

K3 surface

In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means

Dupin hypersurface

In differential geometry, a Dupin hypersurface is a submanifold in a space form, whose principal curvatures have globally constant multiplicities.

Conformal geometry

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of

Symmetry set

In geometry, the symmetry set is a method for representing the local symmetries of a curve, and can be used as a method for representing the shape of objects by finding the topological skeleton. The m

Ruled surface

In geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S. Examples include the plane, the lateral surface of a cylinder or cone, a c

Metric tensor

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that allows defining distances and angles, just as

Hilbert scheme

In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refin

Hedgehog (geometry)

In differential geometry, a hedgehog or plane hedgehog is a type of plane curve, the envelope of a family of lines determined by a support function. More intuitively, sufficiently well-behaved hedgeho

Costa's minimal surface

In mathematics, Costa's minimal surface, is an embedded minimal surface discovered in 1982 by the Brazilian mathematician Celso José da Costa. It is also a surface of finite topology, which means that

Tangent indicatrix

In differential geometry, the tangent indicatrix of a closed space curve is a curve on the unit sphere intimately related to the curvature of the original curve. Let be a closed curve with nowhere-van

Holmes–Thompson volume

In geometry of normed spaces, the Holmes–Thompson volume is a notion of volume that allows to compare sets contained in different normed spaces (of the same dimension). It was introduced by Raymond D.

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