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Space form

In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic

Kosmann lift

In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field on a Riemannian manifold is the canonical projection on the orthonormal frame bundle of its natura

Unit tangent bundle

In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M) or simply UTM, is the unit sphere bundle for the tangent bundle T(M). It is a fiber bundle over M

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

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

Killing–Hopf theorem

In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group

Conformal map

In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let and be open subsets of . A function is called conformal (or angle-preservin

Metric connection

In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those

Weyl tensor

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature t

Wiedersehen pair

In mathematics—specifically, in Riemannian geometry—a Wiedersehen pair is a pair of distinct points x and y on a (usually, but not necessarily, two-dimensional) compact Riemannian manifold (M, g) such

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,

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

Yamabe problem

The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: Let

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point t

Nash embedding theorems

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

Prescribed scalar curvature problem

In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold M and a smooth, real-valued function ƒ on M, construct a Riemann

Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived fr

Biharmonic map

In the mathematical field of differential geometry, a biharmonic map is a map between Riemannian or pseudo-Riemannian manifolds which satisfies a certain fourth-order partial differential equation. A

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

Cheeger constant

In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold M is a positive real number h(M) defined in terms of the minimal area of a hypersurface that divides M into

Geodesic deviation

In general relativity, put another way, if two objects are set in motion along two initially parallel trajectories, the presence of a tidal gravitational force will cause the trajectories to bend towa

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

Parallel transport

In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covarian

Harmonic coordinates

In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian metric on the manifold. They are useful in

Recurrent tensor

In mathematics and physics, a recurrent tensor, with respect to a connection on a manifold M, is a tensor T for which there is a one-form ω on M such that

Essential manifold

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

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

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of t

Fundamental theorem of Riemannian geometry

In the mathematical field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connect

Hodge star operator

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form

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

Second covariant derivative

In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with

Riemannian Penrose inequality

In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalizat

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 manifol

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 prec

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

Berger's sphere

In Riemannian geometry, a Berger sphere, named after Marcel Berger, is a standard 3-sphere with Riemannian metric from a one-parameter family, which can be obtained from the standard metric by shrinki

Sharafutdinov's retraction

In mathematics, Sharafutdinov's retraction is a construction that gives a retraction of an open non-negatively curved Riemannian manifold onto its soul. It was first used by to show that any two souls

Cut locus (Riemannian manifold)

In Riemannian geometry, the cut locus of a point in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from , but it may contain additi

Levi-Civita connection

In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifol

Rayleigh–Faber–Krahn inequality

In spectral geometry, the Rayleigh–Faber–Krahn inequality, named after its conjecturer, Lord Rayleigh, and two individuals who independently proved the conjecture, G. Faber and Edgar Krahn, is an ineq

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

Christoffel symbols

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other mani

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

Orthonormal frame

In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If M is a manifold equipped with a metric g

Curvature invariant

In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the Riemann tensor, the W

Minimal volume

In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a smooth manifold's topology. This topological invariant was introduced by Mikhael G

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

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

Killing spinor

Killing spinor is a term used in mathematics and physics. By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those twistorspinors which are also eigenspinor

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

De Sitter–Schwarzschild metric

In general relativity, the de Sitter–Schwarzschild solution describes a black hole in a causal patch of de Sitter space. Unlike a flat-space black hole, there is a largest possible de Sitter black hol

Einstein notation

In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational conve

Cotton tensor

In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for n = 3 is necessar

Harmonic morphism

In mathematics, a harmonic morphism is a (smooth) map between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphism

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

Isoparametric function

In differential geometry, an isoparametric function is a function on a Riemannian manifold whose level surfaces are parallel and of constant mean curvatures. They were introduced by Cartan.

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

Prescribed Ricci curvature problem

In Riemannian geometry, a branch of mathematics, the prescribed Ricci curvature problem is as follows: given a smooth manifold M and a symmetric 2-tensor h, construct a metric on M whose Ricci curvatu

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.

Scalar curvature

In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns

Cartan–Ambrose–Hicks theorem

In mathematics, the Cartan–Ambrose–Hicks theorem is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by the Riemann curvature tensor, or in other words,

Isoparametric manifold

In Riemannian geometry, an isoparametric manifold is a type of (immersed) submanifold of Euclidean space whose normal bundle is flat and whose principal curvatures are constant along any parallel norm

Contorsion tensor

The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbei

Spherical 3-manifold

In mathematics, a spherical 3-manifold M is a 3-manifold of the form where is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere . All such manifolds are prime, orientable, and clos

Smooth coarea formula

In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains. Let be smooth Riemannian manifolds of respective dimensions

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

Gauss's lemma (Riemannian geometry)

In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let M

Conjugate points

In differential geometry, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole

Ruppeiner geometry

Ruppeiner geometry is thermodynamic geometry (a type of information geometry) using the language of Riemannian geometry to study thermodynamics. George Ruppeiner proposed it in 1979. He claimed that t

Sasakian manifold

In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold equipped with a special kind of Riemannian metric , called a Sasakian 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 v

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

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

Cartan–Hadamard conjecture

In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized

Harmonic map

In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equa

Thurston elliptization conjecture

William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature.

Bishop–Gromov inequality

In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key p

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

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

Metric Structures for Riemannian and Non-Riemannian Spaces

Metric Structures for Riemannian and Non-Riemannian Spaces is a book in geometry by Mikhail Gromov. It was originally published in French in 1981 under the title Structures métriques pour les variétés

Collapsing manifold

In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics gi, such that as i goes to infinity the manifold is close to a k-

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

Clifford module bundle

In differential geometry, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a vector bundle whose fibers are Clifford modules, the representations of Clifford algebras.

Riemannian manifold

In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite in

Killing tensor

In mathematics, a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields. It is a concept in pseudo-Riemannian geomet

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

Ricci soliton

In differential geometry, a complete Riemannian manifold is called a Ricci soliton if, and only if, there exists a smooth vector field such that for some constant . Here is the Ricci curvature tensor

Gromov–Hausdorff convergence

In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

Quaternion-Kähler manifold

In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1) for some . Here Sp(n)

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

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.

Abel–Jacobi map

In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping

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

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

Laplace–Beltrami operator

In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and

Clifford bundle

In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford

Fermi coordinates

In the mathematical theory of Riemannian geometry, there are two uses of the term Fermi coordinates.In one use they are local coordinates that are adapted to a geodesic. In a second, more general one,

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

Killing vector field

In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric

Almost flat manifold

In mathematics, a smooth compact manifold M is called almost flat if for any there is a Riemannian metric on M such that and is -flat, i.e. for the sectional curvature of we have . Given n, there is a

Schouten tensor

In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for n ≥ 3 by: where Ric is the Ricci tensor (defined by contracting the first and third

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

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

Kretschmann scalar

In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann

Conformally flat manifold

A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. In practice, the metric of the manifold has to be co

Hermitian connection

In mathematics, a Hermitian connection is a connection on a Hermitian vector bundle over a smooth manifold which is compatible with the Hermitian metric on , meaning that for all smooth vector fields

Riemannian submersion

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

Schur's lemma (Riemannian geometry)

In Riemannian geometry, Schur's lemma is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a

Bryant surface

In Riemannian geometry, a Bryant surface is a 2-dimensional surface embedded in 3-dimensional hyperbolic space with constant mean curvature equal to 1. These surfaces take their name from the geometer

Spinor bundle

In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold one defines the spinor bundle to be the complex vector bundle associated to the corresponding princip

List of formulas in Riemannian geometry

This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted ot

First variation of area formula

In the mathematical field of Riemannian geometry, every submanifold of a Riemannian manifold has a surface area. The first variation of area formula is a fundamental computation for how this quantity

Lichnerowicz conjecture

In mathematics, the Lichnerowicz conjecture is a generalization of a conjecture introduced by Lichnerowicz. Lichnerowicz's original conjecture was that locally harmonic 4-manifolds are locally symmetr

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

Normal coordinates

In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by appl

Jacobi field

In Riemannian geometry, a Jacobi field is a vector field along a geodesic in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other word

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

De Sitter invariant special relativity

In mathematical physics, de Sitter invariant special relativity is the speculative idea that the fundamental symmetry group of spacetime is the indefinite orthogonal group SO(4,1), that of de Sitter s

Constant curvature

In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining

Geometrization conjecture

In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue

Intrinsic flat distance

In mathematics, the intrinsic flat distance is a notion for distance between two Riemannian manifolds which is a generalization of Federer and Fleming's flat distance between submanifolds and integral

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

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

Macbeath surface

In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface. The automorphism group of the Macbea

Great circle

In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so t

Constraint counting

In mathematics, constraint counting is counting the number of constraints in order to compare it with the number of variables, parameters, etc. that are free to be determined, the idea being that in m

Ricci flow

In the mathematical fields of differential geometry and geometric analysis, the Ricci flow (/ˈriːtʃi/ REE-chee, Italian: [ˈrittʃi]), sometimes also referred to as Hamilton's Ricci flow, is a certain p

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

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

Flat convergence

In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Federer a

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

Pu's inequality

In differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves co

Gauss–Codazzi equations

In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental form

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

Seifert–Weber space

In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahe

Line element

In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, w

Riemannian circle

In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the intrinsic Riemannian metric of a compact one-di

Cartan–Karlhede algorithm

The Cartan–Karlhede algorithm is a procedure for completely classifying and comparing Riemannian manifolds. Given two Riemannian manifolds of the same dimension, it is not always obvious whether they

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