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Ricci curvature

Index Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a small wedge of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. [1]

84 relations: Affine connection, Annals of Mathematics, Ball (mathematics), Bishop–Gromov inequality, Cambridge University Press, Canonical bundle, Chern class, Christina Sormani, Christoffel symbols, Closed and exact differential forms, Cohomology, Comparison theorem, Complex differential form, Complex manifold, Cosmological constant, Curvature form, Curvature of Riemannian manifolds, Determinant, Differential geometry, Einstein field equations, Einstein manifold, Einstein notation, Ellipsoid, Euclidean space, Exterior algebra, Foundations of Differential Geometry, Fundamental group, G-structure on a manifold, Gauss–Codazzi equations, General relativity, Geodesic, Gregorio Ricci-Curbastro, Grigori Perelman, Gromov's compactness theorem (geometry), Harmonic coordinates, Harmonic function, Heat equation, Holonomy, Homotopy, Hypersurface, Introduction to the mathematics of general relativity, Isometry, Jacobi field, John Wiley & Sons, Kähler differential, Kähler manifold, Laplace operator, Laplace–Beltrami operator, Levi-Civita connection, Local coordinates, ..., Metric tensor, Myers's theorem, Normal coordinates, Poincaré conjecture, Principal axis theorem, Principal curvature, Projective differential geometry, Pseudo-Riemannian manifold, Raychaudhuri equation, Ricci calculus, Ricci decomposition, Ricci flow, Ricci-flat manifold, Richard S. Hamilton, Riemann curvature tensor, Riemannian geometry, Riemannian manifold, Scalar curvature, Second fundamental form, Sectional curvature, Space form, Spherical wedge, Splitting theorem, Symmetric bilinear form, Symmetric tensor, Tangent space, Taylor series, Theory of relativity, Trace (linear algebra), Vector field, Volume element, Volume form, William Thurston, World line. Expand index (34 more) »

Affine connection

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.

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Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

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Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

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Bishop–Gromov inequality

In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov.

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Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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Canonical bundle

In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n.

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Chern class

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles.

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Christina Sormani

Christina Sormani is a professor of mathematics at City University of New York affiliated with Lehman College and the CUNY Graduate Center.

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Christoffel symbols

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.

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Closed and exact differential forms

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα.

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Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.

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Comparison theorem

A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects.

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Complex differential form

In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.

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Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

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Cosmological constant

In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) is the value of the energy density of the vacuum of space.

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Curvature form

In differential geometry, the curvature form describes the curvature of a connection on a principal bundle.

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Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.

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Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

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Einstein field equations

The Einstein field equations (EFE; also known as Einstein's equations) comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.

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Einstein manifold

In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric.

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Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity.

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Ellipsoid

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.

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Foundations of Differential Geometry

Foundations of Differential Geometry is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu.

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Fundamental group

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

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G-structure on a manifold

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields.

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Gauss–Codazzi equations

In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds.

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General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

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Geodesic

In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

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Gregorio Ricci-Curbastro

Gregorio Ricci-Curbastro (12January 1925) was an Italian mathematician born in Lugo di Romagna.

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Grigori Perelman

Grigori Yakovlevich Perelman (a; born 13 June 1966) is a Russian mathematician.

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Gromov's compactness theorem (geometry)

In Riemannian geometry, Gromov's (pre)compactness theorem states that the set of compact Riemannian manifolds of a given dimension, with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov–Hausdorff metric.

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Harmonic coordinates

In Riemannian geometry, a branch of mathematics, harmonic coordinates are a coordinate system on a Riemannian manifold each of whose coordinate functions xi is harmonic, meaning that it satisfies Laplace's equation Here Δ is the Laplace–Beltrami operator.

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Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R where U is an open subset of Rn that satisfies Laplace's equation, i.e. everywhere on U. This is usually written as or.

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Heat equation

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

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Holonomy

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported.

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Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

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Hypersurface

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.

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Introduction to the mathematics of general relativity

The mathematics of general relativity is complex.

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Isometry

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

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Jacobi field

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

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John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

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Kähler differential

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.

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Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure.

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Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

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Laplace–Beltrami operator

In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds.

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Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.

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Local coordinates

Local coordinates are measurement indices into a local coordinate system or a local coordinate space.

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Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.

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Myers's theorem

The Myers theorem, also known as the Bonnet–Myers theorem, is a classical theorem in Riemannian geometry.

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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 applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations.

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Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

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Principal axis theorem

In the mathematical fields of geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola.

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Principal curvature

In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point.

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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 submanifolds, that are invariant under transformations of the projective group.

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Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite, but need only be a non-degenerate bilinear form, which is a weaker condition.

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Raychaudhuri equation

In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter.

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Ricci calculus

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields.

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Ricci decomposition

In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties.

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Ricci flow

In differential geometry, the Ricci flow (Italian) is an intrinsic geometric flow.

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Ricci-flat manifold

In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes.

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Richard S. Hamilton

Richard Streit Hamilton (born 1943) is Davies Professor of Mathematics at Columbia University.

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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 method used to express the curvature of Riemannian manifolds.

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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 that varies smoothly from point to point.

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Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

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Scalar curvature

In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold.

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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 \mathrm (read "two").

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Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.

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Space form

In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three obvious examples are Euclidean ''n''-space, the ''n''-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.

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Spherical wedge

In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's base).

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Splitting theorem

The splitting theorem is a classical theorem in Riemannian geometry.

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Symmetric bilinear form

A symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map.

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Symmetric tensor

In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: for every permutation σ of the symbols Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies The space of symmetric tensors of order r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. A related concept is that of the antisymmetric tensor or alternating form.

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Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

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Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

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Theory of relativity

The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity.

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Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.

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Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

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Volume element

In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates.

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Volume form

In mathematics, a volume form on a differentiable manifold is a top-dimensional form (i.e., a differential form of top degree).

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William Thurston

William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.

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World line

The world line (or worldline) of an object is the path that object traces in -dimensional spacetime.

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Ricci Curvature, Ricci curvature tensor, Ricci form, Ricci tensor, Ricci-curvature, Trace-free Ricci tensor.

References

[1] https://en.wikipedia.org/wiki/Ricci_curvature

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