In differential geometry, the

**Ricci curvature tensor**, named after

Gregorio Ricci-CurbastroGregorio Ricci-Curbastro was an Italian mathematician. He was born at Lugo di Romagna. He is most famous as the inventor of the tensor calculus but published important work in many fields....

, represents the amount by which the

volume elementIn 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...

of a

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

ballIn mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....

in a curved

Riemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

deviates from that of the standard ball in

Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean

*n-*space. The Ricci tensor is defined on any

pseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...

, as a trace of the

Riemann curvature tensorIn the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...

. Like the metric itself, the Ricci tensor is a

symmetric bilinear formA symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....

on the

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

of the manifold .

In relativity theory, the Ricci tensor is the part of the curvature of space-time that determines the degree to which matter will tend to converge or diverge in time (via the

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

). It is related to the matter content of the universe by means of the Einstein field equation. In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf.

comparison theoremA comparison theorem is any of a variety of theorems that compare properties of various mathematical objects.-Riemannian geometry:In Riemannian geometry it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.*Rauch...

) with the geometry of a constant curvature

space formIn mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three obvious examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.-Reduction to generalized crystallography:It is a...

. If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold is an

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

, which have been extensively studied (cf. ). In this connection, the

Ricci flowIn differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....

equation governs the evolution of a given metric to an Einstein metric, the precise manner in which this occurs ultimately leads to the solution of the Poincaré conjecture.

## Definition

Suppose that

is an

*n-*dimensional

Riemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

, equipped with its

Levi-Civita connectionIn Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...

. The Riemannian curvature tensor of

is the

tensor defined by

on

vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

s

. Let

denote the

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

of

*M* at a point

*p*. For any pair

of tangent vectors at

*p*, the Ricci tensor

evaluated at

is defined to be the

traceIn 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 of A, i.e.,...

of the linear map

given by

In

local coordinatesLocal coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.Local...

(using the Einstein summation convention), one has

where

In terms of the

Riemann curvature tensorIn the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...

and the

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

, one has

## Properties

As a consequence of the Bianchi identities, the Ricci tensor

of a Riemannian manifold is

symmetric, in the sense that

It thus follows that the Ricci tensor is completely determined by knowing the quantity

for all vectors

of unit length. This function on the set of unit tangent vectors is often simply called

the

**Ricci curvature**, since knowing it is equivalent to knowing the Ricci curvature tensor.

The Ricci curvature is determined by the

sectional curvatureIn Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

s of a Riemannian manifold, but generally contains less information. Indeed, if

is a vector of unit length on a Riemannian

*n-*manifold, then Ric(ξ,ξ) is precisely (

*n*−1) times the average value of the sectional curvature, taken over all the 2-planes containing

. There is an (

*n*−2)-dimensional family of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine the full curvature tensor. A notable exception is when the manifold is given a priori as a

hypersurfaceIn geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...

of Euclidean space. The

second fundamental form, which determines the full curvature via the Gauss-Codazzi equation, is itself determined by the Ricci tensor and the principal directions of the hypersurface are also the eigendirections of the Ricci tensor. The tensor was introduced by Ricci for this reason.

If the Ricci curvature function Ric(ξ,ξ) is constant on the set of unit tangent vectors ξ, the Riemannian manifold is said to have constant Ricci curvature, or to be an

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

. This happens if and only if the Ricci tensor Ric is a constant multiple of the metric tensor

*g*.

The Ricci curvature is usefully thought of as a multiple of the Laplacian of the metric tensor . Specifically, if

*x*^{i} are

harmonicIn 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\Delta x^i = 0.\,...

local coordinates, then

where Δ is the

Laplace-Beltrami operatorIn 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. This more general operator goes by the name Laplace–Beltrami...

regarded here as acting on the functions

*g*_{ij}. This fact motivates, for instance, the introduction of the

Ricci flowIn differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....

equation as a natural extension of the

heat equationThe heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

for the metric. Alternatively, in a

normal coordinate systemIn 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...

based at

*p*, at the point

*p*
## Direct geometric meaning

Near any point

*p* in a Riemannian manifold (

*M*,

*g*), one

can define preferred local

coordinates, called geodesic normal coordinates. These are adapted

to the metric such that geodesics through

*p* corresponds to straight lines through the origin,

in such a manner that the geodesic distance from

*p* corresponds to the Euclidean distance from the origin.

In these coordinates, the metric tensor is well-approximated by the Euclidean metric, in the precise sense that

In fact, by taking the Taylor expansion of the metric applied to a

Jacobi fieldIn Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space...

along a radial geodesic in the normal coordinate system, one has

In these coordinates, the metric

volume elementIn 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...

then has the following expansion at

*p*:

which follows by expanding the square root of the

determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

of the metric.

Thus, if the Ricci curvature Ric(ξ,ξ) is positive in the direction of a vector ξ,

the conical region in

*M* swept out by a tightly focused family of

short geodesic segments emanating from

*p* with initial velocity inside a small cone around ξ

will have smaller volume than the corresponding conical region in Euclidean space, just as the surface a small

spherical wedgeIn geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune . The angle between the radii lying within the bounding semidisks is the dihedral angle of the wedge α...

has lesser area than a corresponding circular sector. Similarly, if the Ricci curvature is negative in the direction of a given vector ξ, such a conical region in the manifold will instead have larger volume than it would in Euclidean space.

The Ricci curvature is essentially an average of curvatures in the planes including ξ. Thus if a cone emitted with an initially circular (or spherical) cross-section becomes distorted into an ellipse (ellipsoid), it is possible for the volume distortion to vanish if the distortions along the principal axes counteract one another. The Ricci curvature would then vanish along ξ. In physical applications, the presence of a nonvanishing sectional curvature does not necessarily indicate the presence of any mass locally; if an initially circular cross-section of a cone of world-lines later becomes elliptical, without changing its volume, then this is due to tidal effects from a mass at some other location.

## Applications

Ricci curvature plays an important role in

general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, where it is the key term in the Einstein field equations.

Ricci curvature also appears in the

Ricci flowIn differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....

equation, where a time-dependent Riemannian

metric is deformed in the direction of minus its Ricci curvature. This system of partial differential equations is a non-linear analog of the

heat equationThe heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

, and was first

introduced by

Richard HamiltonRichard Streit Hamilton is Davies Professor of mathematics at Columbia University.He received his B.A in 1963 from Yale University and Ph.D. in 1966 from Princeton University. Robert Gunning supervised his thesis...

in the early 1980s. Since heat tends to spread through a solid until the body reaches an equilibrium state of constant temperature, Ricci flow may be hoped to produce an equilibrium geometry for a manifold for which the Ricci curvature is constant. Recent contributions to the subject due to

Grigori PerelmanGrigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...

now show that this program works well enough in dimension three to lead to a complete classification of compact 3-manifolds, along lines

first conjectured by

William ThurstonWilliam Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...

in the 1970s.

On a

Kähler manifoldIn mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

, the Ricci curvature determines the first

Chern classIn mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...

of the manifold (mod torsion). However, the Ricci curvature has no analogous topological interpretation

on a generic Riemannian manifold.

## Global geometry and topology

Here is a short list of global results concerning manifolds with positive Ricci curvature; see also classical theorems of Riemannian geometry. Briefly, positive Ricci curvature of a Riemannian manifold has strong topological consequences, while (for dimension at least 3), negative Ricci curvature has

*no* topological implications. (The Ricci curvature is said to be

**positive** if the Ricci curvature function Ric(ξ,ξ) is positive on the set of non-zero tangent vectors ξ.) Some results are also known for pseudo-Riemannian manifolds.

- Myers' theorem states that if the Ricci curvature is bounded from below on a complete Riemannian manifold by , then the manifold has diameter , with equality only if the manifold is isometric
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

to a sphere of a constant curvature *k*. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental groupIn mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

.
- The 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 point in the proof of Gromov's compactness theorem....

states that if a complete *m*-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Euclidean *m*-space. Moreover, if denotes the volume of the ball with center *p* and radius in the manifold and denotes the volume of the ball of radius *R* in Euclidean *m*-space then function is nonincreasing. (The last inequality can be generalized to arbitrary curvature bound and is the key point in the proof of Gromov's compactness theoremIn Riemannian geometry, Gromov's compactness theorem states thatthe set of Riemannian manifolds of a given dimension, with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov-Hausdorff metric. It was proved by Mikhail Gromov....

.)
- The Cheeger-Gromoll splitting theorem states that if a complete Riemannian manifold with contains a
*line*, meaning a geodesic γ such that for all , then it is isometric to a product space . Consequently, a complete manifold of positive Ricci curvature can have at most one topological end. The theorem is also true under some additional hypotheses for complete Lorentzian manifolds (of metric signature (+−−...)) with non-negative Ricci tensor .

These results show that positive Ricci curvature has strong topological consequences. By contrast, excluding the case of surfaces, negative

Ricci curvature is now known to have

*no* topological implications; has shown that any manifold of dimension greater than two admits a Riemannian metric of negative Ricci curvature. (For surfaces, negative Ricci curvature implies negative sectional curvature; but the point

is that this fails rather dramatically in all higher dimensions.)

## Behavior under conformal rescaling

If you change the metric g by multiplying it by a conformal factor

, the Ricci tensor of the new, conformally related metric

is given by

where Δ =

*d*^{∗}*d* is the (positive spectrum) Hodge Laplacian, i.e., the

*opposite* of the usual trace of the Hessian.

In particular, given a point p in a Riemannian manifold, it is always possible to find metrics conformal to the given metric g for which the Ricci tensor vanishes at p. Note, however, that this is only pointwise assertion; it is usually impossible to make the Ricci curvature vanish identically on the entire manifold by a conformal rescaling.

For two dimensional manifolds, the above formula shows that if

*f* is a

harmonic functionIn mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....

, then the conformal scaling

*g* *e*^{2ƒ}*g* does not change the Ricci curvature.

## Trace-free Ricci tensor

In

Riemannian geometryRiemannian 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 which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

and

general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, the

**trace-free Ricci tensor** of a pseudo-Riemannian

manifold

is the tensor defined by

where

is the Ricci tensor,

is the

scalar curvatureIn Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

,

is the

metric tensorIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

, and

is the dimension of

.

The name of this object reflects the fact that its

traceIn 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 of A, i.e.,...

automatically vanishes:

If n

3, the trace-free Ricci tensor vanishes identically if and only if

for some constant

.

In mathematics, this is the condition for

to be an

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

. In physics, this equation

states that

is a solution of Einstein's vacuum field

equations with

cosmological constantIn physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...

.

## Kähler manifolds

On a

Kähler manifoldIn mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

*X*, the Ricci curvature determines the

curvature formIn differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...

of the

canonical line bundleIn mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...

. The canonical line bundle is the top exterior power of the bundle of holomorphic

Kähler differentialIn mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...

s:

The Levi-Civita connection corresponding to the metric on

*X* gives rise to a connection on κ. The curvature of this connection is the two form defined by

where

*J* is the complex structure map of the Kähler manifold. The Ricci form is a closed two-form. Its cohomology class is, up to a real constant factor, the first

Chern classIn mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...

of the canonical bundle, and is therefore a topological invariant of

*X* (for

*X* compact) in the sense that it depends only on the topology of

*X* and the homotopy class of the complex structure.

Conversely, the Ricci form determines the Ricci tensor by

In local holomorphic coordinates

*z*^{α}, the Ricci form is given by

where

is the Dolbeault operator and

If the Ricci tensor vanishes, then the canonical bundle is flat, so the

structure groupIn differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....

can be locally reduced to a subgroup of the special linear group

*SL*(

*n*,

**C**). However, Kähler manifolds already possess

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

in

*U*(

*n*), and so the (restricted) holonomy of a Ricci flat Kähler manifold is contained in

*SU*(

*n*). Conversely, if the (restricted) holonomy of a 2

*n*-dimensional Riemannian manifold is contained in

*SU*(

*n*), then the manifold is a Ricci-flat Kähler manifold .

## Generalization to affine connections

The Ricci tensor can also be generalized to arbitrary

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

s, where it is an invariant that plays an especially important role in the study of the

projective geometryIn mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties that are invariant under the projective group. This is a mixture of attitudes from Riemannian geometry, and the Erlangen program....

(geometry associated to unparameterized geodesics) . If

denotes an affine connection, then the curvature tensor

is the

tensor defined by

for any vector fields

. The Ricci tensor is defined to be the trace:

In this more general situation, the Ricci tensor is symmetric if and only if there exists a parallel

volume formIn mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...

for the connection.

## See also

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

- Scalar curvature
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

- 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...

- Ricci-flat manifold
In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes. In physics, they represent vacuum solutions to the analogues of Einstein's equations for Riemannian manifolds of any dimension, with vanishing cosmological constant...

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

- Basic introduction to the mathematics of curved spacetime
The mathematics of general relativity are very complex. In Newton's theories of motions, an object's mass and length remain constant as it changes speed, and the rate of passage of time also remains unchanged. As a result, many problems in Newtonian mechanics can be solved with algebra alone...

## External links