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In differential geometry, the Riemann curvature tensor, or simply curvature tensor or Riemann tensor, is a generalization of the concept of Gaussian curvature, defined for surfaces, to manifolds of arbitrary dimensions. It represents a measure of the separation of the manifold metric from the Euclidean metric.

It was introduced in 1862 by B. Riemann and developed in 1869 by E. B. Christoffel as a way of completely describing the curvature in any number of dimensions by means of a "small monster": a type tensor (1,3) usually represented by the symbol Rjkli{displaystyle R_{jkl}^{i},!}. The value of any other entity describing the curvature of a variety can be deduced from this tensor. Such is the case of the Ricci tensor (a type tensor (0.2)), of the climbing curvature or of the sectional curvature.

Although in 2 dimensions the curvature can be represented by a scalar at each point (or zero-order tensor), just as the Gaussian curvature did, the geometry of Riemann manifolds with dimension greater than or equal to 3 is too complex as to fully describe it by a number at a given point. Thus, in 3 dimensions the curvature can be represented by a second order tensor (the Ricci tensor). However, for higher dimensions we will need at least a fourth-order tensor (the Riemann tensor).

The curvature tensor has a notable influence on the evolution of the separation of a set of initially close geodesics, via the Hamilton-Jacobi equation. It gives rise to observable curvature effects on tidal forces that appear in general relativity.

Definition

Formally, the curvature tensor is defined for every variety of Riemann, and, more generally, in any variety endowed with an affin connection ► ► {displaystyle nabla } with or without torsion, by the following formula:

R(u,v)w=► ► u► ► vw− − ► ► v► ► uw− − ► ► [chuckles]u,v]w{displaystyle R(u,v)w=nabla _{u}nabla _{v}w-nabla _{v}{u}w-nabla _{[u,v]}w}

where [ ] note the Lie bracket.

This definition leads us to represent the curvature is like a tensor (1.3)-value. In Riemann geometry, the valence of this tensor can be altered: we will often use an equivalent representation as tensor (0.4). Although the definition appears more often, the operator ► ► {displaystyle nabla }historically did not appear until 1954. Meanwhile, Cartan formalism was developed, in which the connection is expressed as a 1-form matrix and the curvature as a 2-form Ω matrix.

Expression in coordinates

Given any basis {e^ ^ i!i=1...... n{displaystyle {{hat {mathbf {e}}_{i}{i}_{i=1dots n}}}defined as a section of tangent fiber, and its dual base {θ θ ^ ^ i!i=1...... n{displaystyle {{hat {boldsymbol {theta}}}}{i}_{i=1dots n}}} the coordinates of the curvature tensor are given by:

Rijkl=θ θ ^ ^ i(R(e^ ^ k,e^ ^ l)e^ ^ j){displaystyle {R^{i}}_{jkl}={hat {boldsymbol {theta }^{i}(R({hat {mathbf {e}}} } },{hat {mathbf {e}}}}}{hat {mathbf {e}}}}{

In a coordinate system associated with a local letter xμ μ {displaystyle x^{mu }} components Riemann curvature tensor are given by:

(♪)Rρ ρ σ σ μ μ .. =dxρ ρ (R(▪ ▪ μ μ ,▪ ▪ .. )▪ ▪ σ σ ){displaystyle {R^{rho }_{sigma mu nu }=dx^{rho }(R(partial _{mu },partial _{nu })partial _{sigma })}}

Where ▪ ▪ μ μ =▪ ▪ /▪ ▪ xμ μ {displaystyle partial _{mu }=partial /partial x^{mu }}} are the vector fields associated with each of the coordinates and which together constitute a natural basis. The expression (♪) may rewrite in terms of Christoffel symbols as follows, using Einstein's summation agreement:

Rρ ρ σ σ μ μ .. =▪ ▪ μ μ Interpreter Interpreter .. σ σ ρ ρ − − ▪ ▪ .. Interpreter Interpreter μ μ σ σ ρ ρ +Interpreter Interpreter μ μ λ λ ρ ρ Interpreter Interpreter .. σ σ λ λ − − Interpreter Interpreter .. λ λ ρ ρ Interpreter Interpreter μ μ σ σ λ λ ####################### #################################################################################################################################################################################################################

Covariant form of the curvature tensor

Yeah. M{displaystyle M} is a riemannian variety, the curvature tensor will be defined from the Levi-Civita connection. The metric tensor g{displaystyle g} can be used to upload or lower curvature tensor indices. In particular, the version completely covariant of the tensor is a type tensor (0.4) given by

Rρ ρ σ σ μ μ .. =gρ ρ α α Rα α σ σ μ μ .. {displaystyle R_{rho sigma mu nu }=g_{rho alpha }{R^{alpha }}{sigma mu nu }{,}

There are different definitions of this tensor, equivalent except in sign, which forces us to have to determine the author's sign convention in each case. In contrast, the other definitions by all authors are adjusted so that the notions of sectional, Ricci, or scalar curvature remain unchanged.

Expression as a set of 2-forms

The mathematical connection of a differential and fixed variety a basis of tangent space at each point {E1,...... ,En!{displaystyle scriptstyle {E_{1},dotsE_{n}} any can be expressed by a 1-form matrix ω ω ij{displaystyle scriptstyle omega _{i}^{j}} which satisfy the following relationship with the covariant derivative:

► ► XEi=ω ω ij(X)Ej{displaystyle nabla _{X}E_{i}=omega _{i}^{j}(X)E_{j}}}}

Where:

X,And{displaystyle X,Y,} are vector fields defined on variety:

It can also be proved that if {φ φ 1,...... ,φ φ n!{displaystyle scriptstyle {phi ^{1},dotsphi ^{n}}}} is the dual base of the former the external differential of the elements of this dual base satisfy:

dφ φ j=␡ ␡ iφ φ i∧ ∧ ω ω ij+Δ Δ j{displaystyle dphi ^{j}=sum _{i}phi ^{i}land omega _{i}^{j}+tau ^{j}}}

Where:

Δ Δ j{displaystyle tau ^{j},} is the set of n 2-forms of torsion

which are null if the Riemannian connection associated to the Riemannian metric of the manifold is used. The 2-forms of curvature are simply given by:

Ω Ω ij=dω ω ij− − ␡ ␡ kω ω ikω ω kj=␡ ␡ k,l12Rikljφ φ k∧ ∧ φ φ l{displaystyle Omega _{i}^{j}=domega _{i}^{j-}sum _{komega _{i}^{k}{k}omega _{k}{j}{j}=sum _{k,l}{frac {1}{2}{2}}R_{ikl}{jphi ^{k}{c}{c}{c}{k}{c}{f}{k}{f}{f}{k}{f}{f}{k}{f}{f}{f}{f}{k}{k}{k}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{

In general, the calculation procedure using the 1-forms of the connection and 2-forms of curvature is more efficient and faster than the direct calculation using the expression in coordinates.

Meaning of the curvature tensor in a Riemann manifold

As a measure of the separation of the metric from the Euclidean metric

An interesting relationship that clarifies the meaning of the curvature tensor is that if considered Normal coordinates {x1,...... ,xn!{displaystyle scriptstyle {x^{1},dotsx^{n}centered on a point p in an environment of such a point the metric of any riemannin variety can be written as:

gij(x)=δ δ ij− − 13Rikljxkxl+O(日本語x日本語3){displaystyle g_{ij}(x)=delta _{ij}-{frac {1{3}}}R_{iklj}x^{k}x^{l}+O(ATAx^{3})}}

In other words, the Riemann tensor gives the deviations of the metric with respect to the flat Euclidean metric up to second order. In a Lorentzian manifold the relationship is similar:

gμ μ .. (x)=MIL MIL μ μ .. − − 13Rμ μ α α β β .. xα α xβ β +O(日本語x日本語3){displaystyle g_{mumu nu }(x)=eta _{mu nu }-{frac {1{3}}R_{mu alpha beta nu }x^{alpha }x^{beta }+O(associatedx^{3}}}}}

As a 2-way linear transformation

To see R as a 2-way linear transformation, consider the sectional curvature, that is, the curvature of a two-dimensional geodesic surface passing through a point - a section, which is the image of a tangent plane under the exponential function. The corresponding tangent plane can be represented by 2-forms. The curvature tensor gives information equivalent to specifying all sectional curvatures. The square norm of a 2-form by the corresponding sectional curvature in fact gives a new quadratic form in a 2-form space, and is given exactly by the linear symmetric operator R. that is, (R(s), s) = k(s)(s, s).

The R operator can be understood in another way. Each 2-shape can be represented by a small rectangular loop (in many ways, but corresponding shape is what matters here). Then parallel transport around this loop gives rise to a tangent space transformation. This is an infinitesimal transformation of tangent space, which can be represented by an element of the Lie algebra corresponding to the Lie group of all linear transformations of tangent space. But this Lie algebra is again a 2-form algebra, and R(s) is precisely this generator. The Lie algebra of all transformations of the loop is the Lie algebra of the holonomy corresponding to the curvature.

Symmetries of the curvature tensor in a Riemann manifold

Fixed a coordinate system at a point of a differentiable manifold, the identities that the tensor satisfies can be written in terms of the components simply as:

• Antisymmetry versus exchange between the first two or the last two indices:

Rabcd=− − Rbacd=− − Rabdc{displaystyle R_{abcd}=-R_{bacd}=-R_{abdc},}!

• Symmetry regarding the exchange of the block formed by the first two indexes with the block formed by the last:

Rabcd=Rcdab{displaystyle R_{abcd}=R_{cdab},!}

• Bianchi's first identity: the sum in three inputs of the curvature tensor (the last three with our sign convention) under circular permutation is annulled

Rabcd+Racdb+Radbc=0{displaystyle R_{abcd}+R_{acdb}+R_{adbc}^{}=0}

which also appears more compactly as Ra[chuckles]bcd]=0{displaystyle R_{a[bcd]}=0}, where the bracket [ ] denotes anti-simetrization on selected components. The 6 terms that appear are reduced to three using the first symmetry.

• Second identity of Bianchi

Rabcd;e+Rabde;c+Rabec;d=0{displaystyle R_{abcd;e}^{}+R_{abde;c}^{{}+R_{abec;d}^{}=0},

or equivalent: Rab[chuckles]cd;e]=0{displaystyle R_{ab[cd;e]}=0;}

Even if the curvature tensor has n4{displaystyle n^{4} components, where n{displaystyle n} is the dimension of the variety where it is defined, the first three relationships reduce the number of independent components to 112n2(n2− − 1){displaystyle {frac {1}{12}}n^{2}(n^{2}-1)}}. In dimensions 2, 3 and 4, the number of independent components will be 1, 6, 20. In case of working in a variety with an arbitrary connection, Bianchi identities adopt a form that generalizes the previous ones and involves the torsion tensor of the connection.

Decomposition of the curvature tensor

Given the complexity of the curvature tensor, it is often convenient to summarize part of the information of this tensor in simpler elements, such as the sectional curvatures, or the combinations thereof that form the Ricci tensor or the same scalar curvature.

Sectional curvature

In the same way that to make a bilinear function more tractable we study it by applying it to two equal vectors (its associated quadratic form), to study a fourth-order tensor such as the curvature tensor we can try to apply it to the minimum number of different vectors. By antisymmetry, with only one vector we would obtain null results. So we must use two different vectors.

Given a plane π π TpM{displaystyle pi subset T_{p}M}and a base {v1,v2!{displaystyle {v_{1},v_{2}}}} of the same, it is shown that the quantity

K=R(v1,v2,v2,v1)g(v1,v1)g(v2,v2)− − g(v1,v2){displaystyle K={frac {R(v_{1},v_{2},v_{2},v_{1})}{g(v_{1},v_{1})g(v_{2},v_{2})-g(v_{1},v.

It does not depend on the chosen base. So, we can say that K only depends on π π {displaystyle pi } and receives the name of the sectional curvature of the plane π π {displaystyle pi }. If we choose an orthonormal basis {e1,e2!{displaystyle {e_{1},e_{2}}}}Your calculation can be simplified, so:

K(π π )=R(e1,e2,e2,e1){displaystyle K(pi)=R(e_{1},e_{2},e_{2},e_{1}),}

Knowledge of all sectional curvatures uniquely determines the curvature tensor. We can think of them as units of information when analyzing the curvature tensor.

Scalar and Ricci curvatures

The Ricci tensor is called the tensor of type (0, 2) whose components are the contraction in a covariant index and another contravariant of the curvature tensor.

Rij=Rikjk{displaystyle R_{ij}=R_{ikj}^{k}}

The diagonal elements of the Ricci tensor can be easily expressed as a combination of sectional curvatures. For example, given an orthonormal basis {e1,...,en!{displaystyle {e_{1},e_{n}{n}}}, R11=␡ ␡ i=2nK(π π (e1,ei)){displaystyle R_{11}=sum _{i=2}^{n}K(pi (e_{1},e_{i})}} (sum of the sectional curvatures of the n-1 orthogonal planes containing the e1{displaystyle e_{1}}).

In addition, the function obtained by metric contraction of the two indices of the tensor of Ricci:

R=gijRij{displaystyle R=g^{ij}R_{ij},}.

If we calculate the contraction using an abnormal base {e1,...,en!{displaystyle {e_{1},e_{n}{n}}}, we will get its development as a sum of sectional curvatures:

<math alttext="{displaystyle R=sum _{ineq j}K(pi (e_{i},e_{j}))=2sum _{iR=␡ ␡ iI was. I was. jK(π π (ei,ej))=2␡ ␡ i.jK(π π (ei,ej)){displaystyle R=sum _{ineq j}K(pi (e_{i},e_{j}))})=2sum _{ineq j}K(pi (e_{i},e_{j}),}<img alt="R=sum _{{ineq j}}K(pi (e_{i},e_{j}))=2sum _{{i

In two dimensions, the curvature tensor is determined by the scalar curvature. In three dimensions, the curvature tensor is specified by the Ricci curvature. This has to do with the fact that the 2-form space is three-dimensional: the same reason we can define the vector product for 3 dimensions (the vector product is precisely the wedge product of two 1-forms composed with the star of Hodge, if we represent vectors with their corresponding 1-forms).

In more dimensions, the full tensor of curvature contains more information than the Ricci curvature. That means that, for a number of dimensions n < 4, the curvature tensor is fully specified if the Ricci tensor is known, but not for n > 3. This has an important consequence in the General Theory of Relativity since the space-time of n = 4 dimensions, but where the gravitational field equations only determine the Ricci tensor. Therefore, the Einstein equations for the gravitational field do not completely determine the total curvature tensor: The part of the curvature not specified by the Einstein equations coincides precisely with the Weyl tensor defined below.

The Weyl Curvature

For dimension n>3, the curvature tensor can be broken down into the part that depends on the Ricci curvature, and the Weyl tensor. If R is the (0, 4)-valent tensor of Riemann curvature, then

R=s/2n(n− − 1)g g+(Ric− − sg/n) g/(n− − 2)+W{displaystyle R=s/2n(n-1)gcirc g+(Ric-sg/n)circ g/(n-2)+W}

where Ric is the (0, 2)-valent version of the Ricci curvature, s is the scalar curvature, and g is the metric tensor (0, 2)-valent and

h k(v1,v2,v3,v4)=h(v1,v3)k(v2,v4)+h(v2,v4)k(v1,v3)− − h(v1,v4)k(v2,v3)− − h(v2,v3)k(v1,v4)(sighs)

is the so-called Kulkarni-Nomizu product of the two (0, 2)-tensors.

The components of the Weyl tensor can be calculated explicitly from the Riemann curvature tensor, the Ricci curvature tensor, and the scalar curvature:

Cabcd=Rabcd− − 2n− − 2(ga[chuckles]cRd]b− − gb[chuckles]cRd]a)+2(n− − 1)(n− − 2)sga[chuckles]cgd]b{displaystyle C_{abcd}=R_{abcd}-{frac {2}{n-2}}}(g_{a[c}R_{d]b}-g_{b[c}R_{d]a})+{frac {2}{(n-1)}s~g_{a[c}g_{d]b}}}

Where:

Rabcd{displaystyle R_{abcd},} are the components of the Riemann tensor.

Rab{displaystyle R_{ab},} are the components of the Ricci tensor.

s{displaystyle s,} It's Ricci's climbing curvature.

[chuckles]]{displaystyle [],} refers to the antisimetric part of a tensor.

If g'=fg for a certain scalar function - the conformal change of the metric - then W'=fW. For constant curvature, the Weyl tensor is zero. On the other hand, W=0 if and only if the metric conforms to the standard Euclidean metric (equal to fg, where g is the standard metric in a certain coordinate frame and f is a certain scalar function). Curvature is constant if and only if W=0 and Ric=s/n