Riemann curvature tensor

Informally

One can see the effects curved space by comparing a tennis court and the Earth. Starting at the lower right corner of the Tennis court with a racket held out towards North, then while walking along the outline at each step one makes sure the Tennis racket currently is kept parallel to it's previous position. Once the loop is complete the Tennis racket will be parallel it's initial starting position. This is because Tennis courts are built so the surface is flat. On the other hand the surface of the Earth is curved, we can complete a loop on the surface of the Earth like so, starting at the equator one points a Tennis racket North (as above the Tennis racket should always remain parallel to it's previous position), for this path first one walks to the North pole then turns 90 degrees and walks down to the equator, finally another 90 degree turn is performed and one can walk back to the start. However now the tennis racket will be pointing backwards (towards the East). This process is akin to parallel transporting a vector along the path and the difference identifies how lines which appear "straight" are only "straight" locally. Each time a loop is completed the Tennis racket will be deflected further from it's initial position by an amount depending on the distance and the curvature of the surface. It is possible to identify paths along a curved surface where parallel transport works as it does on flat space. These are the geodesic of the space, for example any segment of a great circle of a sphere.

The concept of a curved space in mathematics differs from conversational usage. For example if the above process was completed on a cylinder one would find that it is not curved overall as the curvature around the cylinder cancels with the flatness along the cylinder, this is a consequence of Gaussian curvature and the Gauss–Bonnet theorem. A familiar example of this is a floppy pizza slice which will remain rigid along it's length is curved along it's width.

The Riemann curvature tensor is a way to capture a measure of the intrinsic curvature. When you write it down in terms of its components (like writing down the components of a vector), it consists of a multi-dimensional array of sums and products of partial derivatives (some of those partial derivatives can be thought of as akin to capturing the curvature imposed upon someone walking in straight lines on a curved surface).

Formally

When a vector in a Euclidean space is parallel transported around a loop, it will again point in the initial direction after returning to its original position. However, this property does not hold in the general case. The Riemann curvature tensor directly measures the failure of this in a general Riemannian manifold. This failure is known as the non-holonomy of the manifold.

Let x_t be a curve in a Riemannian manifold M. Denote by τ_xt : T_x0M → T_xtM the parallel transport map along x_t. The parallel transport maps are related to the covariant derivative by

∇ x ˙ 0 Y = lim h → 0 1 h ( Y x 0 − τ x h − 1 ( Y x h ) ) = d d t ( τ x t Y ) | t = 0

for each vector field Y defined along the curve.

Suppose that X and Y are a pair of commuting vector fields. Each of these fields generates a one-parameter group of diffeomorphisms in a neighborhood of x₀. Denote by τ_tX and τ_tY, respectively, the parallel transports along the flows of X and Y for time t. Parallel transport of a vector Z ∈ T_x0M around the quadrilateral with sides tY, sX, −tY, −sX is given by

τ s X − 1 τ t Y − 1 τ s X τ t Y Z .

This measures the failure of parallel transport to return Z to its original position in the tangent space T_x0M. Shrinking the loop by sending s, t → 0 gives the infinitesimal description of this deviation:

d d s d d t τ s X − 1 τ t Y − 1 τ s X τ t Y Z | s = t = 0 = ( ∇ X ∇ Y − ∇ Y ∇ X − ∇ [ X , Y ] ) Z = R ( X , Y ) Z

where R is the Riemann curvature tensor.

Coordinate expression

Converting to the tensor index notation, the Riemann curvature tensor is given by

R ρ σ μ ν = d x ρ ( R ( ∂ μ , ∂ ν ) ∂ σ )

where ∂ μ = ∂ / ∂ x μ are the coordinate vector fields. The above expression can be written using Christoffel symbols:

R ρ σ μ ν = ∂ μ Γ ρ ν σ − ∂ ν Γ ρ μ σ + Γ ρ μ λ Γ λ ν σ − Γ ρ ν λ Γ λ μ σ

(see also the list of formulas in Riemannian geometry).

The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector A ν with itself:

A ν ; ρ σ − A ν ; σ ρ = A β R β ν ρ σ ,

since the connection Γ α β μ is torsionless, which means that the torsion tensor Γ λ μ ν − Γ λ ν μ vanishes.

This formula is often called the Ricci identity. This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor. In this way, the tensor character of the set of quantities R β ν ρ σ is proved.

This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows

∇ δ ∇ γ T α 1 ⋯ α r β 1 ⋯ β s − ∇ γ ∇ δ T α 1 ⋯ α r β 1 ⋯ β s = − R α 1 ρ γ δ T ρ α 2 ⋯ α r β 1 ⋯ β s − ⋯ − R α r ρ γ δ T α 1 ⋯ α r − 1 ρ β 1 ⋯ β s + R σ β 1 γ δ T α 1 ⋯ α r σ β 2 ⋯ β s + ⋯ + R σ β s γ δ T α 1 ⋯ α r β 1 ⋯ β s − 1 σ .

This formula also applies to tensor densities without alteration, because for the Levi-Civita (not generic) connection one gets:

∇ μ ( g ) ≡ ( g ) ; μ = 0 , w h e r e g = | d e t ( g μ ν ) | .

It is sometimes convenient to also define the purely covariant version by

R ρ σ μ ν = g ρ ζ R ζ σ μ ν .

Symmetries and identities

The Riemann curvature tensor has the following symmetries:

R ( u , v ) = − R ( v , u ) ⟨ R ( u , v ) w , z ⟩ = − ⟨ R ( u , v ) z , w ⟩ R ( u , v ) w + R ( v , w ) u + R ( w , u ) v = 0 .

Here the bracket ⟨ , ⟩ refers to the inner product on the tangent space induced by the metric tensor. The last identity was discovered by Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the Bianchi identity below. (Also, if there is nonzero torsion, the first Bianchi identity becomes a differential identity of the torsion tensor.) These three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has n 2 ( n 2 − 1 ) / 12 independent components.

Yet another useful identity follows from these three:

⟨ R ( u , v ) w , z ⟩ = ⟨ R ( w , z ) u , v ⟩ .

On a Riemannian manifold one has the covariant derivative ∇ u R and the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) takes the form:

( ∇ u R ) ( v , w ) + ( ∇ v R ) ( w , u ) + ( ∇ w R ) ( u , v ) = 0.

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

Skew symmetry

Interchange symmetry

First Bianchi identity

This is often written where the brackets denote the antisymmetric part on the indicated indices. This is equivalent to the previous version of the identity because the Riemann tensor is already skew on its last two indices.

Second Bianchi identity

The semi-colon denotes a covariant derivative. Equivalently, again using the antisymmetry on the last two indices of R.

The algebraic symmetries are also equivalent to saying that R belongs to the image of the Young symmetrizer corresponding to the partition 2+2.

Ricci curvature

The Ricci curvature tensor is the contraction of the first and third indices of the Riemann tensor.

R a b ⏟ Ricci ≡ R a c b c ⏟ Riemann = g c d R c a d b ⏟ Riemann

Special cases

Surfaces

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor has only one independent component which means the Ricci scalar completely determines the Riemann tensor. There is only one valid expression for the Riemann tensor which fits the required symmetries:

R a b c d = f ( R ) ( g a c g d b − g a d g c b )

and by contracting with the metric twice we find the explicit form:

R a b c d = K ( g a c g d b − g a d g c b )

where g a b is the metric tensor and K = R / 2 is a function called the Gaussian curvature and a, b, c and d take values either 1 or 2. The Riemann tensor has only one functionally independent component. The Gaussian curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by

R a b = K g a b .

Space forms

A Riemannian manifold is a space form if its sectional curvature is equal to a constant K. The Riemann tensor of a space form is given by

R a b c d = K ( g a c g d b − g a d g c b ) .

Conversely, except in dimension 2, if the curvature of a Riemannian manifold has this form for some function K, then the Bianchi identities imply that K is constant and thus that the manifold is (locally) a space form.