Riemann-Christoffel Tensor

Relevant parts to questions...

• Definition::${\nabla }_j{\nabla }_k{A}_i-{\nabla }_k{\nabla }_j{A}_i=R_{ijk}^r{A}_r$ - the failure of second covariant derivatives to commute.
• Formula::$R_{ijk}^r=\frac{\partial {\Gamma }_{ki}^r}{\partial x^j}-\frac{\partial {\Gamma }_{ij}^r}{\partial x^k}+{\Gamma }_{ik}^p{\Gamma }_{pj}^r-{\Gamma }_{ij}^p{\Gamma }_{pk}^r$.
• $R_{ijk}^r=0$ everywhere $⟺$ the space is Euclidean (flat).
• Tensor character comes from the Quotient Rule: $A_r$ is arbitrary.

The Riemann-Christoffel Tensor $R_{ijk}^r$ measures how much a space fails to be Euclidean - equivalently, how much second Covariant Differentiation fail to commute.

Motivation: When Derivatives Stop Commuting

Ordinary partial derivatives always commute: $\frac{{\partial }^{2}f}{\partial x^j\partial x^k}=\frac{{\partial }^{2}f}{\partial x^k\partial x^j}$. In a curved space, however, their covariant counterparts do not:

$${\nabla }_j{\nabla }_k{A}_i\ne {\nabla }_k{\nabla }_j{A}_i$$

The failure is a tensor - it is $R_{ijk}^r{A}_r$.

Definition

$$\boxed{{\nabla }_j{\nabla }_k{A}_i-{\nabla }_k{\nabla }_j{A}_i=R_{ijk}^r{A}_r}$$

Explicitly, the coordinate formula is:

$$\boxed{R_{ijk}^r=\frac{\partial {\Gamma }_{ki}^r}{\partial x^j}-\frac{\partial {\Gamma }_{ij}^r}{\partial x^k}+{\Gamma }_{ik}^p{\Gamma }_{pj}^r-{\Gamma }_{ij}^p{\Gamma }_{pk}^r}$$

Two terms from the derivatives of Christoffel symbols, two bilinear in Christoffel symbols.

Tensor Character

The left-hand side ${\nabla }_j{\nabla }_k{A}_i-{\nabla }_k{\nabla }_j{A}_i$ is a difference of tensors (each covariant derivative is a tensor), hence itself a tensor. Since $A_r$ is arbitrary, the Quotient Rule confirms $R_{ijk}^r$ is a tensor (rank 4: three covariant, one contravariant).

Index placement matters

The signs and ordering in the formula depend on convention. Always check against the definition $R_{ijk}^r{A}_r={\nabla }_j{\nabla }_k{A}_i-{\nabla }_k{\nabla }_j{A}_i$ to verify you have the right sign.

Criterion for Euclidean Space

$$\boxed{R_{ijk}^r=0\text{throughout the space}⟺\text{the space is Euclidean (flat)}}$$

• Euclidean ${\mathbb{R}}^3$ - zero curvature, $R_{ijk}^r=0$.
• Unit sphere - positive curvature, $R_{ijk}^r\ne 0$.
• Hyperbolic space - negative curvature, $R_{ijk}^r\ne 0$.

Properties

• Rank 4 (three lower, one upper), with $3^4=81$ components in 3D, but most vanish in practice due to symmetries of the Christoffel symbols.
• Antisymmetric in the last two indices::$R_{ijk}^r=-R_{ikj}^r$ (immediate from the defining commutator).
• Tensor built from non-tensors::$R_{ijk}^r$ is a tensor even though it is constructed from Christoffel symbols (which are not tensors). The non-tensorial pieces cancel in the combination.
• Contracting gives the Ricci tensor::$R_{ij}=R_{irj}^r$ is a second-rank tensor used in general relativity.

Applications

1. Detecting curvature - in principle, compute $R_{ijk}^r$; if any component is non-zero, the space is curved.
2. Proving a space is Euclidean - harder direction - by showing all components vanish.
3. General relativity::Einstein’s field equations relate the Ricci’s Theorem contraction $R_{ij}$ to stress-energy.

Unit sphere has non-zero curvature

For the unit sphere with coordinates $(\varphi ,\theta )$, position vector $\mathbf{r}=\sin \varphi \cos \theta {\mathbf{i}}_1+\sin \varphi \sin \theta {\mathbf{i}}_2+\cos \varphi {\mathbf{i}}_3$. The metric has $g_{11}=1$, $g_{22}={\sin }^2\varphi$, $g_{12}=0$.

Non-zero Christoffel symbols: ${\Gamma }_{22}^1=-\sin \varphi \cos \varphi$, ${\Gamma }_{12}^2={\Gamma }_{21}^2=\cot \varphi$.

Computing the key component:

$R_{212}^1=\frac{\partial }{\partial \varphi }{\Gamma }_{22}^1-{\Gamma }_{21}^2{\Gamma }_{22}^1=(-{\cos }^2\varphi +{\sin }^2\varphi )+{\cos }^2\varphi =\boxed{{\sin }^2\varphi }$

Since $R_{212}^1\ne 0$, the sphere is not Euclidean - confirming its positive curvature.

Partials commute; covariant derivatives commute iff the space is flat. The gap is $R_{ijk}^r$.