How to Calculate the Riemann Tensor of a 3-Sphere

The Riemann tensor depends on the metric, so the phrase “the 3-sphere” needs a metric to determine a particular tensor. The standard interpretation is the round sphere $S^3$ of radius $a$, with the metric induced from Euclidean space. Its curvature is constant and positive.

Below, the tensor is given both as a coordinate-free expression and in hyperspherical coordinates. A sign convention is stated explicitly because some books define the Riemann tensor with the opposite sign.

The round metric on $S^3$

Use coordinates $0<\chi<\pi$, $0<\theta<\pi$, and $0<\phi<2\pi$. The round metric is

$$ds^2=a^2\left[d\chi^2+\sin^2\chi\left(d\theta^2+\sin^2\theta\,d\phi^2\right)\right].$$

Thus the nonzero metric components are

$$g_{\chi\chi}=a^2,\qquad g_{\theta\theta}=a^2\sin^2\chi,\qquad g_{\phi\phi}=a^2\sin^2\chi\sin^2\theta.$$

These coordinates are singular at the usual spherical-coordinate poles, but the sphere and its curvature are smooth there. The radius $a$ is a length; setting $a=1$ gives the unit 3-sphere.

Riemann tensor convention and result

Use the convention

$$R^\rho{}_{\sigma\mu\nu}=\partial_\mu\Gamma^\rho_{\nu\sigma}-\partial_\nu\Gamma^\rho_{\mu\sigma}+\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}-\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}.$$

With this convention, a round sphere has positive sectional curvature. Lowering the first index gives the tensor

$$R_{\rho\sigma\mu\nu}=\frac{1}{a^2}\left(g_{\rho\mu}g_{\sigma\nu}-g_{\rho\nu}g_{\sigma\mu}\right).$$

This formula is often the quickest way to get the Riemann tensor: the round sphere has constant sectional curvature $K=1/a^2$. It also shows that the tensor is determined entirely by the metric and the radius.

Coordinate calculation

For a direct calculation, the nonzero Christoffel symbols, apart from symmetry in their lower indices, are

$$\begin{aligned} \Gamma^\chi_{\theta\theta}&=-\sin\chi\cos\chi,& \Gamma^\chi_{\phi\phi}&=-\sin\chi\cos\chi\sin^2\theta,\\ \Gamma^\theta_{\chi\theta}&=\cot\chi,& \Gamma^\phi_{\chi\phi}&=\cot\chi,\\ \Gamma^\theta_{\phi\phi}&=-\sin\theta\cos\theta,& \Gamma^\phi_{\theta\phi}&=\cot\theta. \end{aligned}$$

The overall factor $a^2$ cancels in the Christoffel-symbol formula, which is why these symbols do not depend on the radius. As one example, substitute the symbols into the stated convention for $R^\chi{}_{\theta\chi\theta}$:

$$\begin{aligned} R^\chi{}_{\theta\chi\theta} &=\partial_\chi\Gamma^\chi_{\theta\theta} -\partial_\theta\Gamma^\chi_{\chi\theta} +\Gamma^\chi_{\chi\lambda}\Gamma^\lambda_{\theta\theta} -\Gamma^\chi_{\theta\lambda}\Gamma^\lambda_{\chi\theta}\\ &=\partial_\chi(-\sin\chi\cos\chi) -(-\sin\chi\cos\chi)\cot\chi\\ &=\sin^2\chi. \end{aligned}$$

Lowering the first index gives $R_{\chi\theta\chi\theta}=g_{\chi\chi}R^\chi{}_{\theta\chi\theta}=a^2\sin^2\chi$. This agrees with the constant-curvature formula. The independent components with the index order shown are

$$R_{\chi\theta\chi\theta}=a^2\sin^2\chi,\qquad R_{\chi\phi\chi\phi}=a^2\sin^2\chi\sin^2\theta,\qquad R_{\theta\phi\theta\phi}=a^2\sin^4\chi\sin^2\theta.$$

All other components follow from the Riemann tensor’s symmetries or are zero. For example, $R_{abcd}=-R_{bacd}=-R_{abdc}$ and $R_{abcd}=R_{cdab}$.

Ricci and scalar curvature

Contracting the Riemann tensor gives, in three dimensions,

$$R_{\sigma\nu}=g^{\rho\mu}R_{\rho\sigma\mu\nu}=\frac{2}{a^2}g_{\sigma\nu}.$$

Taking the trace gives

$$R=g^{\sigma\nu}R_{\sigma\nu}=\frac{6}{a^2}.$$

For the unit 3-sphere, these reduce to $R_{abcd}=g_{ac}g_{bd}-g_{ad}g_{bc}$, $R_{ab}=2g_{ab}$, and $R=6$.

Common mistakes

• Leaving the radius implicit: The unit sphere and a sphere of radius $a$ have different curvature. Sectional curvature scales as $1/a^2$.
• Mixing sign conventions: Some references define $R^\rho{}_{\sigma\mu\nu}$ with the opposite sign. Their Riemann, Ricci, and scalar curvatures will have the opposite sign; the geometry is unchanged.
• Confusing intrinsic and ambient curvature: $S^3$ is a curved space intrinsically, even though it sits in flat $\mathbb{R}^4$. The tensor above is its intrinsic Riemann tensor, not the ambient space’s tensor.
• Treating coordinate singularities as curvature singularities: The coordinate formulas contain sines and cotangents that behave poorly at poles, but the invariant curvature remains constant there.

If a different metric on $S^3$ is intended—for example, a deformed or left-invariant metric—the constant-curvature formula need not apply. The metric must then be specified before its Riemann tensor can be calculated.