Sectional Curvature

Sectional curvature is a working tool at both ends of physics and mathematics. In cosmology the spatial slices of the Friedmann-Lemaitre-Robertson-Walker universe are homogeneous, isotropic three-spaces of constant sectional curvature, and the single curvature parameter $k$ is exactly what the density parameter $\Omega_k$ measures — the Planck satellite's constraint $|\Omega_k|\lesssim0.001$ says, geometrically, that the sectional curvature of the observable universe's spatial sections is indistinguishable from zero. In the late-time limit the whole spacetime approaches de Sitter space with sectional curvature $k=\Lambda/3$ set by dark energy, the number computed above. In gravitational physics the geodesic-deviation reading of $K$ is what a tidal-tensor measurement returns: the same electric-part-of-Riemann quantity a LIGO interferometer records as a passing wave stretches one section while squeezing the orthogonal one, and the focusing theorem behind the Penrose-Hawking singularity theorems requires appropriate sectional and averaged curvatures to be non-negative along null and timelike congruences. In pure mathematics sectional curvature drives the comparison-geometry program — the Cartan-Hadamard theorem ($K\leq0$ gives a space diffeomorphic to $\mathbb{R}^n$), Bonnet-Myers (positive lower bounds force compactness and finite diameter), and the differentiable sphere theorem of Brendle and Schoen (2007-2009), which upgraded the classical quarter-pinched result from homeomorphism to diffeomorphism using Ricci flow. Every one is a theorem whose hypothesis is a bound on sectional curvature and nothing else.

Curvature, stripped to its essence, is a statement about two-dimensional surfaces — Gauss's Theorema Egregium showed that a surface's curvature is intrinsic, measurable by an ant confined to the surface with no notion of any surrounding space. Riemann's generalization to higher dimensions does not abandon that picture; it multiplies it. At a single point of an $n$-dimensional space there are infinitely many two-dimensional planes of directions, and each one, when geodesics run out along it, sweeps a genuine two-surface whose Gaussian curvature you can measure the Gauss way. Sectional curvature is that measurement, tagged by which plane you chose. The first figure is the construction itself: a plane in the tangent space, exponentiated into a curved sheet, its Gaussian curvature read off.

The sign of $K$ has an immediate, physical reading through the fate of neighbouring geodesics inside the section. Where $K$ is positive the surface curls like a sphere and geodesics that start parallel bend toward each other and cross — the curvature focuses them; where $K$ is negative the surface saddles and geodesics splay apart — it defocuses them; where $K$ vanishes they stay parallel forever, as in flat space. This is not a metaphor. Projected into a section, the geodesic-deviation equation of general relativity becomes exactly the Jacobi equation $\ddot\xi=-K\xi$, the harmonic oscillator whose sign of stiffness is the sign of $K$. The second figure ties the three signs to that law and pins them to real numbers — the sectional curvature of the Earth as a 2-sphere, of the de Sitter vacuum set by the observed cosmological constant, and of flat Minkowski space.

The two figures are one statement told forward and backward. The first builds sectional curvature out of Riemann — contract, normalize by area, get a Gaussian curvature. The remarkable fact developed below is that the arrow runs both ways: the sectional curvatures of all planes, taken together, rebuild the full Riemann tensor. Nothing is lost in the passage to this more pictorial object, which is why comparison geometers use bounds on $K$ — and only $K$ — to control the entire local shape of a space.

The definition is a ratio of two contractions with the tangent bivector $u\wedge v$. The numerator feeds $u$ and $v$ into the four slots of Riemann; the denominator is the squared area of the parallelogram they span, written as a Gram determinant so that the whole quantity is normalized to the plane rather than to the arrows.

where: $K(\sigma)$ — sectional curvature of the plane $\sigma=\mathrm{span}(u,v)$; $R_{abcd}$ — fully covariant Riemann tensor at $p$; $u^a,v^b$ — two linearly independent tangent vectors spanning $\sigma$; $g_{ab}$ — the metric. Reading it: the denominator equals $g(u,u)g(v,v)-g(u,v)^2$, the squared area, so both numerator and denominator are quadratic in the pair $(u,v)$ and the ratio forgets everything except the plane. That denominator deserves its own line, because "area, not the product of lengths" is the whole reason $K$ is a property of the plane. It is a $2\times2$ Gram determinant, equal to $|u|^2|v|^2\sin^2\theta$ in Riemannian signature, and it is exactly the quantity that antisymmetrizes with the antisymmetry already living in Riemann.

where: $|u\wedge v|^2$ — squared area of the parallelogram on $u,v$; $g(u,v)=g_{ab}u^av^b$ — the inner product. Reading it: normalizing by area rather than by $|u|^2|v|^2$ is what makes $K$ invariant when the spanning vectors are sheared, and in Lorentzian signature it is this expression — not a sum of squares — that can vanish or go negative. The invariance is worth stating as an equation in its own right: replacing $u,v$ by any other basis of the same plane leaves $K$ unchanged, provided the change of basis is invertible.

where: $\lambda,\mu,\rho,\sigma$ — entries of an invertible $2\times2$ change-of-basis matrix $A$. Reading it: numerator and denominator both pick up the factor $(\det A)^2=(\lambda\sigma-\mu\rho)^2$, which cancels in the ratio, so $K$ is a well-defined function on the Grassmannian of 2-planes rather than on pairs of vectors. Spaces in which $K$ takes one common value $k$ on every plane are exactly the maximally symmetric ones, and their Riemann tensor has a rigid, metric-built form. This single equation defines the constant-curvature model geometries.

where: $k$ — the common sectional curvature. Reading it: substituting this $R_{abcd}$ into equation (1) returns $K=k$ for every nondegenerate plane; the spheres and de Sitter space ($k>0$), hyperbolic and anti-de Sitter space ($k<0$), and flat space ($k=0$) are the Riemannian and Lorentzian realizations, and in dimension three or more Schur's theorem forces $k$ to be a genuine constant, not a function of position. Sectional curvature also feeds the traces of Riemann directly. Summing the sectional curvatures of all planes spanned by a unit vector and an orthonormal partner gives the Ricci curvature in that direction; summing over all pairs of basis directions gives the scalar curvature.

where: $\{e_i\}$ — an orthonormal basis with $e_1=u$; $\mathrm{Ric}(u,u)=R_{ab}u^au^b$ — Ricci curvature in the direction $u$; $R=g^{ab}R_{ab}$ — Ricci scalar. Reading it: Ricci and scalar curvature are averages of sectional curvature, which is why sectional bounds imply Ricci bounds but not the reverse — the traces discard the off-diagonal information that $K$ retains. Finally, the physical face. The geodesic-deviation (Jacobi) equation of general relativity, restricted to the plane of the geodesic tangent and the separation vector, is a harmonic oscillator whose frequency-squared is the sectional curvature of that plane.

where: $\xi^a$ — separation between neighbouring geodesics; $T^a$ — unit tangent to the reference geodesic; $D/\mathrm{d}\tau=T^b\nabla_b$ — covariant derivative along the worldline; $K$ — sectional curvature of the plane $\mathrm{span}(T,\xi)$. Reading it: with $K>0$ the separation oscillates and geodesics reconverge; with $K<0$ it grows exponentially and they diverge; $K$ is the operational tidal quantity a comoving observer measures.

Sectional curvature collapses to something recognizable whenever the geometry is special, and each collapse is a landmark theorem.

1 · The round 2-sphere, from tensor to number (confidence builder)

Confirm that the round sphere of radius $a$, with metric $\mathrm{d}s^2=a^2(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2)$, has sectional curvature $K=1/a^2$ at every point and in every plane, then evaluate it for the Earth as a 2-sphere.

Step 1. In two dimensions there is a single independent Riemann component, and for this metric it is $R_{\theta\phi\theta\phi}=a^2\sin^2\theta$, computed from the Levi-Civita connection of the round metric. There is only one plane at each point — the whole tangent space is two-dimensional — so $\sigma$ is forced and $K$ is the lone Gaussian curvature. $R_{\theta\phi\theta\phi}=a^2\sin^2\theta$.

Step 2. The Gram denominator is the squared area of the coordinate parallelogram. The metric is diagonal, $g_{\theta\phi}=0$, so $g_{\theta\theta}g_{\phi\phi}-g_{\theta\phi}^2=a^2\cdot a^2\sin^2\theta=a^4\sin^2\theta$. denominator $=a^4\sin^2\theta$.

Step 3. Take the ratio (equation 1): $K=R_{\theta\phi\theta\phi}/(g_{\theta\theta}g_{\phi\phi}-g_{\theta\phi}^2)=a^2\sin^2\theta/(a^4\sin^2\theta)=1/a^2$. The $\sin^2\theta$ cancels, so $K$ is independent of $\theta$ and $\phi$: $K=1/a^2$, constant and positive everywhere — the sphere is a constant-curvature space, and $K$ recovers its classical Gaussian curvature. As a cross-check, the scalar curvature $R=2K=2/a^2$, the standard value for $S^2$.

Step 4. Put in Earth's radius $a=6371$ km $=6.371\times10^6$ m: $K=1/a^2=1/(6.371\times10^6)^2\approx\mathbf{2.5\times10^{-14}\ m^{-2}}$. The corresponding scalar curvature is $R=2/a^2\approx4.9\times10^{-14}\ \mathrm{m^{-2}}$. Tiny, as befits a nearly flat local patch — but strictly positive, which is why geodesic triangles on the globe carry a measurable angle excess proportional to their area times $K$.

2 · The de Sitter vacuum from the cosmological constant (fully worked)

De Sitter spacetime is the maximally symmetric vacuum solution of Einstein's equations with a positive cosmological constant $\Lambda$. Show that its constant sectional curvature is $k=\Lambda/3$, then evaluate $k$ and the de Sitter radius $\ell=1/\sqrt{k}$ for the observed $\Lambda\approx1.1\times10^{-52}\ \mathrm{m^{-2}}$. Use $G$, $c$ only implicitly through $\Lambda$.

Step 1. De Sitter space has constant sectional curvature, so its Riemann tensor takes the maximally symmetric form of equation (4), $R_{abcd}=k(g_{ac}g_{bd}-g_{ad}g_{bc})$. Contracting once gives the Ricci tensor $R_{ab}=k(n-1)g_{ab}$ and the scalar $R=kn(n-1)$; in four dimensions $R=12k$. $R=12k$ for $n=4$.

Step 2. Now impose the vacuum Einstein equation with cosmological constant, $R_{ab}-\tfrac12Rg_{ab}+\Lambda g_{ab}=0$. Taking the trace in four dimensions, $R-2R+4\Lambda=0$, gives $R=4\Lambda$. Combining with Step 1, $12k=4\Lambda$, hence $k=\Lambda/3$ — every nondegenerate 2-plane of de Sitter space has this one sectional curvature.

Step 3. Insert the measured value $\Lambda\approx1.1\times10^{-52}\ \mathrm{m^{-2}}$ (from $\Lambda=3(H_0/c)^2\Omega_\Lambda$ with $H_0\approx67\ \mathrm{km\,s^{-1}Mpc^{-1}}$, $\Omega_\Lambda\approx0.68$): $k=\Lambda/3\approx\mathbf{3.7\times10^{-53}\ m^{-2}}$. This is the sectional curvature the accelerating universe approaches at late times as dark energy dominates.

Step 4. The de Sitter radius is $\ell=1/\sqrt{k}=\sqrt{3/\Lambda}\approx1/\sqrt{3.7\times10^{-53}}\approx\mathbf{1.6\times10^{26}\ m}$, about 17 billion light-years — the curvature scale of empty, dark-energy-dominated space. By Schur's theorem this single number is the same on every plane and at every event, the defining feature of maximal symmetry. $k=\Lambda/3\approx3.7\times10^{-53}\ \mathrm{m^{-2}}$, $\ell\approx1.6\times10^{26}\ \mathrm{m}$, and one may check $R=4\Lambda\approx4.4\times10^{-52}\ \mathrm{m^{-2}}=12k$, closing the loop.

Sectional curvature is one idea with two lives: it is the object Riemann wrote down in 1854 to lift Gauss's surface curvature into arbitrary dimensions, and it is the object twentieth-century geometers turned into the sharpest tool in Riemannian geometry.

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §13.5 and §21.5 · Wald, General Relativity (1984), §3.2 · Carroll, Spacetime and Geometry (2004), §3.7 · Schutz, A First Course in General Relativity, 2nd ed. (2009), §6.5 · Hartle, Gravity (2003), §21 · do Carmo, Riemannian Geometry (1992), ch. 4 · Lee, Riemannian Manifolds (1997), ch. 8.

Provenance. The definition and plane-invariance follow Wald §3.2 and do Carmo ch. 4; the polarization argument that $K$ determines $R_{abcd}$ is Lee, Riemannian Manifolds, Lemma 8.27. A verifier should reproduce the $S^2$ computation ($R_{\theta\phi\theta\phi}=a^2\sin^2\theta$, $K=1/a^2$, $R=2/a^2$, and $K\approx2.5\times10^{-14}\ \mathrm{m^{-2}}$ for $a=6371$ km); confirm the constant-curvature relations $R_{ab}=k(n-1)g_{ab}$, $R=kn(n-1)$ and hence, via the $\Lambda$-vacuum trace $R=4\Lambda$ in 4D, that de Sitter has $k=\Lambda/3\approx3.7\times10^{-53}\ \mathrm{m^{-2}}$ and $\ell=\sqrt{3/\Lambda}\approx1.6\times10^{26}\ \mathrm{m}$ for $\Lambda\approx1.1\times10^{-52}\ \mathrm{m^{-2}}$; and check the trace relations $\mathrm{Ric}(u,u)=\sum_{i\neq1}K(u,e_i)$, $R=\sum_{i\neq j}K(e_i,e_j)$. All curvature symmetries and the first Bianchi identity are recalled by name from the prerequisite articles, not re-derived here.

Prerequisites. Riemann Curvature Tensor · Algebraic Symmetries of the Riemann Tensor · Curvature via Parallel Transport and Holonomy · First (Algebraic) Bianchi Identity. Leads to. Ricci Tensor · Ricci Scalar · Independent Components of the Riemann Tensor · Weyl Conformal Tensor.

Ask the tutor: Want to grind out the sectional curvatures of Schwarzschild or a specific FLRW slice plane by plane, and watch which sections carry the tidal field? Ask and we will compute them together.