Sectional Curvature and Holonomy

Sectional curvature assigns curvature to a nondegenerate tangent two-plane, while holonomy records the linear transformations produced by parallel transport around loops. The small-loop limit recovers the curvature operator; finite loops combine those local generators with topology and path ordering. In indefinite signature the word “nondegenerate” is indispensable: a null two-plane has no sectional curvature through the usual quotient.

Required background. Affine Connections and Covariant Derivatives supplies the general connection and covariant-derivative framework; Parallel Transport Along Curves supplies transport along paths; The Levi-Civita Connection supplies the metric-compatible specialization.

Helpful background. Calculus and Linear Algebra for Relativity supplies bilinear forms and linear maps; the two-plane quotient is developed here. Exterior Algebra and Differential Forms supplies the two-form and wedge algebra used for curvature, while this page fixes the loop orientation.

Curvature of a nondegenerate two-plane

Let $\Pi\subset T_pM$ be spanned by $u$ and $v$. If the induced metric on $\Pi$ is nondegenerate, then

$$K(\Pi) =\frac{R(u,v,u,v)} {g(u,u)g(v,v)-g(u,v)^2}.$$

Here the four-slot notation follows the site’s component order,

$$R(a,b,c,d):=g\bigl(a,R(c,d)b\bigr),$$

so the numerator is $R(u,v,u,v)=g(R(u,v)v,u)$. Stating the slot order is essential: other books attach the four arguments to the curvature operator in another order, which can reverse the displayed numerator while leaving the geometric sectional curvature unchanged.

Changing the basis of $\Pi$ multiplies numerator and denominator by the same squared determinant, so the quotient depends only on the plane. In a Riemannian metric every two-plane is nondegenerate. In Lorentzian signature a plane may be spacelike or timelike and still admit this definition, but a degenerate plane containing a null direction has zero denominator. Assigning it a value by the same formula would be division by zero, not a new curvature invariant.

For a metric of constant sectional curvature $K$ in this component convention,

$$R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc}),$$

which inserted into the quotient returns $K$ for every nondegenerate plane. On the round sphere of radius $a$, $K=1/a^2$. The basis independence and reconstruction of Riemannian curvature from all sectional curvatures are developed in Calegari 2022, §7.3, pp. 55–56, PDF; in indefinite signature, the restriction to nondegenerate planes must be retained.

Holonomy from parallel transport

Fix $p\in M$. Parallel transport around a piecewise smooth loop $\gamma$ based at $p$ defines a linear isometry $P_\gamma:T_pM\to T_pM$. The holonomy group is

$$\operatorname{Hol}_p(\nabla) =\{P_\gamma:\gamma(0)=\gamma(1)=p\}.$$

Changing the base point along a path conjugates this subgroup, so its conjugacy class is the intrinsic datum. The restricted holonomy group $\operatorname{Hol}^0_p$ uses loops homotopic to the constant loop and is the identity component under standard hypotheses. The full holonomy group can contain additional disconnected transformations detected by noncontractible loops.

For a concrete orientation, choose constant coordinate directions $u,v$ near $p$ and follow the small loop $+\epsilon u,+\delta v,-\epsilon u,-\delta v$. Parallel transport satisfies $\dot w+\Gamma(\dot\gamma)w=0$, so

$$P_{\partial\Sigma}w =w-\epsilon\delta\,R(u,v)w +O\!\left((\lvert\epsilon\rvert+\lvert\delta\rvert)^3\right).$$

The sign reverses with loop orientation. The coefficient $\epsilon\delta$ is a parameter area: the leading metric area of a nondegenerate parallelogram is $\lvert\epsilon\delta\rvert\sqrt{\lvert g(u,u)g(v,v)-g(u,v)^2\rvert}$. One may replace it by signed metric area only after the plane basis is normalized. Keeping the two edge vectors explicit also permits the curvature operator to describe degenerate planes for which sectional curvature is undefined.

Thus curvature supplies infinitesimal generators of restricted holonomy. More precisely, for a path $\alpha$ from $p$ to $q$, transport the operator at $q$ back to $p$:

$$P_\alpha^{-1}\circ R_q(u,v)\circ P_\alpha.$$

The Ambrose–Singer theorem identifies the holonomy Lie algebra with the Lie algebra generated by these operators over points, tangent pairs, and paths in the connected manifold. The use of all paths matters: curvature at $p$ alone need not generate it. Two ideas explain the result. First, travel to $q$, make an infinitesimal loop there, and return to $p$; the resulting generator is the transported curvature above. Conversely, deform a contractible loop continuously to a point. The change in its transport is an integral of transported curvature, so the deformation cannot create generators outside that Lie algebra. The precise argument, including the identity-component statement, appears in Ballmann 2022, §3.1, Theorems 3.1.15 and 3.1.22, pp. 24–26, PDF.

Local curvature versus global holonomy

If $R=0$ on a simply connected region, transport there is path independent and restricted holonomy is trivial. Without simple connectivity, $R=0$ does not force the full holonomy group to be trivial. A flat quotient can identify a vector after a deck transformation that includes a rotation or reflection; contractible small loops see no curvature, while a loop representing the quotient topology returns a transformed vector.

The converse distinction is equally important. Curvature at one point does not by itself give the whole finite-loop transformation: curvature values at different points must be parallel transported and path ordered because the corresponding endomorphisms need not commute. Holonomy can also constrain geometric structure. A parallel tensor must have a value at $p$ fixed by every holonomy transformation. Conversely, an invariant tensor at $p$ can be transported to other points independently of the chosen path: two choices differ by a loop, which fixes that tensor. It therefore extends to a parallel tensor. The same reasoning applies to spinors using the lifted spin holonomy group. Thus the particular invariant objects of the holonomy representation, rather than reduced group size alone, determine which parallel structures exist.

Independent checks and scope

On the Euclidean plane, even polar-coordinate connection coefficients produce $R=0$ and trivial transport around every contractible loop. On $S^2_a$, take a positively oriented orthonormal pair $u,v$. The constant-curvature tensor gives $R(u,v)u=-v/a^2$, so the loop above sends $u$ to $u+\epsilon\delta\,v/a^2$ at leading order. Its rotation is positive from $u$ toward $v$ for positive area, with angle $A/a^2+O(A^{3/2})$ for loops shrinking at fixed shape. These checks connect the quotient definition, curvature operator, and loop orientation; the spherical transport example makes the finite-loop geometry visible. The surface relation between holonomy, integrated curvature, and Stokes’ theorem is developed in Sternberg 2003, §4.5.4, pp. 104–105, PDF.

Sectional curvature is a local scalar attached to selected planes; holonomy is a group attached to a connection and a base point. Neither alone is a complete classification of a Lorentzian metric. The physical interpretation of holonomy measurements and the classification of particular spacetimes belong to later volumes.

Exercises

First take orthonormal tangent vectors $u,v$ on a round sphere of radius $a$. Then consider an orthonormal timelike plane in a Lorentzian constant-curvature space, with $g(u,u)=-1$, $g(v,v)=1$, and $g(u,v)=0$. Evaluate the sectional curvature in both cases directly from the constant-curvature tensor.

Solution

On the sphere, the denominator is one. The constant-curvature formula gives

$$R(u,v,u,v)=\frac1{a^2} \left(g(u,u)g(v,v)-g(u,v)^2\right)=\frac1{a^2}.$$

Therefore $K(\operatorname{span}\{u,v\})=1/a^2$, and a sufficiently small positively oriented loop of area $A$ has leading rotation angle $A/a^2$.

For the Lorentzian timelike plane, the denominator is $-1$. If the ambient constant curvature is $K_0$, then

$$R(u,v,u,v)=K_0(-1)=-K_0,$$

so the quotient is again $K_0$. The numerator and denominator both change sign relative to a spacelike orthonormal plane; the sectional curvature does not.

References

• Ballmann, Werner. Vector Bundles and Connections. University of Bonn lecture notes, December 1999, corrected January 2022. PDF.
• Calegari, Danny. Introduction to Differential Geometry. University of Chicago lecture notes, preliminary version dated May 26, 2022. PDF.
• Sternberg, Shlomo. Semi-Riemann Geometry and General Relativity. Course notes, Harvard University, 2003. PDF.