Curvature and What It Means

The Ricci flow equation $\partial_tg = -2\operatorname{Ric}$ says that a metric moves in the direction of minus its Ricci curvature. To understand the equation you need to know what the Ricci tensor is: how it is built from the full curvature tensor, and what it measures. This chapter defines the Riemann curvature tensor, the failure of covariant derivatives to commute, and the quantities built from it: sectional curvature, which generalises the Gauss curvature of 8A.9 The Curvature of Surfaces to each 2-plane; Ricci curvature, an average of sectional curvatures; and scalar curvature, the average of those.

Each has a concrete meaning. Sectional curvature measures how much shorter (or longer) small geodesic circles are than Euclidean ones. Ricci curvature measures how much smaller (or larger) the volume of a thin cone of geodesics is. In general relativity, Ricci curvature measures how quickly a ball of freely falling particles starts to shrink, which is where matter enters. The tides show the curvature that is left in empty space. That leftover curvature changes the shape of a ball but not, to first order, its volume.

By the end of this chapter you will be able to:

• define the curvature tensor, compute it from Christoffel symbols, and use its symmetries and the Bianchi identities;
• define sectional, Ricci and scalar curvature and relate them as averages;
• derive the contracted Bianchi identity $\operatorname{div}\operatorname{Ric} = \frac12dR$;
• state and check the three meanings of curvature: circumferences, volumes and gravity;
• convert a curvature formula from another book to this guide's conventions by testing it on the sphere.

4.1The tides

4.2The curvature tensor

The curvature endomorphism of $(M, g)$ is

and the Riemann curvature tensor is $\operatorname{Rm}(X, Y, Z, W) = \langle R(X, Y)Z, W\rangle$, with components $R_{ijkl} = \operatorname{Rm}(\partial_i, \partial_j, \partial_k, \partial_l)$. Although it is built from derivatives, $R(X, Y)Z$ at $p$ depends only on $X_p$, $Y_p$ and $Z_p$: it is linear over functions in each slot (Exercise 4.2), so it is a tensor. In coordinates, $R(\partial_i, \partial_j)\partial_k = R_{ijk}{}^l\partial_l$ with

What it measures. For coordinate fields, $[\partial_i, \partial_j] = 0$ and $R(\partial_i, \partial_j) = \nabla_i\nabla_j - \nabla_j\nabla_i$: curvature is the failure of covariant derivatives to commute. Geometrically, parallel transport around the small coordinate parallelogram with sides $\varepsilon\partial_i$ and $\varepsilon\partial_j$ changes a vector $v$ by $\pm\varepsilon^2R(\partial_i, \partial_j)v + O(\varepsilon^3)$, the sign depending on the direction of travel: curvature is infinitesimal holonomy (9A.2 Connections). And $\operatorname{Rm} \equiv 0$ exactly when $(M, g)$ is locally flat, locally isometric to $\mathbb{R}^n$ (Lee, chapter 7).

Symmetries. For all $X, Y, Z, W$:

1. $\operatorname{Rm}(X, Y, Z, W) = -\operatorname{Rm}(Y, X, Z, W)$;
2. $\operatorname{Rm}(X, Y, Z, W) = -\operatorname{Rm}(X, Y, W, Z)$ (from metric compatibility);
3. $R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0$, the first Bianchi identity (from torsion-freeness);
4. $\operatorname{Rm}(X, Y, Z, W) = \operatorname{Rm}(Z, W, X, Y)$ (a consequence of 1–3).

In components: $R_{ijkl} = -R_{jikl} = -R_{ijlk} = R_{klij}$ and $R_{ijkl} + R_{jkil} + R_{kijl} = 0$. These cut the $n^4$ components down to $\frac{n^2(n^2 - 1)}{12}$ independent ones: $1$ in dimension $2$, $6$ in dimension $3$, $20$ in dimension $4$. The derivatives satisfy the second Bianchi identity

cyclic in the last three indices $k, l, m$ (Lee, chapter 7). It is the identity behind the contracted Bianchi identity below, and behind the evolution equations of 11A.2 How Curvature Evolves.

4.3Sectional curvature

For linearly independent $X, Y \in T_pM$, the sectional curvature of the plane $\Pi = \operatorname{span}(X, Y)$ is

which depends only on the plane. It is the Gauss curvature at $p$ of the surface swept out by the geodesics from $p$ tangent to $\Pi$ (Lee, chapter 8). The sectional curvatures determine $\operatorname{Rm}$ completely, by a polarisation argument that uses the symmetries above.

A manifold has constant curvature $\kappa$ if $K \equiv \kappa$ for all planes at all points. Then

The model spaces have constant curvature $0$, $1$ and $-1$ (9A.5 Computing Curvature computes this), and the sphere of radius $r$ has $\kappa = \frac{1}{r^2}$. In dimension $2$ there is only one plane at each point, so $\operatorname{Rm}$ is determined by one function, the Gauss curvature $K$ of 8A.9 The Curvature of Surfaces (Exercise 4.3).

4.4Ricci and scalar curvature

The Ricci tensor is the trace $R_{ij} = g^{kl}R_{kijl}$, $\operatorname{Ric}(Y, Z) = \operatorname{tr}\big(X \mapsto R(X, Y)Z\big)$, a symmetric 2-tensor. The scalar curvature is $R = g^{ij}R_{ij}$. In an orthonormal basis $e_1, \dots, e_n$ with $e_1 = v$ a unit vector,

so $\frac{\operatorname{Ric}(v, v)}{n - 1}$ is the average sectional curvature of the planes containing $v$, and $\frac{R}{n(n - 1)}$ is the average over all planes (Exercise 4.4). For the unit sphere, $\operatorname{Ric} = (n - 1)g$ and $R = n(n - 1)$; in dimension $2$, $\operatorname{Ric} = Kg$ and $R = 2K$.

A metric is Einstein if $\operatorname{Ric} = \lambda g$ for a constant $\lambda$. Spaces of constant curvature are Einstein, and in dimension $3$ the converse holds (9A.5 Computing Curvature). Einstein metrics are the fixed points of the Ricci flow up to scaling: if $\operatorname{Ric}(g_0) = \lambda g_0$, then $g(t) = (1 - 2\lambda t)g_0$ (Exercise 4.8).

The identity says that $\operatorname{Ric} - \frac12Rg$, the Einstein tensor, is divergence-free. In general relativity that is the conservation of energy and momentum. In the Ricci flow it is the algebraic shadow of the diffeomorphism invariance of 8A.6 Flows and the Lie Derivative, which keeps the flow from being strictly parabolic, a degeneracy that DeTurck's trick removes (11A.3 Short-Time Existence and Uniqueness). It also gives Schur's lemma: in dimension $n \geq 3$, if $\operatorname{Ric} = fg$ for a function $f$, then $f$ is constant (Exercise 4.5).

4.5Three meanings of curvature

1. Circumferences: sectional curvature. Let $\Pi$ be a plane at $p$ and $C_r$ the curve traced by the points at distance $r$ along the geodesics from $p$ tangent to $\Pi$. Then

On a surface this is the Bertrand–Diguet–Puiseux formula. Positive curvature makes small circles shorter than in the plane, and negative curvature makes them longer. On the model surfaces the circumferences are exactly $2\pi\sin r$, $2\pi r$ and $2\pi\sinh r$ (Figure 4.1). The formula follows from the normal-coordinate expansion of 9A.3 Geodesics and the Exponential Map, or from Jacobi fields (9A.7 Jacobi Fields and Curvature versus Topology).

2. Volumes: Ricci curvature. In normal coordinates at $p$ the metric and its volume element are

(a standard computation with the Jacobi fields of 9A.7 Jacobi Fields and Curvature versus Topology). You check the first on the sphere in the last exercise of 9A.3 Geodesics and the Exponential Map and derive the second from it in Exercise 4.6. So the volume of a thin cone of geodesics leaving $p$ in the direction of a unit vector $v$ is reduced, at distance $r$, by the factor $1 - \frac16\operatorname{Ric}(v, v)r^2$ compared with Euclidean space. Positive Ricci curvature in a direction focuses the geodesics in that direction and shrinks volume. Integrating over all directions,

where $\omega_n$ is the volume of the Euclidean unit ball. Scalar curvature measures the volume of small balls.

This is the intuition for the Ricci flow. Where Ricci curvature is positive the metric has "too little volume", and $\partial_tg = -2\operatorname{Ric}$ shrinks those directions further, as a round sphere shrinks to a point. Where it is negative the metric expands.

3. Gravity: Ricci curvature again. In a spacetime, consider a small ball of test particles, initially at rest relative to each other, falling freely with four-velocity $u$. Its volume $V$ satisfies

(a form of Raychaudhuri's equation). Einstein's equations, for matter that is a perfect fluid at rest relative to the ball, turn this into

in units where $c = 1$: the ball begins to shrink in proportion to the density plus the pressures. In empty space the right-hand side is zero, and the remaining curvature (the Weyl tensor, 9A.5 Computing Curvature) changes the shape of the ball without changing its volume (Figure 4.2). John Baez and Emory Bunn made this statement the starting point of an exposition of general relativity ("The meaning of Einstein's equation", American Journal of Physics, 2005).

4.6Conventions

Books differ in two ways: the overall sign of $R(X, Y)Z$, and the order of the indices in $R_{ijkl}$. This guide follows Lee (8A.7 Tensors and Index Notation):

quantity	this guide	on the round unit $S^n$
curvature endomorphism	$R(X, Y)Z = \nabla_X\nabla_YZ - \nabla_Y\nabla_XZ - \nabla_{[X, Y]}Z$
curvature tensor	$R_{ijkl} = \langle R(\partial_i, \partial_j)\partial_k, \partial_l\rangle$	$R_{ijkl} = g_{il}g_{jk} - g_{ik}g_{jl}$
sectional curvature	$K(X, Y) = \operatorname{Rm}(X, Y, Y, X)/\lvert X\wedge Y\rvert^2$	$K \equiv 1$
Ricci tensor	$R_{ij} = g^{kl}R_{kijl}$	$R_{ij} = (n - 1)g_{ij}$
scalar curvature	$R = g^{ij}R_{ij}$	$R = n(n - 1)$

The last column is the key to converting. Sectional, Ricci and scalar curvature are almost always normalised so that the round sphere has $K = 1$, $\operatorname{Ric} = (n - 1)g$ and $R = n(n - 1)$ (some physics texts flip the sign of the Ricci tensor); the four-index tensor is where books differ most. To translate a formula involving $R_{ijkl}$ from another book, substitute that book's expression for the unit sphere and compare with the column. One substitution fixes both the sign and the index order. When you read a Ricci flow paper, check its formula for $R_{ijkl}$ on the sphere before using any identity in it.

4.7History

Riemann introduced the curvature of a manifold of any dimension in his 1854 lecture, defining it through the curvature of surfaces swept out by geodesics, what is now sectional curvature, and wrote it as a tensor expression in an 1861 essay for the Paris Academy. Christoffel (1869) and Lipschitz derived the four-index tensor from the metric. Gregorio Ricci-Curbastro introduced the contracted tensor that bears his name around 1903–04, and Einstein made it the left-hand side of his field equations in 1915. The second Bianchi identity is named after Luigi Bianchi (1902), though it was found earlier. Amal Kumar Raychaudhuri published his equation for the focusing of geodesics in 1955. The circumference formula is due to Joseph Bertrand, Charles Diguet and Victor Puiseux (1848).

4.8Exercises

Solution

First: $\nabla_{fX}\nabla_YZ - \nabla_Y\nabla_{fX}Z - \nabla_{[fX, Y]}Z = f\nabla_X\nabla_YZ - (Yf)\nabla_XZ - f\nabla_Y\nabla_XZ - f\nabla_{[X, Y]}Z + (Yf)\nabla_XZ = fR(X, Y)Z$. Third slot: $\nabla_X\nabla_Y(fZ) = \nabla_X((Yf)Z + f\nabla_YZ) = (XYf)Z + (Yf)\nabla_XZ + (Xf)\nabla_YZ + f\nabla_X\nabla_YZ$; subtracting the same with $X, Y$ exchanged and $\nabla_{[X, Y]}(fZ) = ([X, Y]f)Z + f\nabla_{[X, Y]}Z$, everything cancels except $fR(X, Y)Z$, because $XYf - YXf = [X, Y]f$.

Solution

In an orthonormal basis, antisymmetry in $ij$ and in $kl$ leaves only components with $\{i, j\} = \{k, l\} = \{1, 2\}$: $R_{1221} = K$, $R_{1212} = R_{2121} = -K$, $R_{2112} = K$. The tensor $K(g_{il}g_{jk} - g_{ik}g_{jl})$ has the same components, so they are equal in every frame. Contracting, $R_{jk} = g^{il}R_{ijkl} = K(2g_{jk} - g_{jk}) = Kg_{jk}$, and $R = 2K$.

Solution

$\operatorname{Ric}(v, v) = \sum_k\operatorname{Rm}(e_k, v, v, e_k) = \sum_{k \geq 2}K(e_k, v)$, the $k = 1$ term vanishing by antisymmetry. Summing over $v = e_j$: $R = \sum_j\sum_{k \neq j}K(e_k, e_j)$. On the unit sphere each $K = 1$, giving $n - 1$ and $n(n - 1)$.

Solution

$R = nf$ and $\nabla^iR_{ij} = \nabla^i(fg_{ij}) = \nabla_jf$. The contracted Bianchi identity gives $\nabla_jf = \frac n2\nabla_jf$, so $(1 - \frac n2)df = 0$ and $df = 0$ when $n \neq 2$. In dimension $2$, $\operatorname{Ric} = Kg$ always (Exercise 4.3), with $K$ any function.

Solution

(a) $\operatorname{tr}h = \delta^{ij}h_{ij} = -\frac13\delta^{ij}R_{iklj}x^kx^l = -\frac13R_{kl}x^kx^l$ (at $p$, $g^{ij} = \delta^{ij}$, and $R_{kl} = g^{ij}R_{iklj}$ contracts the first and last slots). (b) $\operatorname{Vol} = \int_{B_r}(1 - \frac16R_{kl}x^kx^l)\,dx + O(r^{n + 3}) = \omega_nr^n - \frac16R_{kl}\delta^{kl}\frac{\omega_nr^{n + 2}}{n + 2} + \dots = \omega_nr^n(1 - \frac{R}{6(n + 2)}r^2 + \dots)$; the $O(|x|^3)$ term in the volume element is odd and integrates to $O(r^{n + 4})$, hence the $O(r^4)$. (c) $\pi(2r - \sin2r) = \pi(\frac{8r^3}{6} - \frac{32r^5}{120} + \dots) = \frac43\pi r^3(1 - \frac{r^2}{5} + \dots)$, and with $R = 6$, $n = 3$: $\frac{6}{6\cdot5} = \frac15$.

Solution

$\partial_i\partial_j\frac{1}{|x|} = \frac{3x_ix_j - |x|^2\delta_{ij}}{|x|^5}$, so at $(d, 0, 0)$, $\partial_i\partial_j\Phi = -GM\operatorname{diag}(2, -1, -1)/d^3$ and $\ddot\xi = \frac{GM}{d^3}(2\xi^1, -\xi^2, -\xi^3)$. The trace $2 - 1 - 1 = 0$. Numerically, $\frac{2\cdot4.90\times10^{12}\cdot6.37\times10^6}{(3.84\times10^8)^3} \approx 1.1\times10^{-6}$ m/s².

Solution

(a) The Christoffel formula is homogeneous of degree zero in $g$ ($g^{kl}$ scales by $\lambda^{-1}$, the derivatives by $\lambda$), so $\Gamma$, $R(X, Y)Z$ and its trace $\operatorname{Ric}$ are unchanged, and $R = g^{ij}R_{ij}$ scales by $\lambda^{-1}$. (b) $g = r^2g_{S^n}$ and $\operatorname{Ric}(g_{S^n}) = (n - 1)g_{S^n} = \frac{n - 1}{r^2}g$; $R = r^{-2}n(n - 1)$. (c) $\partial_tg = (\rho^2)'g_{S^n}$ and $-2\operatorname{Ric}(g(t)) = -2(n - 1)g_{S^n}$, so $(\rho^2)' = -2(n - 1)$. (d) $\operatorname{Ric}((1 - 2\lambda t)g_0) = \operatorname{Ric}(g_0) = \lambda g_0$, and $\partial_t((1 - 2\lambda t)g_0) = -2\lambda g_0$.