Flows and the Lie Derivative

A vector field on a manifold is a velocity at every point. Following those velocities moves every point at once: the flow of the vector field, a one-parameter family of diffeomorphisms. Flows are how vector fields act on everything else. Differentiating any geometric object along a flow gives its Lie derivative, the rate at which the object changes as it is dragged along. For a function this is just the directional derivative; for another vector field it is the Lie bracket, which measures how two flows fail to commute; and for a metric it is the strain: how the flow stretches lengths.

This chapter ends with the single most important structural fact about the Ricci flow. The Ricci tensor is natural: pulling a metric back by a diffeomorphism pulls back its Ricci tensor. So if $g(t)$ is a Ricci flow, so is $\phi^*g(t)$ for any fixed diffeomorphism $\phi$. That freedom, the diffeomorphism invariance of the equation, has two consequences that run through the rest of the guide: the equation cannot be strictly parabolic, which DeTurck's trick repairs (11A.3 Short-Time Existence and Uniqueness); and there are solutions that move only by diffeomorphisms and scaling, the Ricci solitons, whose defining equation $\operatorname{Ric} + \frac12\mathcal L_Xg = \lambda g$ is derived here (11B.1 Ricci Solitons).

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

• find integral curves and flows of vector fields, and state the fundamental theorem on flows;
• compute Lie brackets in coordinates and explain the bracket as the failure of flows to commute;
• define the Lie derivative of functions, vector fields and tensors, and compute $\mathcal L_Xg$;
• recognise Killing fields and conformal fields;
• state the naturality of the Ricci tensor and derive the soliton equation from it.

6.1How a fluid strains

6.2Integral curves and flows

A vector field $X$ on $M$ assigns to each $p$ a tangent vector $X_p \in T_pM$, smoothly (8A.3 Tangent Vectors and Bundles); in coordinates $X = X^i\partial_i$ with smooth functions $X^i$. An integral curve is a curve $\gamma$ with $\gamma'(t) = X_{\gamma(t)}$: in coordinates, the ODE system $\dot\gamma^i = X^i(\gamma)$.

The existence, uniqueness and smooth dependence on the starting point come from the ODE theory of 2B.10 Ordinary Differential Equations, applied in charts; maximality patches the local solutions together. If $X$ has compact support, every integral curve stays in a compact set, so by the blow-up alternative (2B.10 Ordinary Differential Equations) it exists for all time. A field whose flow exists for all time is complete. On $\mathbb{R}$, the field $x^2\partial_x$ is not: its integral curve $x(t) = \frac{x_0}{1 - x_0t}$ escapes to infinity in finite time.

Time-dependent vector fields $X_t$ have flows too, solving $\frac{d}{dt}\phi_t(p) = X_t(\phi_t(p))$, though they no longer form a group. They are the diffeomorphisms of DeTurck's trick (11A.3 Short-Time Existence and Uniqueness) and of the soliton equation below.

6.3The Lie bracket

The bracket of two vector fields is the vector field

(Exercise 6.4). Second derivatives of $f$ cancel, so $[X, Y]$ is again a first-order operator: a vector field. The bracket is antisymmetric and satisfies the Jacobi identity, and for left-invariant fields on a Lie group it is the bracket of the Lie algebra (8A.5 Lie Groups and Group Actions). Coordinate vector fields commute: $[\partial_i, \partial_j] = 0$.

The geometric meaning. Flow along $X$ for time $t$, then along $Y$ for time $t$, then back along $X$, then back along $Y$. To second order you arrive at

in coordinates (Lee, chapter 9): the bracket measures the failure of the flows to commute. In particular two flows commute, $\theta_t\psi_s = \psi_s\theta_t$, if and only if $[X, Y] = 0$.

6.4The Lie derivative

Let $\theta_t$ be the flow of $X$. The pullback of a covariant tensor field $T$ (a function, a covector field, a metric) by a diffeomorphism $\phi$ is $(\phi^*T)_p(v_1, \dots, v_k) = T_{\phi(p)}(d\phi(v_1), \dots, d\phi(v_k))$; a vector field is pulled back by $(\phi^*Y)_p = (d\phi_p)^{-1}Y_{\phi(p)}$.

It is a tensor of the same type. The cases that recur:

• functions: $\mathcal L_Xf = Xf = df(X)$;
• vector fields: $\mathcal L_XY = [X, Y]$;
• covariant 2-tensors, in particular a metric: in coordinates,

and in Euclidean space with Cartesian coordinates $(\mathcal L_Xg)_{ij} = \partial_iX_j + \partial_jX_i$ (Exercise 6.6); on a Riemannian manifold, in terms of the covariant derivative of 9A.2 Connections, $(\mathcal L_Xg)_{ij} = \nabla_iX_j + \nabla_jX_i$;

• differential forms: $\mathcal L_X\omega = d(\iota_X\omega) + \iota_X(d\omega)$, Cartan's magic formula (8A.8 Differential Forms and Stokes’ Theorem).

The Lie derivative along the flow gives the derivative at every time, not just $t = 0$:

by the group property $\theta_{t+s}^* = \theta_t^*\theta_s^*$ (Exercise 6.9).

Killing and conformal fields. A vector field with $\mathcal L_Xg = 0$ is a Killing field: its flow consists of isometries. On $\mathbb{R}^2$ the rotation field $-y\partial_x + x\partial_y$ is Killing; the dilation $x\partial_x + y\partial_y$ is not, but it satisfies $\mathcal L_Xg = 2g$: a conformal field, whose flow scales all lengths by the same factor (Exercise 6.8).

6.5Naturality and its consequences

The proof is in 9A.4 Curvature and What It Means; the reason is that the Ricci tensor is built from $g$ by a coordinate-free construction (the Levi-Civita connection and its curvature), so it cannot tell a metric from its reparametrisation. Every geometric quantity has this property: scalar curvature, volume, the Laplacian. Two consequences follow, and between them they shape the analysis of the Ricci flow (Figure 6.3).

1. The flow is invariant under diffeomorphisms. If $g(t)$ solves $\partial_tg = -2\operatorname{Ric}(g)$, so does $\phi^*g(t)$ for any fixed $\phi$: $\partial_t\phi^*g = \phi^*\partial_tg = -2\phi^*\operatorname{Ric}(g) = -2\operatorname{Ric}(\phi^*g)$. So solutions come in huge families, one for each diffeomorphism, and the linearised equation has a kernel in the directions $\mathcal L_Xg$. An equation with this degeneracy cannot be strictly parabolic (6A.1 What a PDE Is, 6A.7 Nonlinear Parabolic Equations). DeTurck's trick breaks the symmetry by adding a term $\mathcal L_{W(g)}g$, solves the resulting strictly parabolic equation, and recovers a Ricci flow by pulling back along the time-dependent flow of $W$ (11A.3 Short-Time Existence and Uniqueness).

2. Solitons. The most symmetric solutions change only by diffeomorphisms and scaling:

with $\sigma(t) > 0$, $\sigma(0) = 1$, and $\phi_t$ the flow of a (time-dependent) vector field with $\phi_0 = \operatorname{id}$. Differentiating at $t = 0$, using $\frac{d}{dt}\phi_t^*g_0|_{t=0} = \mathcal L_Xg_0$:

with $\lambda = -\frac12\sigma'(0)$. This is the Ricci soliton equation. The soliton is shrinking, steady or expanding as $\lambda > 0$, $= 0$ or $< 0$. When $X = \nabla f$ is a gradient, $\mathcal L_{\nabla f}g = 2\nabla^2f$ and the equation becomes $\operatorname{Ric} + \nabla^2f = \lambda g$ (Exercise 6.10). Solitons are the fixed points of the Ricci flow modulo symmetry, and they are the models for its singularities (11B.1 Ricci Solitons, 12A.3 The 𝓦-Entropy).

6.6History

Sophus Lie introduced the bracket and the derivative now named after him in his work on transformation groups in the 1870s–80s; the term "Lie derivative" and the modern formulation are due to Władysław Ślebodziński (1931) and David van Dantzig. Wilhelm Killing introduced the fields named after him in 1888. Wei-Liang Chow (1939) and Petr Rashevskii (1938) proved that brackets of the available directions determine the reachable configurations of a nonholonomic system. Richard Hamilton introduced Ricci solitons in the 1980s; Dennis DeTurck's trick appeared in 1983.

6.7Exercises

Solution

$X(Yf) = X^i\partial_i(Y^j\partial_jf) = X^i\partial_iY^j\partial_jf + X^iY^j\partial_i\partial_jf$; subtract the same with $X$ and $Y$ exchanged, and the symmetric second-derivative terms cancel. $[x\partial_y, y\partial_x] = x\partial_y(y)\partial_x - y\partial_x(x)\partial_y = x\partial_x - y\partial_y$.

Solution

$[S, D] = \partial_\phi(\cos\phi)\partial_x + \partial_\phi(\sin\phi)\partial_y = -\sin\phi\,\partial_x + \cos\phi\,\partial_y$. The three are orthonormal in the $(x, y, \phi)$ coordinates. The manoeuvre: after steering, heading $\varepsilon$; driving moves to $(\varepsilon\cos\varepsilon, \varepsilon\sin\varepsilon, \varepsilon)$; steering back gives heading $0$; driving back moves to $(\varepsilon\cos\varepsilon - \varepsilon, \varepsilon\sin\varepsilon, 0) = (-\frac{\varepsilon^3}{2} + \dots, \varepsilon^2 + \dots, 0)$. To leading order the displacement is $\varepsilon^2\partial_y = \varepsilon^2[S, D]$ at $\phi = 0$.

Solution

$\frac{d}{dt}\theta_t^*T = \frac{d}{ds}\Big|_{s=0}\theta_{t+s}^*T = \frac{d}{ds}\Big|_{s=0}\theta_t^*\theta_s^*T = \theta_t^*\frac{d}{ds}\Big|_{s=0}\theta_s^*T$, since $\theta_t^*$ is linear and doesn't depend on $s$.

Solution

(a) $\partial_if = \frac{x_i}{2\tau}$, $\partial_i\partial_jf = \frac{\delta_{ij}}{2\tau}$, and in Euclidean coordinates $(\mathcal L_Yg)_{ij} = \partial_iY_j + \partial_jY_i$, so $\mathcal L_{\nabla f}g = 2\nabla^2f = \frac1\tau g$. (b) Immediate. (c) $\phi_t(x) = \frac{x}{\sqrt{1 - t/\tau}}$ multiplies the Euclidean metric by $\frac{1}{1 - t/\tau}$, so $\sigma(t)\phi_t^*g = g$ for all $t$: the flat metric is constant, as it must be, but it can also be read as shrinking by $\sigma$ while being stretched back by $\phi_t$. Differentiating at $0$: $\sigma'(0) = -\frac1\tau$, so $\lambda = \frac{1}{2\tau}$.