Riemannian Metrics and Model Spaces

The Ricci flow evolves a Riemannian metric, an inner product on each tangent space, varying smoothly from point to point (8A.2 Partitions of Unity, 8A.7 Tensors and Index Notation). A metric is what turns a smooth manifold into a geometric object: it measures lengths of curves and hence distances between points, angles between directions, areas and volumes. Book 8A ended with Gauss's discovery that the curvature of a surface depends only on its metric. Riemann's 1854 lecture took that as a programme for every dimension, and this book carries it out.

This first chapter is a catalogue. It defines the basic notions (length, distance, volume, isometry) and then builds the examples on which everything later is tested: the three model spaces of constant curvature, Euclidean space, the sphere and hyperbolic space, each in several coordinate pictures; and the warped products $dr^2 + \varphi(r)^2g_{S^{n-1}}$, the rotationally symmetric metrics in which the neckpinch, the Bryant soliton and Perelman's standard solution all live.

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

• define Riemannian metrics, lengths, distance and volume, and recognise isometries;
• write the round sphere and hyperbolic space in several models and describe their isometries;
• write rotationally symmetric metrics as warped products, and state the conditions for smoothness at the axis;
• describe products, scalings, conformal changes, flat tori and Berger spheres;
• compute the areas of geodesic discs in the plane, the sphere and the hyperbolic plane.

1.1The shape of the Earth

1.2Metrics, lengths and distances

A Riemannian metric on a smooth manifold $M$ is a smooth symmetric $(0, 2)$-tensor field $g$ that is positive definite at every point: an inner product $g_p = \langle\cdot, \cdot\rangle_p$ on each $T_pM$. In coordinates, $g = g_{ij}\,dx^idx^j$ with $(g_{ij})$ symmetric positive definite (8A.7 Tensors and Index Notation). Every manifold has one (8A.2 Partitions of Unity).

The length of a piecewise smooth curve $\gamma : [a, b] \to M$ is

independent of the parametrisation. The Riemannian distance $d(p, q)$ is the infimum of the lengths of curves from $p$ to $q$. It is a metric in the sense of 2B.1 Metric Spaces, and it induces the manifold's own topology (Lee, chapter 2). The volume form is $dV_g = \sqrt{\det g_{ij}}\,dx^1\cdots dx^n$ (8A.8 Differential Forms and Stokes’ Theorem).

A diffeomorphism $\phi : (M, g) \to (N, h)$ is an isometry if $\phi^*h = g$, that is, $\langle d\phi(v), d\phi(w)\rangle_h = \langle v, w\rangle_g$ for all tangent vectors. Isometries preserve lengths, distances, angles, volumes and every curvature quantity. Two metrics related by a diffeomorphism, $g$ and $\phi^*g$, are geometrically the same: this is the diffeomorphism invariance of 8A.6 Flows and the Lie Derivative.

Basic constructions.

• Induced metrics. A submanifold of a Riemannian manifold inherits a metric by restriction; the first fundamental form of a surface in $\mathbb{R}^3$ is the example (8A.9 The Curvature of Surfaces).
• Products. On $M\times N$, the product metric $g\oplus h$ makes the factors orthogonal.
• Scaling. Replacing $g$ by $\lambda^2g$ multiplies lengths and distances by $\lambda$ and volumes by $\lambda^n$. The Ricci flow's scaling law, $g(t) \mapsto \lambda g(t/\lambda)$, combines this with a rescaling of time (6A.1 What a PDE Is).
• Conformal change. $\tilde g = e^{2u}g$ changes lengths by the factor $e^u$, the same in all directions, and preserves angles (5A.5 Uniformization and the Two-Dimensional Ricci Flow).
• Quotients. If a group $\Gamma$ acts on $(M, g)$ by isometries as a covering space action, $g$ descends to $M/\Gamma$ (7A.6 Covering Spaces): flat tori $\mathbb{R}^n/\Lambda$, real projective spaces, lens spaces, the Poincaré homology sphere.

1.3The model spaces

Euclidean space $\mathbb{R}^n$ with $g = \sum dx^i\,dx^i$. Its isometries are the rigid motions $x \mapsto Ax + b$, $A \in O(n)$.

The sphere $S^n(r)$ of radius $r$, with the metric induced from $\mathbb{R}^{n+1}$. Its isometries are $O(n + 1)$. In stereographic coordinates (8A.1 Smooth Structures) the unit sphere's metric is

Hyperbolic space $\mathbb{H}^n$ can be presented in three equivalent ways, each useful for different purposes.

• The hyperboloid model. In $\mathbb{R}^{n+1}$ with the Minkowski form $\langle x, y\rangle = -x^0y^0 + x^1y^1 + \dots + x^ny^n$, the upper sheet of the hyperboloid $\{\langle x, x\rangle = -1,\ x^0 > 0\}$, with the restriction of this form, which is positive definite on its tangent spaces. Its isometries are the Lorentz transformations preserving the sheet, $O^+(n, 1)$, just as $O(n + 1)$ acts on the sphere.
• The Poincaré ball model. The unit ball with

obtained from the hyperboloid by stereographic projection from $(-1, 0, \dots, 0)$. For $n = 2$ it is the hyperbolic disc of 5A.5 Uniformization and the Two-Dimensional Ricci Flow, whose isometries include the disc automorphisms of 5A.4 Harmonic Functions and Conformal Mapping.

• The upper half-space model. $\{y_n > 0\}$ with $g = \frac{|dy|^2}{y_n^2}$. For $n = 2$, in complex notation, the orientation-preserving isometries are the Möbius maps $z \mapsto \frac{az + b}{cz + d}$ with real coefficients and $ad - bc = 1$ (Exercise 1.1).

All three are conformal to Euclidean space or to Minkowski space restricted to a sheet, and all are isometric to each other. The boundary sphere of the ball is "at infinity": its distance from any interior point is infinite. In 9A.4 Curvature and What It Means the three model spaces are shown to have constant sectional curvature $0$, $+1$ and $-1$.

Geodesic polar coordinates. All three model spaces can be written in a unified way. Around any point, with $r$ the distance from it and $g_{S^{n-1}}$ the round metric on the unit sphere of directions,

for $0 < r < \pi$ on the sphere and all $r > 0$ otherwise (Exercise 1.2). The circumference of the geodesic circle of radius $r$ in the plane, the unit sphere and the hyperbolic plane is $2\pi\operatorname{sn}_k(r)$, and the area of the geodesic disc is

Hyperbolic discs grow exponentially in area: there is far more room far away than in the plane.

1.4Rotationally symmetric metrics

A metric on $\mathbb{R}^n$ or $S^n$ invariant under all rotations about a point (or an axis) can be written as a warped product

where $r$ is the distance along the radial geodesics and $\varphi(r) > 0$ is the radius of the sphere of symmetry at distance $r$. The model spaces are the cases $\varphi = \operatorname{sn}_k$. For the metric to close up smoothly at an end where $\varphi \to 0$, say $r = 0$, the profile must satisfy $\varphi(0) = 0$, $\varphi'(0) = 1$ and $\varphi^{(\text{even})}(0) = 0$ (it must extend to an odd function); otherwise the metric has a cone point there (Exercise 1.3). A metric on $S^n$ needs this at both ends.

These metrics are the stage for the singularity theory of the Ricci flow:

• the round sphere, $\varphi = r_0\sin(r/r_0)$;
• the round cylinder $S^{n-1}\times\mathbb{R}$, with $\varphi$ a positive constant, the model neck;
• a dumbbell, two large round bulbs joined by a thin neck, where $\varphi$ dips to a small minimum (Figure 1.3). Angenent and Knopf proved (2004) that suitable rotationally symmetric dumbbell metrics on $S^n$, $n \geq 3$, develop a neckpinch under the Ricci flow (11B.4 Singularities);
• the Bryant soliton on $\mathbb{R}^n$, a steady soliton with $\varphi$ growing like $\sqrt r$ (11B.1 Ricci Solitons);
• Perelman's standard solution, a capped half-cylinder used in surgery (12B.4 Surgery).

9A.5 Computing Curvature computes their curvature: $-\frac{\varphi''}{\varphi}$ in the radial planes and $\frac{1 - \varphi'^2}{\varphi^2}$ in the planes tangent to the spheres. A thin neck, where $\varphi$ is small and nearly constant, has large positive curvature around it and almost none along it: it is nearly a cylinder.

1.5More examples

Flat tori. $\mathbb{R}^n/\Lambda$ for a lattice $\Lambda$, with the Euclidean metric descended. Different lattices give non-isometric flat tori, though all are diffeomorphic (5A.5 Uniformization and the Two-Dimensional Ricci Flow).

Berger spheres. On $S^3 = SU(2)$, the round metric is left-invariant (8A.5 Lie Groups and Group Actions). Rescale it by a factor $\varepsilon^2$ in the direction of the circles of the Hopf fibration (the orbits of $q \mapsto e^{it}q$, 10A.1 A Zoo of Three-Manifolds) and leave it unchanged in the orthogonal directions: the result is a Berger sphere, homogeneous but not isotropic. As $\varepsilon \to 0$ the Hopf circles shrink to points, the volume tends to $0$, and yet the sectional curvatures stay bounded (they are computed in 9A.5 Computing Curvature). Berger spheres are the first example of collapse with bounded curvature, the phenomenon that Perelman's noncollapsing theorem rules out for the Ricci flow (9B.3 Collapsing and Noncollapsing).

Optical metrics. In a medium with refractive index $n(x)$, light travels at speed $c/n$, and by Fermat's principle its rays are the paths of least travel time, which are the geodesics of the optical metric $n(x)^2|dx|^2$. A mirage over a hot road is a geodesic of the optical metric of air whose index increases with height; graded-index optical fibres guide light because their index is highest on the axis, so the geodesics of the optical metric oscillate about the axis instead of leaving the fibre (Figure 1.4).

1.6History

Riemann introduced Riemannian metrics in his 1854 habilitation lecture, Über die Hypothesen, welche der Geometrie zu Grunde liegen, published in 1868. Hyperbolic geometry was discovered by Lobachevsky (1829) and Bolyai (1832); Beltrami gave its first models in 1868, Klein in 1871 and Poincaré the disc and half-plane models in 1882. Escher made the Circle Limit prints in 1958–60, and Coxeter's analysis of Circle Limit III appeared in 1979. WGS 84 has been the reference system for GPS since the 1980s, with refinements since.

1.7Exercises

Solution

For $w = az + b$: $|f'|^2 = a^2$ and $\operatorname{Im}w = ay$, so the pullback is $\frac{a^2|dz|^2}{a^2y^2}$. For $w = -1/z$: $|f'|^2 = \frac{1}{|z|^4}$ and $\operatorname{Im}(-1/z) = \frac{y}{|z|^2}$, so the pullback is $\frac{|dz|^2/|z|^4}{y^2/|z|^4} = \frac{|dz|^2}{y^2}$.

Solution

(a) $x = (\sin r\cos\theta, \sin r\sin\theta, \cos r)$, $|x_r|^2 = 1$, $x_r\cdot x_\theta = 0$, $|x_\theta|^2 = \sin^2r$. (b) $x_r = (\sinh r, \cosh r\cos\theta, \cosh r\sin\theta)$, with Minkowski norm $-\sinh^2r + \cosh^2r = 1$; $x_\theta = (0, -\sinh r\sin\theta, \sinh r\cos\theta)$, norm $\sinh^2r$; they are orthogonal. (c) $\int_0^{2\pi}\int_0^r\sin s\,ds\,d\theta = 2\pi(1 - \cos r)$ and $2\pi(\cosh r - 1)$.

Solution

The circle $\{r = r_0\}$ is parametrised by $\theta \in [0, 2\pi]$ with speed $cr_0$, so its length is $2\pi cr_0$. Cut the region $r < r_0$ along a ray and unroll it: it is a Euclidean sector of radius $r_0$ and angle $2\pi c$, whose two edges are glued, a cone of angle $2\pi c$. A smooth metric is Euclidean to first order at the centre, where small circles have circumference $2\pi r + O(r^2)$; for $dr^2 + \varphi^2d\theta^2$ the circumference is $2\pi\varphi(r) = 2\pi\varphi'(0)r + O(r^2)$ (since $\varphi(0) = 0$), so $\varphi'(0) = 1$ is necessary. (The even derivatives must also vanish for smoothness of higher order.)

Solution

For each curve, $|\gamma'|_{\lambda^2g} = \lambda|\gamma'|_g$, so $L_{\lambda^2g}(\gamma) = \lambda L_g(\gamma)$, and taking the infimum over curves gives $\lambda d$. In coordinates, $\det(\lambda^2g_{ij}) = \lambda^{2n}\det g_{ij}$, so $dV_{\lambda^2g} = \lambda^ndV_g$. Hence $\operatorname{Vol}(S^n(r)) = r^n\operatorname{Vol}(S^n(1))$, for instance $4\pi r^2$ for $n = 2$.

Solution

Parametrise a ray by Euclidean arc length $s$, so its velocity is $(\cos\alpha, \sin\alpha)$. The metric does not depend on $x$, so the momentum $g(\gamma', \partial_x) = n(y)^2\cos\alpha$ is conserved along geodesics parametrised proportionally to $g$-arc length; converting between the two parametrisations divides by the speed $n(y)$, leaving $n(y)\cos\alpha$ constant. (The conservation of the momentum of a symmetry is Noether's theorem, or Clairaut's relation, 9A.3 Geodesics and the Exponential Map.) A ray descending into lower $n$ must increase $\cos\alpha$ to keep the product fixed, so it flattens; it reaches $\cos\alpha = 1$ at the height where $n(y) = n(y_0)\cos\alpha_0$, and then turns upward. An observer receives rays from the sky arriving from below the horizon, and sees the sky's image on the road.

Solution

$\sin r = r - \frac{r^3}{6} + O(r^5)$ and $\sinh r = r + \frac{r^3}{6} + O(r^5)$, so $2\pi\sin r = 2\pi r(1 - \frac{r^2}{6} + \dots)$, $K = 1$, and $2\pi\sinh r = 2\pi r(1 + \frac{r^2}{6} + \dots)$, $K = -1$.