Riemannian geometry · The Lumenary

MotivationH

Measuring curvature from the inside

In the 1820s Carl Friedrich Gauss led a geodetic survey of the Kingdom of Hanover, triangulating the land with theodolites and measuring the angles of large triangles. Out of that work came a startling theorem: a two-dimensional creature living on a surface, measuring only lengths and angles within the surface, can determine its curvature. Gauss called it the Theorema Egregium — the remarkable theorem. A sheet of paper can be rolled into a cylinder without stretching, so the cylinder is "flat" to its inhabitants; no piece of a sphere can be flattened without distortion, which is why every map of the Earth distorts something.

Riemann turned this into a programme (1854): describe a space of any dimension by an inner product on each tangent space — a way to measure infinitesimal lengths and angles — and derive everything else from it. Sixty years later Einstein used exactly this geometry, in its Lorentzian form, to describe gravity; today the equations of general relativity are corrected for in every GPS receiver. In pure mathematics, Riemannian geometry links local information (curvature) to global information (topology): the total curvature of a closed surface is $2\pi$ times its Euler characteristic, and a complete manifold whose Ricci curvature is bounded below by a positive constant must be compact.

This lesson assumes smooth manifolds and the classical differential geometry of curves and surfaces. For the Gauss–Bonnet theorem it uses the Poincaré–Hopf theorem from differential topology.

IntuitionH

Straight lines and their spreading

On a curved surface there are no straight lines, but there are geodesics: curves that are "as straight as possible", along which an ant walking forward never turns left or right. On a sphere they are the great circles. Two ants that leave the north pole walking straight ahead along different meridians move apart at first, and then — because the sphere curves — come together again at the south pole. On a saddle-shaped surface nearby geodesics spread apart faster than in the plane. That rate of spreading is curvature: positive curvature focuses geodesics, negative curvature disperses them.

A second picture: carry a spear along a closed path, always keeping it "parallel to itself" — never rotating it within the surface. In the plane it returns pointing the same way. On the sphere, carry it from the north pole down a meridian to the equator, along the equator a quarter of the way round, and back up to the pole: it returns rotated by $90^\circ$ , which is the area of the triangle you walked round (it covers one eighth of a sphere of area $4\pi$ ). Curvature is the infinitesimal rotation of parallel transport round small loops, and for geodesic triangles the total turn equals the angle excess $\alpha + \beta + \gamma - \pi$ .

DefinitionsH

Metrics, connections, geodesics and curvature

Definition— Riemannian metric

A Riemannian metric $g$ on a smooth manifold $M$ is a choice of inner product $g_p = \langle\cdot,\cdot\rangle_p$ on each tangent space $T_pM$ , depending smoothly on $p$ : in a chart, $g = \sum_{i,j} g_{ij}\, dx^i\, dx^j$ with $(g_{ij})$ smooth, symmetric and positive definite. The length of a piecewise smooth curve is $L(\gamma) = \int_a^b |\gamma'(t)|\, dt$ , and the Riemannian distance $d(p, q)$ is the infimum of lengths of curves from $p$ to $q$ ; it is a metric inducing the manifold topology. An isometry is a diffeomorphism preserving $g$ .

Every smooth manifold admits Riemannian metrics (glue Euclidean metrics from charts with a partition of unity); a metric is extra structure, not something the manifold has automatically.

Definition— connection, Levi-Civita connection

An (affine) connection is a map $(X, Y) \mapsto \nabla_X Y$ on vector fields that is $C^\infty(M)$ -linear in $X$ , $\R$ -linear in $Y$ , and satisfies the Leibniz rule $\nabla_X(fY) = (Xf)Y + f\nabla_X Y$ . It is torsion-free if $\nabla_X Y - \nabla_Y X = [X, Y]$ , and compatible with $g$ if $X\langle Y, Z\rangle = \langle \nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle$ . The Levi-Civita connection is a torsion-free connection compatible with $g$ . In coordinates $\nabla_{\partial_i}\partial_j = \sum_k \Gamma^k_{ij}\partial_k$ , with Christoffel symbols $\Gamma^k_{ij}$ .

Definition— parallel transport, geodesic, exponential map

A connection differentiates vector fields $V(t)$ along a curve $\gamma$ : $D_tV = \sum_k \big(\dot V^k + \sum_{i,j}\Gamma^k_{ij}\dot\gamma^i V^j\big)\partial_k$ . $V$ is parallel if $D_tV = 0$ ; parallel transport along $\gamma$ is a linear isometry between tangent spaces (for the Levi-Civita connection). A geodesic is a curve with $D_t\dot\gamma = 0$ , i.e.

\ddot\gamma^k + \sum_{i,j}\Gamma^k_{ij}(\gamma)\,\dot\gamma^i\dot\gamma^j = 0 \qquad (k = 1, \dots, n).

For $v \in T_pM$ let $\gamma_v$ be the geodesic with $\gamma_v(0) = p$ , $\dot\gamma_v(0) = v$ ; the exponential map is $\exp_p(v) = \gamma_v(1)$ , defined near $0 \in T_pM$ .

Definition— curvature

The curvature tensor is $R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X,Y]}Z$ ; it is $C^\infty(M)$ -linear in all three arguments, so $R_p$ is a multilinear map on $T_pM$ . Write $R(X, Y, Z, W) = \langle R(X, Y)Z, W\rangle$ . For a $2$ -plane $\sigma \subseteq T_pM$ spanned by $X, Y$ , the sectional curvature is

K(\sigma) = \frac{R(X, Y, Y, X)}{|X|^2|Y|^2 - \langle X, Y\rangle^2},

independent of the basis. The Ricci curvature is $\operatorname{Ric}(Y, Z) = \sum_i R(e_i, Y, Z, e_i)$ for an orthonormal basis $(e_i)$ , so $\operatorname{Ric}(v, v) = \sum_{i \ge 2} K(v, e_i)$ for a unit vector $v = e_1$ ; the scalar curvature is $\sum_i \operatorname{Ric}(e_i, e_i)$ . On a surface all three carry the same information: the Gaussian curvature $K$ .

Definition— Jacobi field, conjugate point

A Jacobi field along a geodesic $\gamma$ is a vector field $J$ along $\gamma$ satisfying the Jacobi equation $D_t^2 J + R(J, \dot\gamma)\dot\gamma = 0$ . Jacobi fields are exactly the variation fields of families of geodesics. A point $\gamma(t_0)$ is conjugate to $\gamma(0)$ if a non-zero Jacobi field vanishes at both $t = 0$ and $t = t_0$ .

Worked examplesH

Model spaces and surfaces of revolution

Example— The three model geometries

Euclidean space $\R^n$ , $g = \sum (dx^i)^2$ : all $\Gamma^k_{ij} = 0$ , geodesics are straight lines, $R = 0$ .
The round sphere $S^n \subset \R^{n+1}$ with the induced metric: geodesics are great circles; sectional curvature $\equiv 1$ . The sphere of radius $r$ has $K \equiv 1/r^2$ .
Hyperbolic space, for example the upper half-plane $\{(x, y) : y > 0\}$ with $g = (dx^2 + dy^2)/y^2$ . Its non-zero Christoffel symbols are $\Gamma^x_{xy} = \Gamma^x_{yx} = -1/y$ , $\Gamma^y_{xx} = 1/y$ , $\Gamma^y_{yy} = -1/y$ , so the geodesic equations are

\ddot x - \frac{2}{y}\dot x\dot y = 0, \qquad \ddot y + \frac{\dot x^2 - \dot y^2}{y} = 0 .

Vertical lines $t \mapsto (0, e^t)$ and semicircles centred on the real axis, such as $t \mapsto (\tanh t, \operatorname{sech} t)$ , are unit-speed geodesics (Lean checks the semicircle below), and $K \equiv -1$ .

In each model space the curvature tensor is $R(X, Y)Z = k\big(\langle Y, Z\rangle X - \langle X, Z\rangle Y\big)$ with $k = 0, 1, -1$ ; then every sectional curvature is $k$ and $\operatorname{Ric} = (n - 1)k\, g$ .

Example— Surfaces of revolution and Clairaut's relation

Rotating a unit-speed profile curve $t \mapsto (f(t), z(t))$ , $f > 0$ , about the $z$ -axis gives the metric $g = dt^2 + f(t)^2\, d\theta^2$ . Meridians ( $\theta$ constant) are geodesics; a parallel ( $t = t_0$ ) is a geodesic exactly when $f'(t_0) = 0$ (the equator of a sphere, the outer and inner equators of a torus). The geodesic equation for $\theta$ is $\ddot\theta + 2(f'/f)\dot t\dot\theta = 0$ , which says precisely that

\frac{d}{ds}\big(f^2\dot\theta\big) = 2ff'\dot t\dot\theta + f^2\ddot\theta = 0 .

For a unit-speed geodesic, $f\dot\theta = \cos\psi$ where $\psi$ is the angle with the parallel, so Clairaut's relation holds: $f\cos\psi$ is constant along every geodesic. A geodesic can therefore never reach the region where $f$ is smaller than this constant — it turns back at the latitude where $f = |f\cos\psi|$ , as the second experiment shows on a torus.

Example— A flat torus and a flat cylinder

$\R^2/\Z^2$ with the metric induced from $\R^2$ is a flat torus: $K \equiv 0$ , and its closed geodesics are the images of lines with rational slope. The cylinder $S^1 \times \R$ is flat too — a rolled-up sheet of paper. The doughnut-shaped torus in $\R^3$ is diffeomorphic to the flat torus but not isometric to it: its curvature is $K = \cos v/\big(r(R + r\cos v)\big)$ , positive on the outside and negative on the inside.

ProofH

The fundamental theorem of Riemannian geometry

Theorem— Theorem 1 (Levi-Civita)

On every Riemannian manifold there is exactly one connection that is torsion-free and compatible with the metric. It is given by the Koszul formula

2\langle \nabla_X Y, Z\rangle = X\langle Y, Z\rangle + Y\langle Z, X\rangle - Z\langle X, Y\rangle - \langle Y, [X, Z]\rangle - \langle X, [Y, Z]\rangle + \langle Z, [X, Y]\rangle,

and in coordinates by $\Gamma^k_{ij} = \tfrac12 \sum_l g^{kl}\big(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}\big)$ .

Proof

Uniqueness. Let $\nabla$ be torsion-free and compatible. Expand three instances of compatibility:

X\langle Y, Z\rangle + Y\langle Z, X\rangle - Z\langle X, Y\rangle = \langle\nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle + \langle \nabla_Y Z, X\rangle + \langle Z, \nabla_Y X\rangle - \langle\nabla_Z X, Y\rangle - \langle X, \nabla_Z Y\rangle .

Use torsion-freeness in the form $\nabla_X Z - \nabla_Z X = [X, Z]$ , $\nabla_Y Z - \nabla_Z Y = [Y, Z]$ and $\nabla_Y X = \nabla_X Y - [X, Y]$ : the right-hand side becomes $2\langle\nabla_X Y, Z\rangle + \langle Y, [X, Z]\rangle + \langle X, [Y, Z]\rangle - \langle Z, [X, Y]\rangle$ , which is the Koszul formula. Its right-hand side does not involve $\nabla$ , and since the metric is non-degenerate, $\langle \nabla_X Y, Z\rangle$ for all $Z$ determines $\nabla_X Y$ . So there is at most one such connection.

Existence. Define $\nabla_X Y$ by the Koszul formula: one checks that the right-hand side is $C^\infty(M)$ -linear in $Z$ , so by non-degeneracy it defines a vector field $\nabla_X Y$ , and then that this is a connection, torsion-free and compatible — each a direct computation with the Leibniz rule $[X, fY] = (Xf)Y + f[X, Y]$ . Coordinates. For $X = \partial_i$ , $Y = \partial_j$ , $Z = \partial_l$ all brackets vanish, leaving $2\sum_k\Gamma^k_{ij}g_{kl} = \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}$ ; multiply by the inverse matrix $g^{kl}$ . ∎

∎

For the half-plane, $g_{ij} = y^{-2}\delta_{ij}$ and $\partial_y g_{ii} = -2y^{-3}$ , which gives the Christoffel symbols listed in the examples; the first experiment computes them symbolically.

ProofH

Geodesics, the exponential map and completeness

Geodesics are solutions of a second-order ODE, so for each $p$ and $v \in T_pM$ there is a unique maximal geodesic $\gamma_v$ , and $\gamma_{cv}(t) = \gamma_v(ct)$ . The exponential map is smooth near $0$ with $d(\exp_p)_0 = \mathrm{id}$ , so by the inverse function theorem it is a diffeomorphism from a ball $B_\varepsilon(0) \subset T_pM$ onto a normal neighbourhood of $p$ .

Theorem— Theorem 2 (geodesics minimise locally; stated)

(Gauss lemma) Radial geodesics from $p$ meet the geodesic spheres $\exp_p(\partial B_r(0))$ orthogonally. Consequently, if $\exp_p$ is a diffeomorphism on $B_\varepsilon(0)$ , then for $|v| < \varepsilon$ the radial geodesic $t \mapsto \exp_p(tv)$ , $0 \le t \le 1$ , is the unique shortest curve (up to reparametrisation) from $p$ to $\exp_p(v)$ . Conversely, every length-minimising curve parametrised by arc length is a geodesic.

(Proofs: do Carmo, Chapter 3; Lee, Introduction to Riemannian Manifolds, Chapter 6.) Locally, then, geodesics are shortest paths. Globally they need not be: a great-circle arc of length greater than $\pi$ on the unit sphere is a geodesic, but the complementary arc is shorter. The obstruction is the conjugate point at distance $\pi$ (Theorem 5 below).

Theorem— Theorem 3 (Hopf–Rinow; stated)

For a connected Riemannian manifold $M$ the following are equivalent: (a) $(M, d)$ is a complete metric space; (b) every geodesic is defined for all time; (c) $\exp_p$ is defined on all of $T_pM$ for some $p$ ; (d) closed bounded subsets of $M$ are compact. If they hold, any two points of $M$ are joined by a geodesic of length $d(p, q)$ .

(Proof: do Carmo, Chapter 7.) Compact manifolds are complete. $\R^2 \setminus \{0\}$ is not: the points $(1, 0)$ and $(-1, 0)$ have distance $2$ , but no curve of length $2$ joins them.

ProofH

The curvature tensor

Theorem— Theorem 4 (symmetries of the curvature tensor)

For all $X, Y, Z, W$ : (i) $R(X,Y,Z,W) = -R(Y,X,Z,W)$ ; (ii) $R(X,Y,Z,W) = -R(X,Y,W,Z)$ ; (iii) $R(X,Y,Z,W) + R(Y,Z,X,W) + R(Z,X,Y,W) = 0$ (first Bianchi identity); and (iv) $R(X,Y,Z,W) = R(Z,W,X,Y)$ .

Proof

(i) is clear from the definition. (ii) is metric compatibility: it suffices to show $R(X, Y, Z, Z) = 0$ , and $XY\langle Z, Z\rangle = 2X\langle\nabla_Y Z, Z\rangle = 2\langle\nabla_X\nabla_Y Z, Z\rangle + 2\langle \nabla_Y Z, \nabla_X Z\rangle$ ; subtract the same with $X, Y$ exchanged and $[X, Y]\langle Z, Z\rangle = 2\langle\nabla_{[X,Y]}Z, Z\rangle$ to get $0 = (XY - YX - [X,Y])\langle Z, Z\rangle = 2R(X, Y, Z, Z)$ . (iii) For coordinate fields (all brackets $0$ ) the cyclic sum is $\nabla_X(\nabla_Y Z - \nabla_Z Y) + \nabla_Y(\nabla_Z X - \nabla_X Z) + \nabla_Z(\nabla_X Y - \nabla_Y X) = 0$ by torsion-freeness, and $R$ is tensorial. (iv) follows from (i)–(iii) alone: write the Bianchi identity (iii) for the four cyclic arrangements $(X,Y,Z,W)$ , $(Y,Z,W,X)$ , $(Z,W,X,Y)$ , $(W,X,Y,Z)$ and add them; using (i) and (ii), eight of the twelve terms cancel in pairs and the rest give $2R(X,Y,Z,W) - 2R(Z,W,X,Y) = 0$ . ∎

∎

Step (iv) is purely algebraic, and Lean checks it below for an arbitrary function $R$ satisfying (i)–(iii) — no linearity needed. By (i)–(iv), the quadratic form $(X, Y) \mapsto R(X, Y, Y, X)$ determines $R$ , so the sectional curvatures determine the curvature tensor. In particular, if all sectional curvatures at $p$ equal $k$ , then $R(X,Y,Z,W) = k\big(\langle Y, Z\rangle\langle X, W\rangle - \langle X, Z\rangle\langle Y, W\rangle\big)$ and $\operatorname{Ric} = (n-1)k\, g$ .

Curvature of a surface from an orthonormal frame. Let $(e_1, e_2)$ be a positively oriented orthonormal frame on an open set of an oriented surface, with dual coframe $(\theta^1, \theta^2)$ , and let $\omega(X) = \langle\nabla_X e_1, e_2\rangle$ be its connection $1$ -form; then $\nabla e_1 = \omega\otimes e_2$ and $\nabla e_2 = -\omega\otimes e_1$ .

Theorem— Lemma (structure equation)

$d\omega = -K\, dA$ , where $dA = \theta^1\wedge\theta^2$ is the area form.

Proof

$d\omega(X, Y) = X\omega(Y) - Y\omega(X) - \omega([X, Y])$ , and $X\omega(Y) = X\langle\nabla_Y e_1, e_2\rangle = \langle\nabla_X\nabla_Y e_1, e_2\rangle + \langle\nabla_Y e_1, \nabla_X e_2\rangle$ . The last term is $\langle \omega(Y)e_2, -\omega(X)e_1\rangle = 0$ . Hence $d\omega(X, Y) = \langle R(X, Y)e_1, e_2\rangle$ , and $d\omega(e_1, e_2) = R(e_1, e_2, e_1, e_2) = -R(e_1, e_2, e_2, e_1) = -K$ . ∎

∎

Example— Curvature of dt² + f(t)² dθ²

Take $e_1 = \partial_t$ , $e_2 = f^{-1}\partial_\theta$ , so $\theta^1 = dt$ , $\theta^2 = f\,d\theta$ . From $d\theta^1 = 0$ and $d\theta^2 = f'\,dt\wedge d\theta$ (Cartan's first structure equations $d\theta^1 = \omega\wedge\theta^2$ , $d\theta^2 = -\omega\wedge\theta^1$ , which encode torsion-freeness) we find $\omega = f'\, d\theta$ . Then $d\omega = f''\, dt\wedge d\theta = -K f\, dt \wedge d\theta$ , so

K = -\frac{f''}{f}.

$f = \sin t$ (round sphere) gives $K = 1$ ; $f = t$ (the plane in polar coordinates) gives $0$ ; $f = \sinh t$ (hyperbolic plane in polar coordinates) gives $-1$ . For a conformal metric $e^{2\varphi}(dx^2 + dy^2)$ a similar computation gives $K = -e^{-2\varphi}\Delta\varphi$ ; for the half-plane, $\varphi = -\log y$ and $K = -y^2 \cdot y^{-2} = -1$ .

Both formulas use only the metric, so the Gaussian curvature is intrinsic. For a surface in $\R^3$ , Gauss's extrinsic curvature (the determinant of the shape operator) equals this $K$ : that identification is the Theorema Egregium.

ProofH

Jacobi fields and conjugate points

If $\gamma_s$ is a smooth family of geodesics, the variation field $J = \partial_s\gamma_s|_{s=0}$ satisfies $D_t^2J = D_tD_s\dot\gamma = D_sD_t\dot\gamma + R(\dot\gamma, J)\dot\gamma = -R(J, \dot\gamma)\dot\gamma$ , the Jacobi equation. So Jacobi fields describe how nearby geodesics spread.

Example— Constant curvature

Let $\gamma$ be a unit-speed geodesic in a space of constant curvature $k$ , $E(t)$ a parallel unit field orthogonal to $\dot\gamma$ , and $J = u(t)E(t)$ . Then $R(J, \dot\gamma)\dot\gamma = k\,u\,E$ , and the Jacobi equation is $u'' + ku = 0$ . With $u(0) = 0$ , $u'(0) = 1$ :

u(t) = \frac{\sin(\sqrt k\, t)}{\sqrt k}\ (k > 0), \qquad u(t) = t\ (k = 0), \qquad u(t) = \frac{\sinh(\sqrt{-k}\, t)}{\sqrt{-k}}\ (k < 0).

For $k > 0$ the Jacobi field vanishes again at $t = \pi/\sqrt k$ : the point $\gamma(\pi/\sqrt k)$ is conjugate to $\gamma(0)$ — on the sphere of radius $1/\sqrt k$ , the antipode. For $k \le 0$ there are no conjugate points.

Theorem— Theorem 5 (Jacobi; stated)

A geodesic that contains a point conjugate to its starting point in its interior is not locally length-minimising: some nearby curve with the same endpoints is shorter.

(Proof via the index form: do Carmo, Chapter 5, or Milnor, Morse Theory, §15.) For the unit sphere this explains the failure of great-circle arcs longer than $\pi$ . In the other direction, Cartan–Hadamard: if $M$ is complete, simply connected and has $K \le 0$ , then $\exp_p : T_pM \to M$ is a diffeomorphism, so $M$ is diffeomorphic to $\R^n$ .

ProofH

The Gauss–Bonnet theorem

Two classical consequences of the structure equation, for a geodesic triangle $T$ (three geodesic sides, interior angles $\alpha, \beta, \gamma$ ) contained in a disc on an oriented surface:

\int_T K\, dA = \alpha + \beta + \gamma - \pi \qquad \text{(local Gauss–Bonnet)}.

(Sketch: $\int_T K\, dA = -\oint_{\partial T}\omega$ by Stokes; along a unit-speed boundary curve making angle $\phi$ with $e_1$ , $\omega(\dot c) = \kappa_g - \dot\phi$ with $\kappa_g = 0$ on geodesic sides; and the total turning of the tangent round $\partial T$ , $\oint d\phi$ plus the exterior angles $\pi - \alpha$ , $\pi - \beta$ , $\pi - \gamma$ , is $2\pi$ by Hopf's Umlaufsatz.) The widget below measures both sides on the sphere and in the hyperbolic plane. The global theorem needs no triangulation if we use a vector field.

Theorem— Theorem 6 (Gauss–Bonnet)

For every closed oriented Riemannian $2$ -manifold $M$ ,

\int_M K\, dA = 2\pi\,\chi(M).

Proof

(Following Chern.) Choose a smooth vector field $X$ on $M$ with finitely many zeros $p_1, \dots, p_k$ (for example the gradient of a Morse function). On $M^* = M \setminus \{p_1, \dots, p_k\}$ let $e_1 = X/|X|$ , let $e_2$ complete it to a positive orthonormal frame, and let $\omega$ be the connection form, so $d\omega = -K\, dA$ on $M^*$ . Let $B_i$ be small closed geodesic discs of radius $\varepsilon$ round $p_i$ , and $M_\varepsilon = M \setminus \bigcup_i \operatorname{int} B_i$ . Its boundary is $\bigcup_i \partial B_i$ with the orientation opposite to that of $\partial B_i$ as the boundary of $B_i$ . By Stokes,

\int_{M_\varepsilon} K\, dA = -\int_{M_\varepsilon} d\omega = -\int_{\partial M_\varepsilon}\omega = \sum_{i=1}^k \oint_{\partial B_i}\omega ,

each circle now traversed anticlockwise. On the disc $B_i$ choose any positive orthonormal frame $(f_1, f_2)$ , with connection form $\omega_f$ , smooth on all of $B_i$ . On $B_i \setminus \{p_i\}$ write $e_1 = \cos\phi\, f_1 + \sin\phi\, f_2$ with a locally defined angle $\phi$ . Differentiating, $\nabla e_1 = (\omega_f + d\phi)\otimes e_2$ , so $\omega = \omega_f + d\phi$ . As we go once round $\partial B_i$ , $\phi$ increases by $2\pi$ times the number of turns of $X$ relative to the frame $f$ , which is $2\pi\operatorname{ind}_{p_i}(X)$ . Hence

\oint_{\partial B_i}\omega = \oint_{\partial B_i}\omega_f + 2\pi\operatorname{ind}_{p_i}(X).

As $\varepsilon \to 0$ , $\oint_{\partial B_i}\omega_f \to 0$ (a smooth form integrated over a curve of length $\to 0$ ) and $\int_{M_\varepsilon} K\, dA \to \int_M K\, dA$ . Therefore $\int_M K\, dA = 2\pi\sum_i\operatorname{ind}_{p_i}(X) = 2\pi\chi(M)$ by the Poincaré–Hopf theorem. ∎

∎

Consequences: a metric on the sphere must have positive curvature somewhere, and a metric on a surface of genus $g \ge 2$ must have negative curvature somewhere; a closed orientable surface with $K > 0$ everywhere is a sphere, and one with $K \equiv 0$ is a torus. The left-hand side is local geometry, the right-hand side global topology. Chern's generalisation (1944) expresses the Euler characteristic of a closed even-dimensional manifold as the integral of a polynomial in the curvature (the Pfaffian).

ProofH

The Bonnet–Myers theorem

We need one more tool. For a geodesic $\gamma : [0, L] \to M$ with unit speed and a proper variation $\gamma_s$ (endpoints fixed) with variation field $V \perp \dot\gamma$ , the energy $E(c) = \frac12\int_0^L |\dot c|^2\, dt$ satisfies the second variation formula (do Carmo, Chapter 9):

\frac{d^2}{ds^2}\Big|_{s=0} E(\gamma_s) = \int_0^L \Big( |D_tV|^2 - R(V, \dot\gamma, \dot\gamma, V)\Big)\, dt .

Theorem— Theorem 7 (Bonnet–Myers)

Let $M$ be a complete connected Riemannian $n$ -manifold ( $n \ge 2$ ) with $\operatorname{Ric} \ge (n-1)k\, g$ for a constant $k > 0$ . Then $\operatorname{diam} M \le \pi/\sqrt k$ ; hence $M$ is compact, and its fundamental group is finite.

Proof

Let $p, q \in M$ . By Hopf–Rinow there is a unit-speed geodesic $\gamma : [0, L] \to M$ from $p$ to $q$ with $L = d(p, q)$ . Suppose $L > \pi/\sqrt k$ . Choose parallel fields $E_1, \dots, E_{n-1}$ along $\gamma$ that together with $\dot\gamma$ are orthonormal, and let $V_i(t) = \sin(\pi t/L)E_i(t)$ , proper variation fields with $D_tV_i = (\pi/L)\cos(\pi t/L)E_i$ . Summing the second variations,

\sum_{i=1}^{n-1} E''_{V_i}(0) = \int_0^L \Big( (n-1)\frac{\pi^2}{L^2}\cos^2\frac{\pi t}{L} - \sin^2\frac{\pi t}{L}\,\operatorname{Ric}(\dot\gamma, \dot\gamma) \Big) dt \le (n - 1) \int_0^L \Big(\frac{\pi^2}{L^2}\cos^2\frac{\pi t}{L} - k\sin^2\frac{\pi t}{L}\Big) dt .

The last integral equals $\frac{L}{2}\big(\frac{\pi^2}{L^2} - k\big)$ , which is negative because $L > \pi/\sqrt k$ . So some $E''_{V_i}(0) < 0$ , and the variation $\gamma_s$ in the direction $V_i$ has $E(\gamma_s) < E(\gamma)$ for small $s \ne 0$ . By Cauchy–Schwarz $L(\gamma_s)^2 \le 2L\,E(\gamma_s) < 2L\,E(\gamma) = L(\gamma)^2$ , so $\gamma_s$ is a shorter curve from $p$ to $q$ , contradicting $L = d(p, q)$ . Hence $d(p, q) \le \pi/\sqrt k$ for all $p, q$ .

$M$ is complete and bounded, so it is compact by Hopf–Rinow (d). Its universal cover, with the pulled-back metric, is complete and satisfies the same Ricci bound, so it is compact too; a covering of a compact space by a compact space has finitely many sheets, so $\pi_1(M)$ is finite. ∎

∎

The bound is sharp: the unit sphere has $\operatorname{Ric} = (n-1)g$ and diameter $\pi$ (Cheng's theorem says it is the only case of equality). The hypothesis needs a positive lower bound: the paraboloid $z = x^2 + y^2$ is complete with $K > 0$ everywhere but $K \to 0$ at infinity, and it is not compact. Bonnet–Myers is the first of the comparison theorems — Rauch, Toponogov, Bishop–Gromov — which compare a manifold with the model space of the same curvature bound.

Formal mathematicsF

Riemannian geometry in Lean and Mathlib

Mathlib's Riemannian geometry is young. It has Riemannian metrics on vector bundles, the class IsRiemannianManifold I M (the distance equals the infimum of lengths of $C^1$ paths), the Riemannian distance riemannianEDist and its basic properties, and the theorem that an inner product space is a Riemannian manifold. We cite two of these:

euclidean_riemannian_distanceLean 4Applies a library theoremFormally verified

open Bundle in
/-- **In Euclidean space, the Riemannian distance is the usual distance**: the infimum of the
lengths of `C¹` paths from `x` to `y` equals `‖x - y‖`, attained by the straight segment. This is
Mathlib's instance `IsRiemannianManifold 𝓘(ℝ, F) F` for an inner product space `F`. -/
theorem euclidean_riemannian_distance {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
    (x y : F) : edist x y = riemannianEDist (modelWithCornersSelf ℝ F) x y :=
  IsRiemannianManifold.out x y