Connections

To differentiate a function you compare its values at nearby points. To differentiate a vector field you would compare vectors at nearby points, but they live in different tangent spaces, and a manifold gives no way to compare them. In $\mathbb{R}^n$ you differentiate component by component, because all tangent spaces are identified with $\mathbb{R}^n$. On a manifold, the partial derivatives $\partial_jX^i$ of the components depend on the coordinates and do not form a tensor. The Lie derivative of 8A.6 Flows and the Lie Derivative compares vectors by dragging them along a flow, but it depends on the whole vector field, not just its direction at a point.

A connection is the extra structure that fixes this, and a Riemannian metric determines one: the Levi-Civita connection. It gives a covariant derivative $\nabla$ of every tensor field, the $\nabla_i$ in every formula of the Ricci flow. It also gives parallel transport, a way of carrying a vector along a curve without "turning" it. On a curved manifold, parallel transport around a loop returns a vector rotated, and the rotation measures the curvature inside the loop. Physics shows this in the Foucault pendulum, in gyroscopes orbiting the Earth, and in light twisting through a coiled optical fibre.

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

• state the axioms of a connection and compute with Christoffel symbols;
• derive the Levi-Civita connection from the Koszul formula, and its Christoffel symbols from the metric;
• differentiate tensor fields covariantly, in index notation;
• solve the parallel transport equation along a curve, and compute holonomy around a circle on the sphere;
• explain why the difference of two connections, and hence the time derivative of the Christoffel symbols under the Ricci flow, is a tensor.

2.1The Foucault pendulum

2.2Connections

An affine connection on $M$ is a map $(X, Y) \mapsto \nabla_XY$ taking two vector fields to a vector field, such that

• $\nabla_XY$ is linear over functions in $X$: $\nabla_{fX + hX'}Y = f\nabla_XY + h\nabla_{X'}Y$;
• $\nabla_XY$ is linear over $\mathbb{R}$ in $Y$ and satisfies the product rule $\nabla_X(fY) = (Xf)\,Y + f\,\nabla_XY$.

The first property means $(\nabla_XY)(p)$ depends only on $X_p$, so $\nabla_vY$ makes sense for a single vector $v$: it is the derivative of $Y$ in the direction $v$. The second makes it a derivative. In coordinates, set

the $n^3$ functions $\Gamma_{ij}^k$ are the Christoffel symbols, and the axioms give

The Christoffel symbols are the correction to the naive componentwise derivative. They are not the components of a tensor: they depend on the coordinates, and they can be made to vanish at any one point by a choice of coordinates (9A.3 Geodesics and the Exponential Map). But the difference of two connections, $\nabla_XY - \tilde\nabla_XY$, is linear over functions in both $X$ and $Y$ (the $(Xf)Y$ terms cancel), so it is a $(1, 2)$-tensor field (8A.7 Tensors and Index Notation).

Covariant derivatives of tensors. A connection extends uniquely to all tensor fields so that it agrees with $Xf$ on functions, obeys the product rule for tensor products, and commutes with contractions. On a 1-form, $(\nabla_X\omega)(Y) = X(\omega(Y)) - \omega(\nabla_XY)$. The total covariant derivative $\nabla T$ of a $(k, l)$-tensor is a $(k, l + 1)$-tensor, with components written $\nabla_iT^{\cdots}_{\cdots}$ (8A.7 Tensors and Index Notation):

with one $+\Gamma$ for each upper index and one $-\Gamma$ for each lower index. Second derivatives are $\nabla^2T = \nabla(\nabla T)$, with components $\nabla_i\nabla_jT$; for a function, $\nabla_i\nabla_jf = \partial_i\partial_jf - \Gamma_{ij}^k\partial_kf$ is the Hessian (9A.6 The Laplacian and the Bochner Formula).

2.3The Levi-Civita connection

A connection on a Riemannian manifold is

• compatible with the metric if $X\langle Y, Z\rangle = \langle\nabla_XY, Z\rangle + \langle Y, \nabla_XZ\rangle$, that is, $\nabla g = 0$;
• torsion-free (symmetric) if $\nabla_XY - \nabla_YX = [X, Y]$, that is, $\Gamma_{ij}^k = \Gamma_{ji}^k$ in every chart.

From now on $\nabla$ is always the Levi-Civita connection. Isometries preserve it: if $\phi$ is an isometry, $\phi_*(\nabla_XY) = \nabla_{\phi_*X}\phi_*Y$, because the Koszul formula involves only the metric and brackets.

The geometric picture. For a submanifold $M \subset \mathbb{R}^N$ with the induced metric, the Levi-Civita connection is

the ordinary derivative of $Y$ in $\mathbb{R}^N$ followed by orthogonal projection to $TM$ (Exercise 2.5). On the sphere: differentiate in space, then throw away the part normal to the sphere. The thrown-away part is the second fundamental form of 8A.9 The Curvature of Surfaces (9A.8 Submanifolds and Minimal Surfaces).

Examples. In Euclidean coordinates all $\Gamma_{ij}^k = 0$. In polar coordinates on the plane, $g = dr^2 + r^2d\theta^2$, the nonzero symbols are $\Gamma_{\theta\theta}^r = -r$ and $\Gamma_{r\theta}^\theta = \Gamma_{\theta r}^\theta = \frac1r$. For the warped product $dr^2 + \varphi(r)^2d\theta^2$, which includes the sphere ($\varphi = \sin r$, $r$ the colatitude),

and all others vanish (Exercise 2.3).

2.4Parallel transport

Along a curve $\gamma : I \to M$, a vector field along $\gamma$ is a smooth assignment $V(t) \in T_{\gamma(t)}M$; the velocity $\gamma'$ is one. A connection gives its covariant derivative along $\gamma$,

which equals $\nabla_{\gamma'}\tilde V$ for any extension $\tilde V$ of $V$. $V$ is parallel along $\gamma$ if $D_tV = 0$. This is a linear system of ordinary differential equations for the components $V^k(t)$, so by 2B.10 Ordinary Differential Equations each initial vector $V(t_0)$ extends uniquely to a parallel field along the whole curve. The map

taking $V(t_0)$ to $V(t_1)$ is parallel transport. It is linear, invertible, and, for the Levi-Civita connection, an isometry: if $V$ and $W$ are parallel, $\frac{d}{dt}\langle V, W\rangle = \langle D_tV, W\rangle + \langle V, D_tW\rangle = 0$.

Parallel transport recovers the connection: $\nabla_vY = \frac{d}{dt}\big|_{t = 0}P^\gamma_{t\to0}Y(\gamma(t))$ for any curve with $\gamma'(0) = v$. Covariant differentiation is differentiation after using parallel transport to bring the vectors into one tangent space.

Holonomy. Parallel transport around a closed loop at $p$ is an isometry of $T_pM$, the holonomy of the loop. In $\mathbb{R}^n$ it is always the identity. On the sphere it is a rotation: around a geodesic triangle, by the triangle's area (Figure 2.1); around a simple loop on any surface, by $\iint_DK\,dA$, the total curvature enclosed. This is Gauss–Bonnet (8A.9 The Curvature of Surfaces) in another form, and it is the first of the meanings of curvature in 9A.4 Curvature and What It Means: curvature is the infinitesimal holonomy.

The holonomy can be nontrivial even where the curvature vanishes, if the loop surrounds a singular point or a hole. A flat cone with cone angle $\alpha$ has holonomy rotation $2\pi - \alpha$ around its apex. This is how the Foucault pendulum is usually explained: the cone tangent to the Earth along a circle of latitude has the same parallel transport along that circle as the sphere, and the cone can be unrolled flat (Figure 2.2).

2.5History

Elwin Bruno Christoffel introduced the symbols named after him in 1869, studying when two quadratic differential forms are equivalent. Gregorio Ricci-Curbastro and his student Tullio Levi-Civita developed the "absolute differential calculus" of tensors in a joint memoir of 1900, the calculus Einstein learned for general relativity. Parallel transport was introduced by Levi-Civita in 1917; Hermann Weyl and Élie Cartan generalised connections in the following years, and Cartan introduced holonomy in the 1920s. The Koszul formula is named after Jean-Louis Koszul. Foucault's pendulum dates from 1851; Berry's geometric phase from 1984.

2.6Exercises

Solution

$g_{rr} = 1$, $g_{\theta\theta} = r^2$, and only $\partial_rg_{\theta\theta} = 2r$ is nonzero. $\Gamma_{\theta\theta}^r = \frac12g^{rr}(-\partial_rg_{\theta\theta}) = -r$, and $\Gamma_{r\theta}^\theta = \frac12g^{\theta\theta}\partial_rg_{\theta\theta} = \frac{2r}{2r^2} = \frac1r$. A tensor that vanishes in one coordinate system vanishes in all (its components transform linearly), so the $\Gamma$ cannot be one.

Solution

The only nonconstant component is $g_{\theta\theta} = \varphi^2$, with $\partial_rg_{\theta\theta} = 2\varphi\varphi'$. Then $\Gamma_{\theta\theta}^r = -\frac12\partial_rg_{\theta\theta} = -\varphi\varphi'$ and $\Gamma_{r\theta}^\theta = \frac{1}{2\varphi^2}\cdot2\varphi\varphi' = \frac{\varphi'}{\varphi}$; every other combination involves a derivative of a constant component. On the sphere: $\Gamma_{\theta\theta}^r = -\sin r\cos r$ and $\Gamma_{r\theta}^\theta = \cot r$.

Solution

With $V = A\,\partial_r + B\,\partial_\theta$ and $\gamma' = \partial_\theta$, $D_tV = (A' + \Gamma_{\theta\theta}^rB)\partial_r + (B' + \Gamma_{\theta r}^\theta A)\partial_\theta$, so $A' = \sin r_0\cos r_0\,B$ and $B' = -\cot r_0\,A$. With $a = A$, $c = B\sin r_0$: $a' = \cos r_0\,c$ and $c' = -\cos r_0\,a$, a rotation at rate $\cos r_0$ (clockwise relative to the frame $(\partial_r, \partial_\theta/\sin r_0)$). After time $2\pi$ the angle is $2\pi\cos r_0$ in that sense, and $-2\pi\cos r_0 \equiv 2\pi - 2\pi\cos r_0 = 2\pi(1 - \cos r_0)$ modulo $2\pi$. With $\cos r_0 = \sin\varphi$: the swing plane turns by $2\pi\sin\varphi$ relative to the meridian frame, which is the floor's frame, in one day.

Solution

The connection axioms are inherited from $\bar D$, since projection is linear over functions. Compatibility: for $Y, Z$ tangent, $X\langle Y, Z\rangle = \langle\bar D_XY, Z\rangle + \langle Y, \bar D_XZ\rangle$, and since $Z$ and $Y$ are tangent, only the tangential parts contribute: $= \langle\nabla_XY, Z\rangle + \langle Y, \nabla_XZ\rangle$. Torsion: $\nabla_XY - \nabla_YX = (\bar D_XY - \bar D_YX)^\top = [X, Y]^\top = [X, Y]$. By uniqueness in Theorem 2.1, $\nabla$ is the Levi-Civita connection.

Solution

Away from the apex the cone is flat and the sector is an isometric chart in which the Christoffel symbols vanish, so parallel fields have constant components: translation. Carry a vector around: in the sector it stays fixed, but the two edges are glued by a rotation through $\alpha$ about the apex, so the vector arrives at the glued edge rotated by $\alpha$, that is, by $-(2\pi - \alpha)$ modulo $2\pi$, relative to where it would have to be to match. The tangent cone along colatitude $r_0$ has slant distance $\tan r_0$ from the circle to the apex and circle circumference $2\pi\sin r_0$, so its development is a sector of radius $\tan r_0$ and angle $\frac{2\pi\sin r_0}{\tan r_0} = 2\pi\cos r_0$. The holonomy, $2\pi - 2\pi\cos r_0$, matches Exercise 2.4, because the cone and the sphere have the same tangent planes along the circle, and hence the same parallel transport along it.

Solution

$\nabla f = df$ has components $\partial_jf$, and the rule for a 1-form gives $\nabla_i(\partial_jf) = \partial_i\partial_jf - \Gamma_{ij}^k\partial_kf$. Both terms are symmetric in $i, j$, the second because the Levi-Civita connection is torsion-free. At a critical point, $\partial_kf = 0$, so the $\Gamma$ term vanishes.

Solution

$\partial_t\Gamma = \lim_{s\to0}\frac1s(\Gamma(t + s) - \Gamma(t))$ is a limit of differences of connections, which are tensors, so it is a tensor. Differentiate $\Gamma_{ij}^k = \frac12g^{kl}(\partial_ig_{jl} + \partial_jg_{il} - \partial_lg_{ij})$ in $t$: the term with $\partial_tg^{kl}$ multiplies $\Gamma$, which vanishes at $p$, leaving $\frac12g^{kl}(\partial_ih_{jl} + \partial_jh_{il} - \partial_lh_{ij})$. At $p$, where $\Gamma = 0$, partial and covariant derivatives agree, so this is $\frac12g^{kl}(\nabla_ih_{jl} + \nabla_jh_{il} - \nabla_lh_{ij})$. Both sides are tensors that agree at $p$ in one coordinate system, so they agree everywhere. Substituting $h = -2\operatorname{Ric}$ gives the stated formula.