Elements of differential geometry, by hand

Companion to the notes on Chapter 5 of Amari's Information Geometry and Its Applications. Back to the notes. Each widget is one idea of the chapter made touchable: a tangent vector as a random variable, what the Christoffel symbols measure, geodesics under three different connections, parallel transport and holonomy, the loop formula for curvature, and the difference between intrinsic and embedding curvature. The numbers shown are the ones printed by code/diffgeo.py.

1. A tangent vector is a derivative, an arrow, and a random variable

On the Gaussian manifold with coordinates $\xi=(\mu,\sigma)$, a tangent vector $A=A^\mu\partial_\mu+A^\sigma\partial_\sigma$ is an arrow in the parameter plane. The book also identifies $\partial_i$ with the score $\partial_i\log p(x;\xi)$ (5.10), so $A$ is the random variable $f(x)=A^i\partial_i\log p$. It has mean $0$, and its second moment is $A^\mathsf{T}gA$: the squared Fisher length of the arrow. Left: the arrow, drawn in coordinates $(\mu/\sqrt2,\sigma)$ in which the Fisher metric is conformal, so the dashed unit Fisher circle really is a circle; an arrow reaching beyond it has length above 1. Right: the density, the density moved a little along $A$, and below it $f(x)$ built from the two basis scores (dashed) and, as dots, built a second way from the components in the natural coordinates $\theta$. The dots land on the curve: components change contravariantly, basis vectors covariantly, the vector does not change.

2. What a Christoffel symbol measures: the polar frame turning

In the plane with polar coordinates $(r,\vartheta)$ the basis vectors $e_r,e_\vartheta$ change from point to point; $e_\vartheta$ even has length $r$. The book's definition (5.18)–(5.20) is: map the frame at the nearby point back to the first tangent space; the change is $de_i=\Gamma_{ki}{}^je_j\,d\xi^k$. Drag the point and the step. The picture shows $e_r,e_\vartheta$ at $P$ (solid), the frame at $P'=P+d\xi$ moved back to $P$ (dashed), and the difference (green). The table solves for the exact components of that difference in the basis at $P$, and compares them with the first-order prediction from $\Gamma_{\vartheta\vartheta}{}^r=-r$ and $\Gamma_{r\vartheta}{}^\vartheta=\Gamma_{\vartheta r}{}^\vartheta=1/r$ (everything else vanishes). The plane is flat; these numbers are non-zero only because the coordinates are curved.

$e_r$$e_\vartheta$ (length $r$)frame at $P'$, moved back$de_i$

3. Geodesics depend on the connection, not just the manifold

The geodesic equation $\ddot\xi^k+\Gamma_{ij}{}^k\dot\xi^i\dot\xi^j=0$ (5.54) needs a connection. On the Gaussian manifold there are three natural ones. The e-connection declares the coordinate lines of $\theta=(\mu/\sigma^2,-1/2\sigma^2)$ straight, the m-connection those of $\eta=(\mu,\mu^2+\sigma^2)$, and the Levi-Civita connection is the one tied to the Fisher metric; it is their average. Start all of them at the same point with the same velocity. The black dashed curve is a connection in between, $(1-s)\Gamma^e+s\Gamma^m$. The picture uses $(\mu/\sqrt2,\sigma)$ so that Levi-Civita geodesics are true semicircles. Only the Levi-Civita curve keeps its speed, and only it is also the shortest curve between its endpoints (Theorem 5.2): compare the Fisher length of each curve with the Fisher–Rao distance between its endpoints.

emLevi-Civita$(1-s)\Gamma^e+s\Gamma^m$

4. Parallel transport round a loop: holonomy

Carry a vector round the boundary of a coordinate rectangle $P\to Q\to R\to S\to P$ by solving (5.58). On a flat manifold it comes back unchanged. On the sphere it comes back rotated by the enclosed area times $K=1$; on the Gaussian manifold by area times $K=-1/2$, the other way round. In the plane with polar coordinates the components change all along the way, but the Cartesian arrow does not move: all the arrows are parallel and the loop does nothing. On the Gaussian manifold choose the connection with $s$: $s=0$ is e, $s=1$ is m, $s=\tfrac12$ is Levi-Civita. For e and m the loop returns the vector exactly (the connections are flat), but its Fisher length changes on the way; only $s=\tfrac12$ preserves it, and only $s=\tfrac12$ gives a pure rotation. Also read the last line: the two routes from $P$ to $R$ arrive at different vectors (path dependence, (5.59)).

starttransported along the wayafter the loop

5. The loop formula: no factor 2

The book derives curvature from a tiny quadrilateral $P,Q,R,S$ with sides $d_1\xi$ and $d_2\xi$: transport $A$ from $P$ to $R$ along $PQR$ and along $PSR$ and subtract. Its (5.65) reads $A_{21}-A_{12}=R_{jkl}{}^iA^l(d_1\xi^jd_2\xi^k-d_1\xi^kd_2\xi^j)$. Here the experiment is done honestly with the Runge–Kutta transport of widget 4 and the curvature tensor of (5.66). Shrink $\varepsilon$: the measured $(A_{21}-A_{12})/\varepsilon^2$ (green) converges to $R_{jkl}{}^iA^l\,u_1^ju_2^k$ (black), half of the printed right-hand side (red dashed), because the printed bracket counts each pair $(j,k)$ twice. The e-connection has $R=0$: everything collapses to zero.

measured $(A_{21}-A_{12})/\varepsilon^2$$R_{jkl}{}^iA^l u_1^ju_2^k$printed (5.65): twice that

6. Flat inside, curved outside: unrolling a cylinder

Bend a flat sheet into a cylinder. The sheet's coordinates $(s,z)$ never change, the metric stays $\mathrm{diag}(1,1)$, the induced connection stays $0$, and a straight line stays a geodesic: the intrinsic (Riemann–Christoffel) curvature is $0$ throughout. What changes is the embedding: the acceleration of the geodesic in space is purely normal, with size $\kappa=u/R$ per unit speed squared, the embedding curvature $H_{ss}$ of (5.101)–(5.102). The Gauss curvature $K=\det H/\det g=\kappa\cdot0=0$. Move the bend $u$ from 0 (flat sheet) to 1 (full cylinder of radius $R$), and turn the view. Compare the sphere, where $H_{ab}=-g_{ab}/\rho$ in both directions, $\det H\neq0$, and $K=1/\rho^2$: no bending of a sheet can produce it.

geodesic (straight in the sheet)acceleration in space (normal)