Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
Home/Resources/Geometry, Dynamics, and Operators for Economic Models/III-04
Local Geometry and Policy Paths · Chapter III-04

Curvature, Transport, and Holonomy

When two policy directions are lifted to the state space, applying them in different orders can produce different latent outcomes. Their commutator is the leading local measure of this order effect. Transport also changes target covectors, so a closed path in reported variables may alter both the hidden state and the response assigned to the next policy.

Conceptual map

  1. III-04.01Vector fields and order
  2. III-04.02Curvature of a lift
  3. III-04.03Target changes along paths
  4. III-04.04Covector transport
  5. III-04.05Levels of closure

1. Lifted policy directions

Two policies can generate the same closed path in reported variables and leave different latent states because their lifted state motions fail to commute. The leading discrepancy is a Lie bracket evaluated at the starting state.

Let \(h\): \(M\to \mathbb{R}^{2}\) be a report map and let \(X\), \(Y\) be smooth vector fields satisfying \(Dh X=e_{1}\) and \(Dh Y=e_{2}\). Their flows are denoted \(\Phi ^{t}_{X}\) and \(\Phi ^{t}_{Y}\). Fix the bracket convention \([X,Y]=DY\cdot X-DX\cdot Y\).

Definition 1 · Small policy loop

Starting from \(x\), apply \(X\), then \(Y\), then their reversals for duration \(\varepsilon\). The loop map is \(L_{\varepsilon}=\Phi ^{-\varepsilon}_{Y}\circ\Phi ^{-\varepsilon}_{X}\circ\Phi ^{\varepsilon}_{Y}\circ\Phi ^{\varepsilon}_{X}\). Its report returns to the starting value through order \(\varepsilon ^{2}\) when the two report directions are constant.

2. The commutator expansion

Proposition 1 · Leading loop displacement

If \(X\) and \(Y\) are three times continuously differentiable near \(x\), then

\[L_{\varepsilon }(x)-x = \varepsilon ^{2}[X,Y](x)+O(\varepsilon ^{3}).\](1)

If \(Dh X\) and \(Dh Y\) are constant coordinate directions, \(Dh[X,Y]=0\). The leading displacement is vertical and remains hidden from the report.

Derivation. Expand each flow as \(\Phi ^{\varepsilon}_{X}(x)=x+\varepsilon X(x)+\frac{1}{2}\varepsilon ^{2}DX(x)X(x)+O(\varepsilon ^{3})\) and compose the four expansions. First-order terms cancel. The remaining cross terms are \(DY\cdot X-DX\cdot Y\). Applying \(Dh\) and differentiating the constant identities \(Dh X=e_{1}\), \(Dh Y=e_{2}\) gives the verticality statement. ∎

For a scalar target \(\tau\), the leading target change is \(\varepsilon ^{2}D\tau_{x}[X,Y]\). This is a target-specific curvature statistic. A state displacement can be economically silent for one target and active for another.

3. A loop with an exact vertical residue

Use state coordinates (\(r_{1},r_{2},z\)) and report \(h(r_{1},r_{2},z)=(r_{1},r_{2})\). Let

\[X=(1,0,0), \quad Y=(0,1,r_{1}).\](2)

The bracket is \([X,Y]=(0,0,1)\). Starting at the origin, the four legs give (\(\varepsilon\),0,0), (\(\varepsilon,\varepsilon,\varepsilon ^{2}\)), (0,\(\varepsilon,\varepsilon ^{2}\)), and finally (0,0,\(\varepsilon ^{2}\)). The report closes exactly while the latent coordinate increases by \(\varepsilon ^{2}\). Reversing the loop changes the sign.

For target \(\tau=z\), the area-normalized contrast equals one at every \(\varepsilon\) in this example. For target \(\tau=r_{1}+r_{2}\), it equals zero. The policy loop identifies a directional and target-indexed object.

4. Transport of future responses

A future marginal response is a covector \(\alpha_{x}\in T_{x}^{*}M\). Moving the state changes both its evaluation point and the coordinate representation used to compare responses. Pulling covectors back along the flow places them in the starting cotangent space. The loop difference of pulled-back covectors records response holonomy.

Failure case · Declaring closure at the wrong level

A closed report path establishes \(h(L_{\varepsilon}x)=h(x)\) for the chosen loop. It leaves open the latent state, a future response, and the full equilibrium correspondence. Each closure claim needs its own map and tolerance.

5. Implementation, exercises, and sources

  1. Integrate both loop orientations from the same initial state and solver tolerances.
  2. Record report closure, latent displacement, and target displacement separately.
  3. Divide the orientation difference by \(2\varepsilon ^{2}\) and compare it with the analytic bracket.
  4. Repeat over an \(\varepsilon\) grid to diagnose higher-order and integration errors.
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Exercises

  1. Compute the bracket in (2) from the Jacobians of the two vector fields.
  2. Add a term \(r_{2}^{2}\) to the vertical component of \(X\) and derive the base-point bracket.
  3. Construct a target whose first-order loop response vanishes and whose next nonzero term is cubic.
Partial solutions

1. \(DX=0\), while the derivative of \(Y\) in the \(r_{1}\) direction is (0,0,1). Hence \(DY\cdot X=(0,0,1)\). 2. For \(X\)=(1,0,\(r_{2}^{2}\)), the added derivative \(DX\cdot Y\) has vertical component \(2r_{2}\); at the origin this term vanishes, so the base-point bracket remains (0,0,1).

  1. John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Chapters 8–9.Flows, Lie brackets, and commuting vector fields.
  2. Velimir Jurdjevic, Geometric Control Theory, Chapter 2.Vector fields, flows, and control-generated directions.
  3. R. W. Brockett (1972), “System Theory on Group Manifolds and Coset Spaces,” SIAM Journal on Control 10, 265–284.Geometric control interpretation of noncommuting motions.

6. Audit checkpoint

State the bracket convention, loop orientation, lifted fields, closure map, target, base point, scale, integration method, and remainder rate. Orientation and convention determine the sign of the reported curvature.

7. Scope boundary

The chapter treats local commutator effects of smooth policy lifts. Global holonomy groups, principal bundles, and Riemannian curvature tensors are beyond the calculation used here.

Prerequisites