Curvature and Circulation

Definitions

• Let \(L\) be a small rectangle in the \((u,v)\) plane, \(\hat{L}\) its image on \(H^2\).
• Let \(V = (P,Q)\) be a vector field on the \((u,v)\) plane induced by the metric on \(H^2\):
  • \(P = \frac{\partial_v(A)}{B}\)
  • \(Q = -\frac{\partial_u(B)}{A}\)
• The circulation of \(V\) around the boundary of \(L\) is defined as: \[ C_V(L) = \oint_{L} V \cdot d\vec{s} = \oint_L P du + Q dv \]

Calculation

\[ C_V(L) = \oint_L V \cdot dr = \oint_L \frac{\partial_v(A)}{B} du -\frac{\partial_u(B)}{A} dv \]

In the hyperbolic plane, \(A = B = \frac{1}{v}\), so

\[ C_V(L) = \oint_L v \partial_v\left(\frac{1}{v}\right) du - v \partial_u\left(\frac{1}{v}\right) dv \]

and this equals

\[ C_V(L) = \oint_L -\frac{1}{v} du + 0 dv = \oint_L -\frac{1}{v} du \]

This reduces the circulation calculation to a line integral of a single-variable function around the rectangle \(L\).

Let \(L\) have corners at \((u,v)\), \((u+\delta u, v)\), \((u+\delta u, v+\delta v)\), and \((u, v+\delta v)\). Then these integrals become

\[\begin{align*} C_V(L) &= \int_{u}^{u+\delta u} -\frac{1}{v} du + \int_{v}^{v+\delta v} 0 dv + \int_{u+\delta u}^{u} -\frac{1}{v+\delta v} du + \int_{v+\delta v}^{v} 0 dv \\ &= -\frac{\delta u}{v} + \frac{\delta u}{v+\delta v} \\ &= \delta u \left( \frac{1}{v+\delta v} - \frac{1}{v} \right) \\ &= \delta u \\left( \frac{v - (v+\delta v)}{v(v+\delta v)} \right) \\ &= -\frac{\delta u \delta v}{v(v+\delta v)} \end{align*}\]