---
title: "Notes on The Proof of The Metric Curvature Formula"
subtitle: "Based on Needham's VDGAF"
format: html
---

# The Metric Induced Vector Field
Given a surface with a metric we define a metric induced vector field (MIVF) whose natural properties encode the holonomy, and thus the curvature of the surface.

Assume the (u,v) coordinates refer to (horizontal, vertical) coordinates on the surface.  The first coordinate of the MIVF is P and is, roughly, the rate of change of the length of a horizontal vector as we move vertically.  The second coordinate of the MIVF is Q and is, roughly, the rate of change of the length of a vertical vector as we move horizontally.

For a small loop $\hat{R}$ with sides of length $A du$ and $B dv$ the holonomy is determined by the cumulative change of a parallel translated vector over each edge of the loop.  It intuitively makes sense that the change in the first (horizontal) edge is given by the rate of change of a vertical vector as we move horizontally.  This gives $$ \frac{\partial_v A}{B}$$ from the geometric argument of Needham.  The chain rule explains the presence of $\partial_v A \delta v$ and the $\delta_u$ determines the length of the magnification of change over the edge.  Similarly, the second edge contributes $$ \frac{\partial_u B}{A} $$ but in the opposite direction since we are moving "backwards" along that edge.  

Here it is important to note that we are building the *holonomy element*, (like the area element that we integrate to get area, or the arc length element we integrate to get arc length) which is the infinitesimal contribution to holonomy from a small loop.  Thus we multiply each rate of change by the length of the edge over which it is applied (reminder, this is happening in the (u,v)  map), giving

$$\text{holonomy element} = \left( \frac{\partial_v A}{B} dv - \frac{\partial_u B}{A} du \right) $$

The holonomy of a general loop on the surface is thus the integral of the holonomy element in the coordinate plane over the region enclosed by the pre-image of the loop.

$$\text{holonomy(R)} = \iint_{R} \left( \frac{\partial_v A}{B} dv - \frac{\partial_u B}{A} du \right)  $$


This integral over a small region can be thought of as the limit of a sum over a small rectangle, where we we approximate the value vector field over the edges of the loop by evaluating at the midpoints (a,b,c,d) of each edge.  Thus the holonomy can be computed as 


$$ \text{holonomy(R)} \asymp \sum \left( \frac{\partial_v A(a,b)}{B(a,b)} \delta v - \frac{\partial_u B(b,c)}{A(b,c)} \delta u + \frac{\partial_v A(c,d)}{B(c,d)} (-\delta v) - \frac{\partial_u B(d,a)}{A(d,a)} (- \delta u) \right) .$$

Rearranging terms gives
$$ \asymp  \sum \left( \frac{\partial_v A(a,b)}{B(a,b)} - \frac{\partial_v A(c,d)}{B(c,d)} \right) \delta v + \left( \frac{\partial_u B(d,a)}{A(d,a)} - \frac{\partial_u B(b,c)}{A(b,c)} \right) \delta u $$

which by the Mean Value Theorem can be expressed as
$$ \asymp \sum  \left( \partial_u \left( \frac{\partial_v A (e,f)}{B (e,f)} \right)  \delta u \, \delta v - \partial_v \left( \frac{\partial_u B  (g,h) }{A(g,h)} \right) \delta u \, \delta v \right) $$

for some points (e,f) and (g,h) in the rectangle.  As the rectangle shrinks, these points approach the same point (u0,v0) in the rectangle, giving

$$  \asymp \sum \left( \partial_u \left( \frac{\partial_v A (u_0,v_0)}{B (u_0,v_0)} \right) - \partial_v \left( \frac{\partial_u B  (u_0,v_0) }{A(u_0,v_0)} \right) \right) \delta u \, \delta v $$.

Suppressing the (u0,v0) notation for clarity, and taking the limit we have
$$ \text{holonomy(R)} = \iint_R \left( \partial_u \left( \frac{\partial_v A }{B } \right) - \partial_v \left( \frac{\partial_u B   }{A} \right) \right) du \, dv $$.


This last part is nothing other than the curl of the MIVF evaluated at the point (u0,v0) times the area element du dv.  Thus we have shown that the holonomy element is given by the curl of the MIVF times the area element.

$$ \text{holonomy(R)} = \iint_R \left( \text{curl (MIVF)} \right) du \, dv $$

Now imagine $R$ is a loop shrinking to a central point $p$.  Then the curl(MIVF) becomes independint of $u,v$ since in the limit it approaches a constant (the curvature at point p).  Thus, it factors out of the integral and we have:

$$ \text{holonomy(R)}  = \text{curl (MIVF)} \cdot \text{area(R)} = \text{curl (MIVF)} \iint_R du \, dv $$  


We should note that , by Green's Theorem - which states that the circulation of a vector field around a small loop is given by the curl of the vector field times the area element, and the work above we can define the holonomy as the circulation of the MIVF around the loop.

$$\text{holonomy(R)} = \oint_{\partial R} \text{(MIVF)} \cdot ds = \iint_R \text{curl (MIVF)} \, du \, dv $$
