---
title: Curvature via Curl
subtitle: ma541 - f25 - assignment 8
format: 
    pdf:
        include-in-header: 
            file: ../latex-header.tex
---


## Find the curvature of the standard model spaces via their curl as follows.

1. Beginning with the metric for the sphere of radius \(R\):
$$ds^2 = R^2 (\sin^2\phi \, d\theta^2 + d\phi^2)$$ find the metric-induced vector field $V = (P,Q)$ induced by the metric on $S^2$ where

$$ V = \left(\displaystyle\frac{\partial_v(A)}{B}, - \displaystyle\frac{\partial_u(B)}{A} \right). $$

Let $L$ be a small rectangle in the $(u,v)$ plane, $\hat{L}$ its image on $S^2$.  Use the circulation of $V$ around the boundary of $L$ to find an expression for the curvature at a point $(\theta, \phi)$ on the sphere.

2. Repeat the process for the metric of the torus of revolution with parameters $R$ and $r$:
$$ ds^2 = (R + r \cos v)^2 du^2 + r^2 dv^2 $$ to find an expression for the curvature at a point $(u, v)$ on the torus.  
    i) What is the sign of the curvature at points where $v = 0$ or $v = \pi$?   
    ii) What is the sign of the curvature at points where $u = 0$ or $u = \pi/2$?
    iii) Explain how these signs relate to the orientation of the Gauss map at these points.

3. Repeat the process for the hyperbolic plane with metric:
$$ ds^2 = \frac{1}{v^2} (du^2 + dv^2) $$ to verify that the curvature is -1 at all points.

4. (Optional) Use R/Python/MATLAB to visualize the vector fields on the sphere, torus & hyperbolic plane.  Numerically approximate the curvature at various points.  Compare your numerical results to your analytical expressions from parts 1-3.  


