3
$\begingroup$

I am currently doing some exercises to prepare for an Oral next month and have found myself stuck on computing Christoffel symbols for a certain metric $h$ defined below. The setup is as follows:

We have a chart $(\theta^1,\theta^2)$ on $T^2$ and a map $\Phi : T^2 \to \mathbb{R}^3$ \begin{align*} \Phi(\theta^1,\theta^2) = \left((2+\cos\theta^1)\cos\theta^2, (2+\cos\theta^1)\sin\theta^2,\sin \theta^1\right). \end{align*} We are asked to define a metric $h$ on $T^2$ as the pullback of the Euclidean metric $$h= \Phi^*(dx^i \otimes dx^i)$$ and then compute the Christoffel symbols for $\nabla$ on $(T^2,h)$. My progress thus far has been to compute $$ h(\xi,\eta)=\left( \xi^j \frac{\partial\Phi^m}{\partial\theta^j} \right) \left( \eta^j \frac{\partial\Phi^m}{\partial\theta^j}\right), $$ with summation over $m,j$ implicit. Writing out the matrix for $h$ is tractable. I find $$ \begin{bmatrix} 5-4\sin\theta^1& 2 (2-\sin \theta^1)(2+\cos \theta^1)\sin \theta^2 \cos \theta^2\\ 2 (2-\sin \theta^1)(2+\cos \theta^1)\sin \theta^2 \cos \theta^2 & \left(2+\cos\theta^1\right)^2 \end{bmatrix} $$ However, when I start computing the Christoffel symbols I run into what feels like a computational nightmare, mainly due to the rather ugly inverse matrix for $h$. Is there any way around this or should I just suck it up (also, did I make a computation error?)?

Edit: Let me just add that Mathematica gives me the following inverse matrix (to motivate my lack of computational morals with $x=\theta^1,y=\theta^2$)

$$ \left( \begin{array}{cc} \frac{(\cos (x)+2)^2}{(\cos (x)+2)^2 (5-4 \sin (x))-4 (\cos (x)+2)^2 \cos ^2(y) (2-2 \sin (x))^2 \sin ^2(y)} & -\frac{2 (\cos (x)+2) \cos (y) (2-2 \sin (x)) \sin (y)}{(\cos (x)+2)^2 (5-4 \sin (x))-4 (\cos (x)+2)^2 \cos ^2(y) (2-2 \sin (x))^2 \sin ^2(y)} \\ -\frac{2 (\cos (x)+2) \cos (y) (2-2 \sin (x)) \sin (y)}{(\cos (x)+2)^2 (5-4 \sin (x))-4 (\cos (x)+2)^2 \cos ^2(y) (2-2 \sin (x))^2 \sin ^2(y)} & \frac{5-4 \sin (x)}{(\cos (x)+2)^2 (5-4 \sin (x))-4 (\cos (x)+2)^2 \cos ^2(y) (2-2 \sin (x))^2 \sin ^2(y)} \\ \end{array} \right) $$

$\endgroup$
  • 2
    $\begingroup$ Yeah, you have a computation error. See win.tue.nl/~rvhassel/Onderwijs/Tensor-ConTeX-Bib/…. $\endgroup$ – levap Feb 25 at 13:45
  • $\begingroup$ That looks like a very useful resource. Thank you for the link. $\endgroup$ – afightingchance Feb 25 at 13:59
  • 1
    $\begingroup$ It's correct. For me, it's simpler to write (the matrix representing) $h$ as $\begin{pmatrix} \left< \frac{\partial \Phi}{\partial \theta^1}, \frac{\partial \Phi}{\partial \theta^1} \right> & \left< \frac{\partial \Phi}{\partial \theta^2}, \frac{\partial \Phi}{\partial \theta^1} \right> \\ \left< \frac{\partial \Phi}{\partial \theta^2}, \frac{\partial \Phi}{\partial \theta^1} \right> & \left< \frac{\partial \Phi}{\partial \theta^2}, \frac{\partial \Phi}{\partial \theta^2} \right> \end{pmatrix}$ where $\left< \cdot, \cdot \right>$ is the standard inner product on $\mathbb{R}^3$. $\endgroup$ – levap Feb 25 at 15:20
  • 1
    $\begingroup$ This is equivalent to your expression. $\endgroup$ – levap Feb 25 at 15:21
  • 1
    $\begingroup$ Thanks again. Your expression indeed looks neater. I kind of derived mine without knowing about fundamental forms (I read up on this after your first link). I mustve made a mistake typing it into mathematica then... $\endgroup$ – afightingchance Feb 25 at 15:34
1
$\begingroup$

Your $h_{22}$ is right, but the mixed 12 and 21 components should be 0, since the torus coordinates are orthogonal, and $h_{11}$ is 1. This is also obvious since you are changing latitude on a "tube" with unit radius.

Since you use Mathematica you should perhaps check the xAct tensor package, www.xact.es

With the xPrint GUI for xAct, which I developed, learning to use xAct is much quicker. Please see my profile for direct link (I have problems with my mobile browser right now).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.