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- Jan 26, 2012

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I've posted a bunch of analysis questions as of late. I'm going to change things up a little bit and ask something that involves manifold theory. Here's this week's problem:

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\[\begin{aligned}d\theta^1 &= -\omega\wedge\theta^2\\ d\theta^2 &= \omega\wedge\theta^1\end{aligned}\]

to compute the curvature $K(r)$ of a surface with metric $ds^2=dr^2+f(r)^2d\theta^2$.

(ii) Verify your answer by finding $f$ in the cases where $K=constant$ and checking it with the literature (i.e. show that you end up with spherical, Euclidean, or hyperbolic geometries depending on the value of the constant).

I'll provide suggestions on how to do the problem.

(i) First write the metric in the form $ds^2 = (\theta^1)^2 + (\theta^2)^2$, where $\theta^1$ and $\theta^2$ are coframes (or 1-forms). Curvature is then defined by

\[K(r) = \frac{d\omega}{\theta^1\wedge\theta^2}\]

where $\omega$ is determined from the structure equations

\[\begin{aligned}d\theta^1 &= -\omega\wedge\theta^2\\ d\theta^2 &= \omega\wedge\theta^1.\end{aligned}\]

(ii) Consider the cases $C>0$, $C=0$ and $C<0$; then solve the appropriate differential equations with intial conditions $f(0)=0$ and $f^{\prime}(0)=1$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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**Problem**: (i) Let $\omega$ be a 1-form. Use the structure equations\[\begin{aligned}d\theta^1 &= -\omega\wedge\theta^2\\ d\theta^2 &= \omega\wedge\theta^1\end{aligned}\]

to compute the curvature $K(r)$ of a surface with metric $ds^2=dr^2+f(r)^2d\theta^2$.

(ii) Verify your answer by finding $f$ in the cases where $K=constant$ and checking it with the literature (i.e. show that you end up with spherical, Euclidean, or hyperbolic geometries depending on the value of the constant).

I'll provide suggestions on how to do the problem.

\[K(r) = \frac{d\omega}{\theta^1\wedge\theta^2}\]

where $\omega$ is determined from the structure equations

\[\begin{aligned}d\theta^1 &= -\omega\wedge\theta^2\\ d\theta^2 &= \omega\wedge\theta^1.\end{aligned}\]

(ii) Consider the cases $C>0$, $C=0$ and $C<0$; then solve the appropriate differential equations with intial conditions $f(0)=0$ and $f^{\prime}(0)=1$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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