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Let D be an open disc centred at the origin in $ \Bbb R^2 $. Give D a Riemannian metric of the form $ (dx^2 + dy^2)/f(r)^2 $, where $ r = \sqrt{x^2 + y^2} $ and $ f(r) > 0 $. Show that the Gauss curvature of this metric is $ K = f f'' - (f')^2 + f f' / r $.

This looks like it shouldn't be too bad, but I can't work out how to do it! My supervisor said that it isn't too bad, just pretty messy, and we didn't have time to do it in the supervision. If someone would be able to help me get (substantially) underway, then I'd be most appreciative! (I don't expect a full typset solution since this would probably take quite a long time!)

This is an example sheet question, completely non-examinable!

Thanks! :)

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Cartesian coordinates are isothermal for your metric, so writing $\lambda = 1/f(r)$ you have the standard formula $K = -(\Delta \log \lambda)/\lambda^{2}$. (The computation is indeed a bit messy, but the formula comes out as stated.)

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  • $\begingroup$ Thanks for your response! :) Is $\Delta$ in "$K = -(\Delta \log \lambda)/\lambda^{2}$" the Laplacian ($\nabla^2$)? This isn't a formula that I have been given in lectures... is there some more elementary other (still straightforward, but messy) way of doing it? $\endgroup$
    – Sam OT
    Mar 7, 2014 at 8:42
  • $\begingroup$ The formulae that I have for 2D are all in terms of geodesic polar coordinates - using the ratio of the determinants of the two fundamental forms is only given in 3D and I can't see how to make this work for 2d... $\endgroup$
    – Sam OT
    Mar 7, 2014 at 8:44
  • $\begingroup$ You can do it calculating the christoffel symbols $\Gamma_{ij}^k$ and with these the connection and curvature $\endgroup$ Mar 7, 2014 at 8:48
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    $\begingroup$ @SmileySam: Yes, $\Delta$ denotes the ordinary Laplacian. Your expression for the metric is intrinsic, and not every such metric embeds isometrically in $\mathbf{R}^{3}$ (much less as a surface of rotation), so you'll have to use an intrinsic expression for $K$. The formula proposed is elementary (in the realm of a first course in differential geometry) and arguably the simplest for your situation; my crystal ball says it's the approach your professor has in mind. :) $\endgroup$ Mar 7, 2014 at 12:19
  • $\begingroup$ Afterthought: If it turns out you must deduce an intrinsic formula for the Gaussian curvature from first principles, I'd recommend using connection forms (learning them if necessary; they're explicit and simple to work with in two dimensions) rather than Christoffel symbols. $\endgroup$ Mar 7, 2014 at 12:28

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