I know how to calculate the Gaussian curvature of an embedded surface using first and second fundamental forms, but how does one calculate the curvature of a non-embedded surface like the hyperbolic plane? Is there some general method to employ? The example I have in mind is the open unit disc in $\mathbb{R^2}$ equipped with the metric $(dx^2+dy^2)/(f(r))^2$ where $r=(x^2+y^2)^{1/2}$ as usual, and $f$ is suitably constrained, but I would prefer general hints rather than a detailed solution.

  • $\begingroup$ Gaussian curvature is a special case of sectional curvature when the manifold is two-dimensional. If you know how to calculate sectional curvature intrinsically, you should also be able to calculate Gaussian curvature in the same way. Is that enough of a hint without giving it all away? $\endgroup$ – Muphrid Apr 6 '13 at 15:26
  • $\begingroup$ You know how to calculate the Gaussian curvature using first and second fundamental forms, but the so-called Theorema Egregium (by Gauss) says that really only the first fundamental form is needed. That's why it makes perfect sense to calculate the surface curvature from the "intrinsic" metric itself, without having the surface embedded in some three- or four-dimensional space. $\endgroup$ – Jeppe Stig Nielsen Apr 6 '13 at 15:26
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    $\begingroup$ @Muphrid If the asker knew of curvature of general $n$-dimensional Riemannian manifolds, and knew about sectional curvature, I don't think he would have asked this question. $\endgroup$ – Jeppe Stig Nielsen Apr 6 '13 at 15:29
  • $\begingroup$ What is your definition of Gaussian curvature? $\endgroup$ – treble Apr 7 '13 at 1:29

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