I am looking for some intuition into a way of looking at the Gauss curvature on a surface that describes it as the divergence of a potential function - at least locally.(adsbygoogle = window.adsbygoogle || []).push({});

I am not sure exactly what the intuition is - but this way of looking at things seems suggestive. Any insight is welcome.

In isothermal coordinates on a 2 dimensional Riemannian manifold the metric takes the form

e[itex]^{f}[/itex](dx[itex]^{2}[/itex] + dy[itex]^{2}[/itex])

The Gauss curvature is -Divergence of gradf , that is it equals

e[itex]^{-2f}[/itex]([itex]\partial^{2}[/itex]f/[itex]\partial[/itex]x[itex]^{2}[/itex] + [itex]\partial^{2}[/itex]f/[itex]\partial[/itex]y[itex]^{2}[/itex])

The Divergence Theorem in 2 dimensions now says that the total curvature inside the region if the total flux of grad f through the boundary.

So the Gauss curvature is analogous to the source of a potential field.

What is the intuition/picture for this ways of interpreting the Gauss curvature? Maybe there is an interpretation of this in Physics?

If one does not like that this as a purely local result things can be improved. Take any meromorphic 1 form on the surface. In each isothermal coordinate system it looks like

gdz where g is a meromorphic function. Away from the singularities of these g's

the ratios e[itex]^{f}[/itex]/|g|[itex]^{2}[/itex] fit together across isothermal charts to give a function on the entire remainder of the surface. Apply the Divergence theorem to the log of this function. Note that log|g| is harmonic. (I read about this construction in a survey paper on the Gauss Bonnet Theorem for Complex manifolds.)

I also wonder whether using the merormorphic 1 form in this way adds some intuition to this way of looking a Gauss curvature.

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# Curvature as the source of a field on a two dimensional Riemannian manifold

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