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

lavinia

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.

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.

lavinia

A little reading about heat conduction suggests that the Gauss curvature can be thought of as the strength of a heat source and f, 1/2 log metric scale function in isothermal coordinates ,can be thought of as a steady state temperature. The isothermals would be the curves where f is a constant.

So it seems that given a steady state temperature distribution in a domain arising from a source with strength, K(x,y), then K is the curvature of the metric whose scale function is exp(f).

Feynmann in Lectures on Physics Volume 2 uses an example where a heated plate stretches a ruler, the hotter parts stretching it more, so that the shortest path as measured by a blind bug on the plate will not be a Euclidean straight line. The stretch factor is the metric scale factor, exp(f)I guess. What is the physics that translates the temperature into exp(f) for the stretch factor?

In Feynmann's example the curvature is the source of the temperature gradient.

TrickyDicky

Some come to mind, first in Yang-Mills and EM theory force (the 2-form curvature field Maxwell tensor) is curvature associated to surfaces obtained by using the 4-potential as connection to derive the field force.
Also IIRC I read that in condensed matter physics models are used in which the defects in a substrate are modelled as varying Gaussian curvature in wich the topological defects are treated in a point-like particle way and a geometrical potential V(x) appears related to the gaussian curvature source by its Laplacian: ΔV(x)=-G(x).