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Update: I was able to figure this out on my own. The problem was that I assumed the $\phi$ coordinate runs from $0$ to $2\pi$, but the metric so defined has a conical singularity. To eliminate the conical singularity, $\phi$ must run from 0 to $2\pi(1+B^2)$. Including this factor in the Gauss-Bonnet formula makes everything work out. Neat.

Trying to learn some geometry, I would like to understand how the Gauss-Bonnet theorem works for a simple surface, a 2-sphere with metric

$f(\theta) d\theta^2 + f(\theta)^{-1} \sin^2\theta d\phi^2$,

where

$f(\theta)=1+B^2 \cos(\theta)^2$

and $B$ is a parameter (a real number between 0 and 1, say).

When I compute the Gaussian curvature, $K$, I find that it depends on $B$. But the area element, $dA=\sqrt{g}d\theta d\phi = \sin\theta d\theta d\phi$ is independent of $B$. So $\int K dA$ depends on $B$. But this can't be right, because Gauss-Bonnet says $\int K dA=4\pi$ is a constant. I must be missing something basic... I need a hint!

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You are trying to understand Gauss-Bonnet Theorem using geodesic polar coordinates, fair enough. But do not begin with an arbitrary metric. You can use only valid general metric coefficients from first fundamental form isometric creatures for a sphere or a constant K surface. At this stage it even need not have rotational symmetry. We have

$ ds^2 = du^2 +G( u,\phi) d\phi^2 $ , then go through Christoffel symbols valid for chosen geodesic polar coordinates.

Refer to Equns (3-4) to (3-6), pp132, Ch 4-3, Lectures on classical Differential Geometry by D.J.Struik Second Edition for this treatment.

Standard form is ( I am more comfortable with classical u notation rather than $\theta$ for radial lines as it designates a linear distance from center to the curved circle periphery u = constant, or along meridian of a surface of revolution as the case may be). Letting $ K = 1/a^2, $

$ ds^2 = du^2 +( a \sin( u/a) d\phi) ^2 $. Now Gauss-Bonnet tallies perfectly.

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