I want to find for which cases this metric can define an sphere: $$\frac{1}{P^2}\left(\mathrm d\theta^2+\sin^2 \theta\; \mathrm d\phi^2\right)$$ where $P=\sin^2 \theta+K\cos^2 \theta$, with $K$ the gaussian curvature (constant).
I know from the post Spherical metric multiply by a function that the curvature doesn't change if you multiply the metric by a differentiable function, so I suppose that the only condition is that $K$ has to be positive, right?
HINT:
There should be something on the left hand side of a metric and physical dimensions should match.
The metric of a sphere $ \phi=$latitude, $\theta = $longitude.
$$ ds^2 = a^2 d\phi^2 + a^2 \cos^{2}\phi \,d\theta^2 ; K = 1/a^2 \,; $$
$$ K ds^2 = d\phi^2 + \cos^{2}\phi \,d\theta^2 $$