Gaussian curvature for a given metric

[mahdisadjadi]

Homework Statement

Assume that we have a metric like:
$$ds^{2}=f dr^{2}+ g d\theta^{2}+ h d\varphi^{2}$$

where $$r,\theta , \varphi$$ are spherical coordinates.
f,g and h are some functions of r and theta but not phi.

Homework Equations

How can I calculate Gaussian curvature in r-theta, r-phi and theta-phi plane(2D)?
And also how for original metric(3D)?

The Attempt at a Solution

 

[HallsofIvy]

See the "Brioschi formula" on this page:
http://en.wikipedia.org/wiki/Gaussian_curvature#Alternative_Formulas

 

[mahdisadjadi]

@ HallsofIvy

Thanks!

 

[mahdisadjadi]

After 1 month work on this problem, I found out following remarks:

1. If we tend to use "Brioschi formula" to calculate Gaussian curvature of a surface, we should embed it into a 2D space. For example, in the given metric, we should take r=constant to achieve a surface in theta-phi 2D space.

2. It we like to calculate Gaussian curvature of higher dimensional space, we can use Riemann Curvature Tensor and determinant of metric, as follows:
$$K=\frac{R_{1212} }{g}$$
where $g$ is determinant of metric matrix and $R_{1212}$ is a component of Riemann Curvature Tensor. This $K$ gives Gaussian curvature of the plane which is perpendicular to third axis.

 

[HallsofIvy]

I was kind of hoping we wouldn't have to calculate the Riemann tensor! Yes, the Brioschi formula only works for a two dimensional surface embeded in three dimensions- which was exactly your situation.