Difference between "intrinsic"and "parametric" curvature

[hideelo]

I am studying differential geometry of surfaces. I am trying to understand some features of the first fundamental form. The first fundamental form is given by

ds² = α_ijdx_idx_j

Now if the α_ijs are all constants (not functions of your variables) then I think (correct me if I'm wrong) that the surface is not curved. However if they are not then it depends. I am making the following distinction (which might not be legitimate) between what I am calling parametric curvature and intrinsic curvature. Let me give examples of the two.

The plane ℝ² can be parametrized with polar coordinates in which case ds² = dr² + r²dθ². However although the coefficients are not all constants, it is only the coordinates which are curved and not the surface. Most importantly, with the right reparametrization the curvature will go away.

On the other hand the sphere S² with spherical coordinates has as its first fundamental form ds² = dΦ² + cos²θ dθ². In this case the curvature is clearly a property of the surface and is intrinsic and (I'm guessing) will not disappear with simply reparameterizing.

My questions are:

1 How can I distinguish between these?
2 In such cases where reparameterization will help eliminate the curvature, is there a simple way to generate this uncurved parameterization?

TIA

 

[Matterwave]

I would suggest you do not use the term "parametric curvature". This is not a standard terminology, and it suggests the presence of curvature when there is none. A flat manifold is flat.

1) You can calculate the Riemann curvature tensor. If it is 0, then the manifold is actually flat.

2) There's no "general rule" you can find for transforming an arbitrarily complex curvilinear coordinate system back into Cartesian coordinates that I am aware of.

 

[Ben Niehoff]

Since you're specifically studying the differential geometry of surfaces (I assume 2-dimensional surfaces in $\mathbb{R}^3$?), then you will soon learn about the second fundamental form, which measures the extrinsic curvature of the embedding. And I assume shortly after that you will learn about the Gauss curvature and the Theorema Egregium.

The Gauss curvature measures the intrinsic curvature of a 2-dimensional surface. Therefore, you can find "Cartesian" coordinates if and only if the Gauss curvature vanishes.

Actually finding such coordinates typically requires a bit of guessing, although in 2 dimensions there are some methods, such as normal coordinates and isothermal coordinates, that should give you a Cartesian system if your surface is intrinsically flat.

The Riemann curvature tensor comes into play in higher-dimensional manifolds. In 2 dimensions, you require only one quantity (the Gauss curvature) to measure the intrinsic curvature; but in higher dimensions, you require $d^2 (d^2-1)/12$ independent quantities, which are organized into the Riemann curvature tensor. The Riemann curvature tensor vanishes if and only if a manifold is intrinsically flat (in which case, it will be possible to find local Cartesian coordinates).