- #1
hideelo
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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 = αijdxidxj
Now if the αijs 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
ds2 = αijdxidxj
Now if the αijs 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
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