when we consider some regular surface with constant gaussian curvature.
Then my question is: Why and how the curvature does changes, when i multiply the givin scalar product by a constant .
I'm cannot see, why a different 1.fundamental form changes the determinant of the shape-map? For instance, what is wrong with the following ideas:
Let (S,I) be a geometric surface with curvature K and a given 1.Fundamental form I.
Now i change my quadratic form by . Then i get the same curvature, because i have:
here the vectors e_1 and e_2 are the corresponding eigenvectors, which form an orthonormal basis.
I hope, someone can help me...
Thank you for your help!
So the idea is to fix II and just change I. Because if i change I and II by I'=a*I and II'=a*II, then the curvature keeps the same! Is this statement correct?
My second question is, why is it possible to change just I, but let II fixed?
I was thinking, that the definition of II on a given Surface is in terms of I. So if i change I, then we have ??
Thank you a lot!!
Let's clearly define what does "changing the first fundamental form" means here.
If we consider only the intrinsic geometry here, since the Gaussian curvature depends only on intrinsic geometry, the meaning of "changing the first fundamental form" is clear: let g be the metric tensor, then defines a new metric tensor, where is a positive real number. The vector length is changed by a ratio . In this case the Gaussian curvature changes to . You can easily see this since the area element becomes , and areas are changed by a ratio of , then use Gauss-Bonnet theorem on a small geodesic triangle you'll find the Gaussian curvature turns to .
If we instead consider an embedded surface in , the meaning of "changing the first fundamental form" could be defined as follows:
If is the parameterization of the surface, then defines a new surface, with a corresponding first fundamental form of and the corresponding second fundamental form is , so the normal curvature is changed to