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...