Hello everybody,

when we consider some regular surface$\displaystyle S\subset \mathbb{R}^3$ 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 $\displaystyle a\neq 0$.

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 $\displaystyle I'(x):=a\cdot I(x)$. Then i get the same curvature, because i have:

$\displaystyle K=k_1 \cdot k_2=II(e_1)\cdot II(e_2)=a\cdot <dN(\frac{e_1}{\sqrt{a}}) , \frac{e_1}{\sqrt{a}} >\cdot a<dN(\frac{e_2}{\sqrt{a}}) , \frac{e_2}{\sqrt{a}} >=II'(\frac{e_1}{\sqrt{a}})\cdot II'(\frac{e_2}{\sqrt{a}})=k'_1 \cdot k'_2 =K' $

here the vectors e_1 and e_2 are the corresponding eigenvectors, which form an orthonormal basis.

I hope, someone can help me...