I'm using the Gauss-Bonnet theorem on the following exercise and I would like to see if what I've done is correct.
Given a compact, connected surface $M$ with area $A(M)=1984707$ and Gauss curvature $K=\frac{-31658} {10^6}$, determine the surface.
So, using Gauss-Bonnet to determine the Euler characteristic $χ(M)$ of $M$, I get:
$K*A(M)=2π*χ(M)$ which of course will gives a non integer characteristic.
I'm assuming it's just the numbers given but I thought to ask.
So, did I use Gauss-Bonnet correctly or this is not how it's done? Also, if I did everything right, is there a way to determine if this surface is orientable or not with the information that the exercise gives?
Again, I'm sorry if this is trivial