For a parameterized smooth embedded surface $\sigma: V \rightarrow U \subset \mathbf{R}^{3}$, where $V$ is an open domain in $\mathbf{R}^{2}$, define the first fundamental form, the second fundamental form, and the Gaussian curvature $K$. State the Gauss-Bonnet formula for a compact embedded surface $S \subset \mathbf{R}^{3}$ having Euler number $e(S)$.

Let $S$ denote a surface defined by rotating a curve

$$\eta(u)=(r+a \sin u, 0, b \cos u) \quad 0 \leq u \leq 2 \pi$$

about the $z$-axis. Here $a, b, r$ are positive constants, such that $a^{2}+b^{2}=1$ and $a<r$. By considering a smooth parameterization, find the first fundamental form and the second fundamental form of $S$.