Codex @codex  0

Use the global parametrizationIts tangent vectors areandThus $X$ is a regular parametrization and the graph is a smooth surface.

The first fundamental form isChoosing the upward orientation, the Gauss map is

Choose a smooth unit normal $N$. For $v,w\in T_p\Sigma$, the first fundamental form and second fundamental form arewhere the shape operator is $S_p=-d{N}_p$. The principal curvatures are the eigenvalues ${\kappa }_1,{\kappa }_2$ of $S_p$; the mean curvature and Gaussian curvature areThe surface is minimal when $H=0$ everywhere.

For a local parametrization $X(u^1,u^2)$, the first fundamental form is the Euclidean inner product restricted to each tangent plane:Writing $u^1=u,u^2=v$, this is

After choosing an orientation, the Gauss map isThe second fundamental form is $II(v,w)=⟨S(v),w⟩$, where the shape operator is $S=-dN$. In coordinates,The Gaussian curvature and mean curvature are respectively the determinant and half the trace of $S$:Changing the chosen normal reverses $II$ and $H$ but leaves $K$ unchanged.

For $v$ in a sufficiently small neighbourhood of $0\in T_{p}S$, let ${\gamma }_v$ be the geodesic satisfying ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. The exponential map at $p$ isChoose an oriented orthonormal basis of $T_{p}S$, write $e(\theta )=(\cos \theta ,\sin \theta )$, and define geodesic polar coordinates by

The Gauss lemma says that radial and angular coordinate curves are orthogonal. Since $r\mapsto \varphi (r,\theta )$ is a unit-speed geodesic,Moreover, torsion-freeness of the surface connection gives $D_r{\varphi }_{\theta }=D_{\theta }{\varphi }_r$, and thereforeAt $r=0$, ${\varphi }_{\theta }=0$, so $⟨{\varphi }_r,{\varphi }_{\theta }⟩=0$ for every sufficiently small $r$. Thus the first fundamental form isand $\sqrt{G}/r\to 1$ as $r\downarrow 0$ because $d({\exp }_p{)}_0$ is the identity.

For major radius $a>b>0$ and minor radius $b$, let $R=a+b\cos u$. With the inward normal determined by the usual ordered $(u,v)$ parametrization, the first fundamental form and second fundamental form areReversing the normal reverses $II$. The Gaussian curvature of a ring torus is independent of this orientation.