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The tangent vectors ofareHence the first fundamental form has coefficientsPutThe upward unit normal is $N=(-f_u,-f_v,1)/W$, so the second fundamental form has coefficientsThus the two forms are

The Gaussian curvature isSince $EG-F^2=W^2$, the graph formula isas in Gaussian curvature of a graph surface.

For the final claim, fix a point of $\gamma$ and make a rigid motion taking $P$ to the plane $z=0$. The common tangent plane is horizontal, so locally $\Sigma$ is the graph $z=f(u,v)$. Along the projected curve $c(s)$ of tangency,Differentiating the second identity givesBecause $c$ is a smooth curve, $c^{\prime }(s)\ne 0$, so the Hessian is singular. Its determinant is zero, and the graph formula gives $K=0$ at every point of $\gamma$. This is tangency to a plane along a curve forces zero Gaussian curvature.

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.

Let $\overline{\nabla }$ be the Euclidean connection and split the ambient tangent bundle along the embedded submanifold as $T{\mathbb{R}}^{n+m}{|}_M=TM\oplus NM$. The second fundamental form is the normal-bundle-valued bilinear formIt is symmetric because $\overline{\nabla }$ is torsion-free and the Lie bracket of tangent vector fields remains tangent:This is the symmetry of the second fundamental form.

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.

If a compact regular surface lies in a closed Euclidean ball of radius $R$, maximize distance from the ball's centre. At a maximizing point, comparison of the second fundamental form with the tangent supporting sphere makes both principal curvatures have magnitude at least $R^{-1}$ and the same sign. The Gaussian curvature there is therefore at least $R^{-2}$.

The second fundamental form of an embedded submanifold of a Riemannian manifold is symmetric. Its ambient Levi-Civita connection is a torsion-free connection, soThe Lie bracket of vector fields tangent to the submanifold is tangent and therefore has zero normal component. This proof applies to a curved ambient manifold as well as Euclidean space.

For a nondegenerate induced metric tensor, a totally geodesic submanifold has vanishing second fundamental form; equivalently, every ambient geodesic initially tangent to it remains in it. This extends the notion of a totally geodesic hypersurface to arbitrary codimension.