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The Gauss map sends $p$ to the chosen unit normal $N(p)$. Differentiating $⟨N,N⟩=1$ shows that $d{N}_p$ maps $T_{p}S$ into itself. The shape operator $S_p=-d{N}_p$ is self-adjoint because its associated second fundamental form is symmetric. Its eigenvalues $k_1,k_2$ are the principal curvatures. A point is umbilical when $k_1=k_2$, and the surface is minimal when its mean curvature $H=(k_1+k_2)/2$ vanishes everywhere.

In an orthonormal principal basis, $d{N}_p$ has diagonal matrix $\mathrm{diag}(-k_1,-k_2)$. It is conformal exactly whenwhich is equivalent to $k_1^2=k_2^2\ne 0$. At a nonumbilical point this says $k_1=-k_2$, exactly the condition $H=0$. Hence, when there are no umbilical points, the surface is minimal if and only if its Gauss map is conformal.

No. If the image lies in a great circle, there is a fixed unit vector $a$ with $⟨N,a⟩=0$, so $a$ is tangent to $S$ everywhere. Since the ambient derivative of this constant vector is zero, the shape operator satisfies $S(a)=0$. One principal curvature is therefore zero. Minimality makes their sum zero, so both vanish and $S=0$. The Gauss map is then locally constant, and connectedness makes it constant on all of $S$, contradicting its image being a great circle.

Choose the orientation of a surface represented by a smooth Gauss map $N:S\to {\mathbb{S}}^2$. The shape operator at $p$ isIf its principal curvatures are $k_1,k_2$, then by definition

Let $E(x)=Ax+b$ be a proper Euclidean motion of Euclidean three-space, so $A\in SO(3)$. Orient $\tilde{S}=E(S)$ by transporting the normal:The chain rule gives the orthogonal conjugacyThe determinant and trace are invariant under similarity, so the Euclidean invariance of the shape operator yields

If the surface orientation is reversed, then $N$ and $S_p$ are multiplied by $-1$. Therefore the orientation reversal of surface curvature says

The principal curvatures are the two eigenvalues $k_1,k_2$ of the shape operator. Thus $K=k_1{k}_2$ and $H=(k_1+k_2)/2$.

A principal direction is an eigenvector of the shape operator. The normal curvature in a unit principal direction is the corresponding principal curvature.

For the ambient Levi-Civita connection, a normal field $\eta$, and tangent fields $X,Y$, differentiate $⟨\eta ,Y⟩=0$ using metric compatibility. The tangential part of ${\nabla }_{X}Y$ is orthogonal to $\eta$, giving the displayed identity. It controls how an ambient derivative of a normal vector can develop a tangential component, and is the vector-valued version of the shape operator relation.

An umbilical point of a surface is a point where the two principal curvatures are equal, equivalently where the shape operator is a scalar multiple of the identity.