= Solution
For an oriented surface with unit normal $N$, the <Gauss map> sends $p$ to $N(p)\in S^2$. The <shape operator> is $-dN$ on the tangent plane; its eigenvalues are the <principal curvatures> $k_1,k_2$. The <Gaussian curvature> is $K=k_1k_2$ and the <mean curvature> is $H=(k_1+k_2)/2$. Reversing orientation reverses both principal curvatures and $H$, but preserves $K$. The <Theorema Egregium> says that <Gaussian curvature> is determined by the <first fundamental form>, so is invariant under local isometries.
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