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We know for a 2-dimensional surface there are two orthogonal principal directions at every point, where the principal curvatures $\kappa_1$ and $\kappa_2$ are the two ends of the curvature spectrum over all directions at that point. Gauss and mean curvatures are defined as $K=\kappa_1\kappa_2$ and $H=(\kappa_1+\kappa_2)/2$ respectively.

My question is why we use these two specific combination of principal curvatures to quantify a surface's deviation from being "flat" at that point? why not use for example $\kappa_1^2+\kappa_2^2$? A single number can never describe the curvedness in all directions, so what's the advantage of $K$ and $H$? More generally, what criterions are used to decide whether a definition of curvature is "good"?

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The functions $2H = \operatorname{tr}(S)$ and $K = \det(S)$ are elementary symmetric polynomials in the eigenvalues of the shape operator $S$, hence "natural". Other prospective measures of non-flatness, such as $$ \kappa_{1}^{2} + \kappa_{2}^{2} = 4H^{2} - K, $$ can be expressed as polynomials in $H$ and $K$.

Separately, $K$ turns out to be intrinsic, (invariant under local isometry, detectable by measurements within the surface), so it's a particularly important geometric quantity. (The principal curvatures themselves and the mean curvature are not intrinsic; they can change under local isometries.)

The mean curvature $H$ has geometric meaning as well: it's the derivative of area under normal deformations.

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