Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why?

Question

What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this?

Answer 1

The idea seems to be that Gaussian curvature is intrinsic because it is invariant over isometric embedding. That is, if you apply an isometric embedding to a surface, the mean curvature of the result may be different (making mean curvature an extrinsic property), but the Gaussian curvature (and any other intrinsic measurement) would have to remain the same.

Answer 2

I like to point out that you can find out these numbers, at the origin only, for $z = a x^2 + b y^2.$ The Hessian matrix of second partials is diagonal, and the first derivatives are zero (at the origin only). The Gauss curvature is, say, one quarter the determinant, $ab.$ If we flip it upside down, $z = -a x^2 - b y^2,$ we get $(-a)(-b)= ab$ which is unchanged.

Meanwhile, the mean curvature of the original is $a+b$ or half of that, depends on the author. If we flip it upside down, it is $-a-b,$ which is not the same as the original.

There are always choices of constants; worth working this out on your own.