Intrinsic vs Extrinsic geometry of convex surfaces

Question

By Alexandrov's isometric embedding theorem, any locally convex metric prescribed on the sphere admits a realization as a convex surface in Euclidean 3-space, which, by Pogorelov's rigidity result, is unique up to an isometry of the ambient space.

Thus, the intrinsic geometry of a convex surface completely determines its extrinsic geometry, and in principle it should be possible to describe all geometric quantities in terms of the intrinsic metric. On the other hand, Alexandrov's proof is not constructive, and does not provide any hints as to how one might be able to do this. Hence my first question is:

Question 1: Is there any way to compute, recognize, or characterize intrinsically any of the geometric quantities or features of a convex surface which are not invariant under local isometric deformations, e.g., mean curvature, principal directions, principal curvatures, or umbilic points?

I recognize this might be too much to ask, because the procedures which the above question asks for would somehow need to take into account the whole metric, not just a neighborhood of a point. So let me try to make things a bit more concrete and maybe a little more accessible.

It is well known that any (smooth) convex surface must have at least one umbilic, i.e., a point where the principal curvatures are equal (if not then the direction of the larger principal curvature yields a nonvanishing line field on the surface which violates the Poincare-Hopf index theorem). Can one prove this without reference to the ambient space, or quantities that we only know how to compute extrinsically:

Question 2: Is it possible to prove the existence of an umbilic point of a smooth convex surface in a purely intrinsic way? If so, can one also find or approximate the location of the umbilics?

One motivation behind this questions is the famous conjecture of Caratheodory, which states that each convex surface must have at least two umbilics. The extrinsic or local approaches to this problem have always been problematic.

Adendum: As Robert Bryant correctly points out below, the topological argument for the existence of at least one umbilic described above is really intrinsic and independent of the notion of principal directions. So the main point of Question 2 is its second part, i.e., to somehow get a handle on the location of the umbilic. See also this related question.

Answer 1

But the existence of an umbilic point on the sphere follows from topological considerations: The sum of the Hopf indices of the umbilics is 1 (by a theorem of Hopf) so there has to be at least one umbilic.

Put another way: If there were no umbilics, then union of the principal directions at each point would define a 4-fold covering space of the sphere, which would, because the sphere is simply connected, have a section, which would give a nowhere vanishing vector field on the sphere. Thus, there must be at least one umbilic. I don't see why this isn't 'intrinsic'.

As for locating the umbilics, I don't see how you could expect to do that in general.

Answer 2

"Umbilic points" have no obvious meaning in Alexandrov's framework (since his theory is not restricted to smooth surfaces), but there has been a fair amount of progress on making the theory more effective (it was always constructive: you could approximate the metric by a polyhedral metric. A polyhedral metric could be embedded in finite time [slowly, true], and then results would converge to the embedding of the surface you started with. The speed of convergence was already addressed in the '50s by A. Volkov). The last word on effectivizing Alexandrov for polyhedra is:

Bobenko, Alexander I.; Izmestiev, Ivan, Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes, Ann. Inst. Fourier 58, No. 2, 447-505 (2008). ZBL1154.52005.