I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some measure of global geometry. One of such a measure is second fundamental form for embedded (in $R^n$) manifolds. The first fundamental form is a measure of local geometry and therefore insufficient to run the algorithm efficiently. For example I tried using the induced Riemannian metric (metric induced by Euclidean geometry on the hypersurface), but it worked well in some cases - I assume it is because the information it contains is basically about the local behaviour of the manifold. Secod fundamental form works quite well. But I wonder if there are any other measures that would be able to grasp the global geometry of the embedding? Surely, Riemannian or Ricci curvature tensors are such measures, but their evaluation is computationally intensive. Hence, my question - what are other measures of global geometry?


What I meant (incorrectly, as the comments below have indicated) by "measure of the global geometry" is the measure of how hypersurfaces are embedded in $R^n$, i.e. the extrinsic quantity. It was my mistake to invoke the notion of global geometry and I apologize for that.

  • $\begingroup$ Riemannian or Ricci curvature tensors are intrinsic. They can not grasp the global geometry of the embedding neither. By the way, could you explain more about the stochastic algorithm $\endgroup$
    – shu
    Mar 14 '15 at 10:21
  • $\begingroup$ Nothing so far you've mentioned is a measure of "global geometry" The second fundamental form is a measure of the "local geometry of the embedding." It's not totally clear what you're asking, but e.g. the Willmore functional could be relevant if you're interested in surfaces: en.wikipedia.org/wiki/Willmore_energy $\endgroup$ Mar 14 '15 at 17:19
  • $\begingroup$ @OtisChodosh Yes, you are absolutely right. In a hurry, I wrongly invoked the "global geometry" notion. All that I meant is the extrinsic quantities/measures, i.e. how the hypersurface is embedded into $R^n$. I'll look into the Willmore energy functional $\endgroup$
    – Tomas
    Mar 14 '15 at 18:16

If I understand correctly, you're interested in measuring the "extrinsic geometry" of the embedding. In some sense, the first and second fundamental forms are exactly what you need to know to understand this:

From a theoretical viewpoint, if you know the first and second fundamental forms, then you know everything that you need to know about the local structure of the embedding. For example, any intrinsic curvature, e.g. the Riemannian curvature tensor is derived from the induced metric (first fundamental form) and its derivatives. The rest of the data is encoded in the second fundamental form and its derivatives.

As you might know, there are some compatibility conditions between the first/second fundamental forms (and their derivatives), the Gauss and Codazzi equations (see wikipedia, or your favorite Riemannian geometry book). It turns out that these conditions are also sufficient for $(g,k)$ to be the first and second fundamental form of some embedding into $\mathbb{R}^n$. Moreover, this embedding is unique up to isometries of $\mathbb{R}^n$. See Robert Bryant's answer here, for example.

So, what this says is that the first and second fundamental form's (and their derivatives) tell you everything about the local structure of the extrinsic (and also intrinsic) geometry.

Of course, this may not be very useful for you in practice, but its not clear from your question what exactly you're looking for!f


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