# Global geometry measures for Riemannian manifolds

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?

EDIT

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.

• 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
– shu
Mar 14 '15 at 10:21
• 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 Mar 14 '15 at 17:19
• @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 Mar 14 '15 at 18:16

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.