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.