Oriented Grassmannian Bundle, Normal Curvature Reduction, and Persistent Homology

Abstract

We consider a smooth closed orientable submanifold $M \subset \mathbb{R}^D$ with narrow cycles. We embed $M$ into a scaled oriented Grassmannian bundle via the Gauss map in order to enlarge the scale of these cycles. Under mild assumptions, we show that this embedding reduces the normal curvature of the embedded submanifold in directions where the original normal curvature is large. For smooth closed hypersurfaces, we further show that this construction increases the distance between antipodal points of narrow cycles for fixed volume. We then obtain an explicit range of radii for which the ambient \v{C}ech complex on this Grassmannian bundle is homotopy equivalent to the embedded manifold, yielding lower bounds on the scales at which the \v{C}ech filtration recovers the homology of $M$. Since the distance induced by the embedding depends on both positions and oriented tangent spaces, we work with Whitney $C^1$ convergence of embeddings and prove that the associated \v{C}ech persistent homology is stable with respect to the interleaving distance. Finally, we describe a procedure for computing a distance matrix for a finite subset with respect to this embedding and illustrate the construction on several examples, including an approximate quasi-halo orbit in the Saturn--Enceladus system.