Focal radius, rigidity, and lower curvature bounds

Abstract

We prove a new comparison lemma for Jacobi fields that exploits Wilking's transverse Jacobi equation. In contrast to standard Riccati and Jacobi comparison theorems, there are situations when our technique can be applied after the first conjugate point. Using it, we show that the focal radius of any submanifold N of positive dimension in a manifold M with sectional curvature greater than or equal to 1 does not exceed π2. In the case of equality, we show that N is totally geodesic in M and the universal cover of M is isometric to a sphere or a projective space with their standard metrics, provided that N is closed. Our results also hold for kth intermediate Ricci curvature, provided that the submanifold has dimension ⩾k. Thus, in a manifold with Ricci curvature ⩾n−1, all hypersurfaces have focal radius ⩽π2, and space forms are the only such manifolds where equality can occur, if the submanifold is closed. Example and Remark show that our results cannot be proven using standard Riccati or Jacobi comparison techniques.

Turing approximations, toric isometric embeddings & manifold convolutions

“…Let us now recall the geometric notion of reach, first introduced by Federer [17] (see also [28]). The reach of a manifold's embedding measures how it departs from being convex.…”

Gehring Link Problem, Focal Radius and Over-torical width

“…In fact if a closed surface Σ 2 in S 3 has focal radius equal to π/2 then Σ 2 has to be the equatorial 2-sphere, cf. [GW18]. The following question is asked in [Gro18]: Let Σ n be a smoothly embedded n-torus in S 2n−1 , what is the largest possible focal radius?…”

Volume Estimates for Tubes Around Submanifolds Using Integral Curvature Bounds

“…The earliest global results using k-Ricci lower bounds as a partial positivity condition for curvature were obtained by Wu [23], Shen [19], and Shen-Wei [20], though the relationship between k-Ricci curvature and volume had been considered previously by Bishop and Crittenden [2, p. 253]. A signficant literature has since developed which bridges a gap between the global results based on sectional curvature bounds and those based on Ricci curvature bounds [21,16,24,7,9].…”

“…Let γ : [0, t 0 ) → M be any geodesic segment satisfying r(γ(t)) = t. If Ric k (γ, ·) ≥ kH for some constant H then for any orthonormal k-frame In Section 3 we give a slightly more general version of this theorem which also treats the question of rigidity when equality holds. This comparison theorem should be compared with that of Guijarro-Wilhelm which gives comparison along a family of k-dimensional subspaces determined by Jacobi fields rather than parallel subspaces [7,Lemma 2.23]. That comparison is based on Wilking's transverse Jacobi equation; by contrast, Theorem 1.6 above is based on the comparison theory for a Riccati differential equation and thus yields an elementary proof of the volume comparison of Section 4.…”