Positive Intermediate Ricci Curvature on Fibre Bundles

TL;DR

This work extends the canonical variation paradigm to positive intermediate Ricci curvature, showing that a submersion with base and fibre metrics satisfying $Ric_{k_1}>0$ and $Ric_{k_2}>0$ respectively yields a total-space metric with $Ric_k>0$ for all $k \ge \max\{k_1+p,k_2+q\}$ after small fibre scaling. The authors derive curvature transformation rules under fibre scaling, relate $Ric_k>0$ to eigenvalue sums of the endomorphism $R_X$, and use these to prove the main theorems. They then augment this by developing plumbing/surgery constructions that preserve $Ric_k>0$, producing broad new classes of manifolds (including highly connected manifolds and exotic spheres) carrying such metrics. Finally, they extend moduli-space results to $Ric_k>0$, showing infinitely many path components in the relevant cases by adapting and extending previous Ricci-curvature deformation frameworks to the intermediate curvature setting. Overall, the paper blends curvature-operator analysis with topological constructions to significantly widen the catalog of manifolds admitting positive intermediate Ricci curvature metrics and to illuminate the topology of their metric moduli spaces.

Abstract

We prove a canonical variation-type result for submersion metrics with positive intermediate Ricci curvatures. This can then be used in conjunction with surgery techniques to establish the existence of metrics with positive intermediate Ricci curvatures on a wide range of examples which had previously only been known to admit positive Ricci curvature, such as highly connected manifolds and exotic spheres. Further, we extend results of the second author on the moduli space of metrics with positive Ricci curvature to positive intermediate Ricci curvatures.

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1.2
• Lemma 2.1
• Lemma 2.2
• Corollary 2.3