Preserving Positive Intermediate Curvature

Abstract

Consider a compact manifold N (with or without boundary) of dimension n. Positive m-intermediate curvature interpolates between positive Ricci curvature (m = 1) and positive scalar curvature (m = n-1), and it is obstructed on partial tori N^n = M^{n-m} times mathbb {T}^m. Given Riemannian metrics g, {bar{g}} on (N, partial N) with positive m-intermediate curvature and m-positive difference h_g - h_{{bar{g}}} of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive m-intermediate curvature interpolating between g and {bar{g}}. Moreover, we apply this result to prove a non-existence result for partial torical bands with positive m-intermediate curvature and strictly m-convex boundaries.