Preserving positive intermediate curvature

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.