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 × T m . Given Riemannian metrics g , g ¯ on ( N , ∂ N ) with positive m -intermediate curvature and m -positive difference h g - h 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 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.