Preserving Positive Intermediate Curvature

Abstract

Abstract Consider a compact manifold N (with or without boundary) of dimension n. Positive m-intermediate curvature interpolates between positive Ricci curvature (

$$m = 1$$

m = 1

) and positive scalar curvature (

$$m = n-1$$

m = n - 1

), and it is obstructed on partial tori

$$N^n = M^{n-m} \times \mathbb {T}^m$$

N n

=

M

n - m

×

T

m

. Given Riemannian metrics

$$g, {\bar{g}}$$

g ,

g

¯

on

$$(N, \partial N)$$

( N , ∂ N )

with positive m-intermediate curvature and m-positive difference

$$h_g - h_{{\bar{g}}}$$

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

$${\bar{g}}$$

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.