Scalar curvature under the collapse of metric

Abstract

We prove a formula involving the scalar curvature of a Riemannian manifold endowed with a distribution in terms of an adapted orthonormal frame for its tangent bundle. Using the formula, we then investigate the effect of collapsing the metric along the distribution on the scalar curvature. This result contributes to the question of finding a positive scalar curvature metric on a Riemannian manifold.