Scalar curvature under the collapse of metric

TL;DR

This paper derives a formula for scalar curvature involving distributions on Riemannian manifolds and studies how collapsing the metric along these distributions affects scalar curvature, contributing to the understanding of positive scalar curvature metrics.

Contribution

It introduces a new formula for scalar curvature with respect to distributions and analyzes the impact of metric collapse along distributions.

Findings

Derived a scalar curvature formula involving distributions

Analyzed the effects of metric collapse on scalar curvature

Contributed to the problem of positive scalar curvature metrics

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.