# A note about scalar curvature on the total space of a vector bundle

## TL;DR

This paper constructs complete Riemannian metrics with positive scalar curvature on the tangent bundles of orientable closed surfaces (excluding tori), addressing a question posed by Gromov.

## Contribution

It provides explicit constructions of complete positive scalar curvature metrics on tangent bundles of certain surfaces, extending known results in geometric analysis.

## Key findings

- Tangent bundles of orientable closed surfaces (except tori) admit complete uniformly PSC-metrics.
- The construction offers a partial positive answer to Gromov's question on scalar curvature.
- The methods may inform future research on scalar curvature on vector bundles.

## Abstract

We construct complete Riemannian metrics to show that the total space of tangent bundles of orientable closed surfaces (except torus) admits complete uniformly PSC-metrics. It gives a partial positive answer to one of Gromov's question.