Curvature forms and Curvature functions for 2-manifolds with boundary

TL;DR

This paper demonstrates that any 2-form and smooth function on 2-manifolds with boundary can be realized as the curvature form and Gaussian curvature of some Riemannian metric, respectively.

Contribution

It establishes the realization of arbitrary curvature forms and functions on 2-manifolds with boundary, expanding understanding of geometric structures.

Findings

Any 2-form can be realized as a curvature form.

Any smooth function can be realized as Gaussian curvature.

Results apply to 2-manifolds with boundary.

Abstract

We obtained that any 2-form and any smooth function on 2-manifolds with boundary can be realized as the curvature form and the gaussian curvature function of some Riemmanian metric, respectively.