Prescribed curvature tensor in locally conformally flat manifolds

TL;DR

This paper investigates the Prescribed Curvature Tensor problem in locally conformally flat manifolds, providing explicit solutions and examples, especially for complete metrics on Rn, advancing understanding of conformal geometry.

Contribution

It offers explicit solutions to the Prescribed Curvature Tensor problem in locally conformally flat manifolds, including complete metrics on Rn, and constructs concrete examples of such metrics.

Findings

Explicit solutions for the Prescribed Curvature Tensor problem in special cases.

Construction of explicit conformal metrics solving the problem.

Examples of complete metrics on Rn satisfying the prescribed curvature conditions.

Abstract

Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifolds. As applications of these results we exhibit explicit examples of metrics g, conformal to g they are solutions for this problem.