Growth conditions for conformal transformations preserving Riemannian completeness

TL;DR

This paper investigates how the growth rate of conformal factors affects the preservation of completeness in Riemannian metrics after conformal transformation, providing conditions to ensure the transformed metric remains complete.

Contribution

It establishes new growth conditions on conformal factors that guarantee the preservation of Riemannian completeness after conformal transformation.

Findings

Derived growth conditions for conformal factors

Provided criteria for completeness preservation

Enhanced understanding of conformal transformations in Riemannian geometry

Abstract

For a complete Riemannian metric, a pointwise conformal transformation may lead to a complete or incomplete transformed Riemannian metric, depending on the behavior of the conformal factor. We establish conditions on the growth of the conformal factor towards the infinity of the Riemannian metric, such that the conformally transformed Riemannian metric remains complete.