Surfaces of revolution admitting strongly convex slope metrics

TL;DR

This paper characterizes the geometric conditions under which surfaces of revolution can admit strongly convex slope metrics, providing necessary and sufficient inequalities involving derivatives in Cartesian and polar coordinates.

Contribution

It establishes the precise conditions for surfaces of revolution to support strongly convex slope metrics, extending geometric understanding in this area.

Findings

Derived inequalities for derivatives ensuring strong convexity

Applied conditions to well-known surfaces of revolution

Provided a comprehensive criterion for slope metric convexity

Abstract

This paper discusses the geometry of a surface endowed with a slope metric. We obtain necessary and sufficient conditions for any surface of revolution to admit a strongly convex slope metric. Such conditions involve certain inequalities for the derivative of the associated function on the Cartesian coordinate and the polar coordinate. In particular, we apply this result to certain well-know surface of revolution.