# Monotonicity on homogeneous spaces

## TL;DR

This paper extends the concept of monotonicity to homogeneous spaces by introducing invariant cone fields and differential positivity, enabling more tractable analysis of ordered structures in applications.

## Contribution

It formulates monotonicity on homogeneous spaces using invariant cone fields and proposes invariant differential positivity as a natural generalization.

## Key findings

- Provides a theoretical framework for invariant cone fields on homogeneous spaces
- Lists examples from information engineering and applied mathematics
- Introduces invariant differential positivity as a generalization of monotonicity

## Abstract

This paper presents a formulation of the notion of monotonicity on homogeneous spaces. We review the general theory of invariant cone fields on homogeneous spaces and provide a list of examples involving spaces that arise in applications in information engineering and applied mathematics. Invariant cone fields associate a cone with the tangent space at each point in a way that is invariant with respect to the group actions that define the homogeneous space. We argue that invariance of conal structures induces orders that are tractable for use in analysis and propose invariant differential positivity as a natural generalization of monotonicity on such spaces.