  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.
