  This paper is the third and final component of a three-part effort on
observers contracting a Riemannian distance between the state of the system and
its estimate. In Part I, we showed that such a contraction property holds if
the system dynamics and the Riemannian metric satisfy two key conditions: a
differential detectability property and a geodesic monotonicity property. With
the former condition being the focus of Part II, in this Part III, we study the
latter condition in relationship to the nullity of the second fundamental form
of the output function. We formulate sufficient and necessary conditions for it
to hold. We establish a link between it and the infinite gain margin property,
and we provide a systematic way for constructing a metric satisfying this
condition. Finally, we illustrate cases where both conditions hold and propose
ways to facilitate the satisfaction of these two conditions together.
