  We consider observers contracting a Riemannian distance between the state of
the system and its estimate. As shown in [1], such a contraction property holds
if the system dynamics and the Riemannian metric satisfy two conditions: a
differential detectability property (Condition A2 in [2]), and a geodesic
monotonicity property (Condition A3 in [2]). Condition A2 is thoroughly studied
in [2]. In this paper, we study Condition A3 in relationship to the nullity of
the second fundamental form of the output function. We formulate sufficient and
necessary conditions for Condition A3 to hold. We establish a link between
Condition A3 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 the first and the second conditions hold and propose ways to
facilitate the satisfaction of these two conditions together.
