  We study how convergence of an observer whose state lives in a copy of the
given system's space can be established using a Riemannian metric. We show that
the existence of an observer guaranteeing the property that a Riemannian
distance between system and observer solutions is nonincreasing implies that
the Lie derivative of the Riemannian metric along the system vector field is
conditionally negative. Moreover, we establish that the existence of this
metric is related to the observability of the system's linearization along its
solutions. Moreover, if the observer has an infinite gain margin then the level
sets of the output function are geodesically convex. Conversely, we establish
that, if a complete Riemannian metric has a Lie derivative along the system
vector field that is conditionally negative and is such that the output
function has a monotonicity property, then there exists an observer with an
infinite gain margin.
