  In the past few years, there has been considerable activity around a set of
quantum bounds on transport coefficients (viscosity) and chaos (Lyapunov
exponent), relevant at low temperatures. The interest comes from the fact that
Black-Hole models seem to saturate all of them. The goal of this work is to
gain physical intuition about the quantum mechanisms that enforce these bounds
on simple models. To this aim, we consider classical and quantum free dynamics
on curved manifolds. These systems exhibit chaos up to the lowest temperatures
and - as we discuss - they violate the bounds in the classical limit. First of
all, we show that the quantum dimensionless viscosity and the Lyapunov exponent
only depend on the de Broglie length and a geometric length-scale, thus
establishing the scale at which quantum effects become relevant. Then, we focus
on the bound on the Lyapunov exponent and identify three different ways in
which quantum effects arise in practice. We illustrate our findings on a toy
model given by the surface of constant negative curvature - a paradigmatic
model of quantum chaos - glued to a cylinder. By exact solution and numerical
investigations, we show how the chaotic behaviour is limited by the quantum
effects of the curvature itself. Interestingly, we find that at the lowest
energies the bound to chaos is dominated by the longest length scales, and it
is therefore a collective effect.
