  We address the problem of sensing the curvature of a manifold by performing
measurements on a particle constrained to the manifold itself. In particular,
we consider situations where the dynamics of the particle is quantum mechanical
and the manifold is a surface embedded in the three-dimensional Euclidean
space. We exploit ideas and tools from quantum estimation theory to quantify
the amount of information encoded into a state of the particle, and to seek for
optimal probing schemes, able to actually extract this information. Explicit
results are found for a free probing particle and in the presence of a magnetic
field. We also address precision achievable by position measurement, and show
that it provides a nearly optimal detection scheme, at least to estimate the
radius of a sphere or a cylinder.
