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.
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