  For manifolds $\cal M$ of noncompact type endowed with an affine connection
(for example, the Levi-Civita connection) and a closed 2-form (magnetic field)
we define a Hilbert algebra structure in the space $L^2(T^*\cal M)$ and
construct an irreducible representation of this algebra in $L^2(\cal M)$. This
algebra is automatically extended to polynomial in momenta functions and
distributions. Under some natural conditions this algebra is unique. The
non-commutative product over $T^*\cal M$ is given by an explicit integral
formula. This product is exact (not formal) and is expressed in invariant
geometrical terms. Our analysis reveals this product has a front, which is
described in terms of geodesic triangles in $\cal M$. The quantization of
$\delta$-functions induces a family of symplectic reflections in $T^*\cal M$
and generates a magneto-geodesic connection $\Gamma$ on $T^*\cal M$. This
symplectic connection entangles, on the phase space level, the original affine
structure on $\cal M$ and the magnetic field. In the classical approximation,
the $\hbar^2$-part of the quantum product contains the Ricci curvature of
$\Gamma$ and a magneto-geodesic coupling tensor.
