  We consider the Laplace-Beltrami operator in tubular neighbourhoods of curves
on two-dimensional Riemannian manifolds, subject to non-Hermitian parity and
time preserving boundary conditions. We are interested in the interplay between
the geometry and spectrum. After introducing a suitable Hilbert space framework
in the general situation, which enables us to realize the Laplace-Beltrami
operator as an m-sectorial operator, we focus on solvable models defined on
manifolds of constant curvature. In some situations, notably for non-Hermitian
Robin-type boundary conditions, we are able to prove either the reality of the
spectrum or the existence of complex conjugate pairs of eigenvalues, and
establish similarity of the non-Hermitian m-sectorial operators to normal or
self-adjoint operators. The study is illustrated by numerical computations.
