  We study the influence of the intrinsic curvature on the large time behaviour
of the heat equation in a tubular neighbourhood of an unbounded geodesic in a
two-dimensional Riemannian manifold. Since we consider killing boundary
conditions, there is always an exponential-type decay for the heat semigroup.
We show that this exponential-type decay is slower for positively curved
manifolds comparing to the flat case. As the main result, we establish a sharp
extra polynomial-type decay for the heat semigroup on negatively curved
manifolds comparing to the flat case. The proof employs the existence of
Hardy-type inequalities for the Dirichlet Laplacian in the tubular
neighbourhoods on negatively curved manifolds and the method of self-similar
variables and weighted Sobolev spaces for the heat equation.
