  Here we describe a weakly interacting Bose gas on a curved manifold, which is
embedded in the three-dimensional Euclidean space.~To this end we start by
considering a harmonic trap in the normal direction of the manifold, which
confines the three-dimensional Bose gas in the vicinity of its
surface.~Following the notion of dimensional reduction as outlined in
[L.~Salasnich et al., Phys.~Rev.~A {\bf 65}, 043614 (2002)], we assume a large
enough trap frequency so that the normal degree of freedom of the condensate
wave function can be approximately integrated out. In this way we obtain an
effective condensate wave function on the quasi-two-dimensional surface of the
curved manifold, where the thickness of the cloud is determined
self-consistently. For the particular case when the manifold is a sphere, our
equilibrium results show how the chemical potential and the thickness of the
cloud increase with the interaction strength.~Furthermore, we determine within
a linear stability analysis the low-lying collective excitations together with
their eigenfrequencies, which turn out to reveal an instability for attractive
interactions.
