  Recently there is a surge of interest in network geometry and topology. Here
we show that the spectral dimension plays a fundamental role in establishing a
clear relation between the topological and geometrical properties of a network
and its dynamics. Specifically we explore the role of the spectral dimension in
determining the synchronization properties of the Kuramoto model. We show that
the synchronized phase can only be thermodynamically stable for spectral
dimensions above four and that phase entrainment of the oscillators can only be
found for spectral dimensions greater than two. We numerically test our
analytical predictions on the recently introduced model of network geometry
called Complex Network Manifolds which displays a tunable spectral dimension.
