  This paper extends the possibility to examine the underlying curvature of
data through the lens of topology by using the Betti curves, tools of
Persistent Homology, as key topological descriptors, building on the clique
topology approach. It was previously shown that Betti curves distinguish random
from Euclidean geometric matrices - i.e. distance matrices of points randomly
distributed in a cube with Euclidean distance. In line with previous
experiments, we consider their low-dimensional approximations named integral
Betti values, or signatures that effectively distinguish not only Euclidean,
but also spherical and hyperbolic geometric matrices, both from purely random
matrices as well as among themselves. To prove this, we analyse the behaviour
of Betti curves for various geometric matrices -- i.e. distance matrices of
points randomly distributed on manifolds of constant sectional curvature,
considering the classical models of curvature 0, 1, -1, given by the Euclidean
space, the sphere, and the hyperbolic space. We further investigate the
dependence of integral Betti signatures on factors including the sample size
and dimension. This is important for assessment of real-world connectivity
matrices, as we show that the standard approach to network construction gives
rise to (spurious) spherical geometry, with topology dependent on sample
dimensions. Finally, we use the manifolds of constant curvature as comparison
models to infer curvature underlying real-world datasets coming from
neuroscience, finance and climate. Their associated topological features
exhibit a hyperbolic character: the integral Betti signatures associated to
these datasets sit in between Euclidean and hyperbolic (of small curvature).
The potential confounding ``hyperbologenic effect'' of intrinsic low-rank
modular structures is also evaluated through simulations.
