  Motivated by computational aspects of persistent homology for Vietoris-Rips
filtrations, we generalize a result of Eliyahu Rips on the contractibility of
Vietoris-Rips complexes of geodesic spaces for a suitable parameter depending
on the hyperbolicity of the space. We consider the notion of geodesic defect to
extend this result to general metric spaces in a way that is also compatible
with the filtration. We further show that for finite tree metrics the
Vietoris-Rips complexes collapse to their corresponding subforests. We relate
our result to modern computational methods by showing that these collapses are
induced by the apparent pairs gradient, which is used as an algorithmic
optimization in Ripser, explaining its particularly strong performance on
tree-like metric data.
