If it was topology we wouldn't bother to warp the manifold so we can do similarity search. No, it's geometry, with a metric. Just as in real life, we want to be able to compare things.

Topological transformation of the manifold happens during training too. That makes me wonder: how does the topology evolve during training? I imagine it violently changing at first before stabilizing, followed by geometric refinement. Here are some relevant papers:

* Topology and geometry of data manifold in deep learning (https://arxiv.org/abs/2204.08624)

* Topology of Deep Neural Networks (https://jmlr.org/papers/v21/20-345.html)

* Persistent Topological Features in Large Language Models (https://arxiv.org/abs/2410.11042)

* Deep learning as Ricci flow (https://www.nature.com/articles/s41598-024-74045-9)