In a file at this Website, I quote Cornelius Lanczos in saying that analytical mechanics in a curved space is more topological than geometric.

Remarkable effects of curved space have been theorized in general relativity mechanics and often experimentally confirmed. But they are primarily treated as geometric. Since geometry is simply topology with a metric, all of these remarkable aspects of general relativity are topological. But the implications of this appear not to have been developed in the literature.