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- “Softness” vs “rigidity” in Geometry 3 answers

While listening to some lecture of Alain Connes about noncommutative geometry, he spoke about various generalizations of the classical concepts from geometry and divided it into "soft" and "hard" part. The audience seemed to know what is all about but for me it was not clear what is the distinction. In particular he mentioned vector bundles, differential forms, connections and curvatures and characteristic classes as being a "soft" part, while Riemannian metric, Riemannian curvature and Pontryagin classes was classified as the "hard" part. The rather vaque question would be:

**what is the criterion of being soft/hard part of geometry?**

More particular question:

**Why Pontryagin classes are part of hard geometry and the other characteristic classes are soft?**