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?
1 Answer
In the context of geometry, a distinction between "soft" and "hard" was introduced by Gromov, as explained here and applied to Soft and Hard Symplectic Geometry.
In Gromov's words, 'hard' refers to a strong and rigid structure of a given object, while 'soft' suggests some weak general property of a vast class of objects. Riemannian geometry is hard, while symplectic geometry is soft "because all symplectic forms are locally diffeomorphic". (There is no analog of curvature in a symplectic structure.) The "hard" part of a symplectic structure is Gromov's non-squeezing theorem.
Local invariants are "hard", which is presumably why one calls the Pontryagin class "hard".
In the context of analysis, the distinction between "soft" and "hard" seems to be more widely used and more precise, as explained in this blog, with a table of correspondences:
