Given an oriented real smooth vector bundle, the Euler class assigns a differential form on the base to each metric and metric-compatible connection on the bundle. Is there a reason that it is not possible to naturally extend this to define a differential form for each connection on the bundle? I am aware of why the Pfaffian process in particular fails, since the connection's curvature does not necessarily have the necessary skew-symmetry.
2026-03-28 06:40:56.1774680056
Euler form of a non-metric connection
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