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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
5
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distributional Hessian for semiconvex functions on non-smooth manifolds
In $C^1$ manifolds I guess one could define measure-valued Hessian at least for $C^1$ functions by working in charts and throwing only one derivative on the $C^1$ test function.
In general $CD(K,N)$ …