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As an outsider to both, manifolds of bounded geometry and bandlimited functions appear rather connected: for example, bounded geometry is defined in terms of bounds on curvature and its derivatives, while if the derivative of a bandlimited function has $\mathit{L}^1$ Fourier transform (and hence is bounded) then so do all higher derivatives. Isaac Pesenson et al. seem to have done some work connecting these subjects, which I've just started looking at.

It would be nice if this work could be formulated in the setting of a category. For example, we could imagine setting a universal length scale with corresponding frequency bandlimit, and then compact Riemannian manifolds (satisfying the appropriate bounds) would naturally carry a finite number of degrees of freedom for maps to other manifolds.

My naive attempt at trying to define a suitable category failed since, for example, a wave with frequency close to the bandlimit cannot be postcomposed with anything interesting without exceeding it. I have tried thinking in terms of concrete submanifolds of Euclidean space etc., but quickly became mired in laborious inequalities without any intuition for what was going on.

I'd be very grateful for any suggestions on how to proceed.

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