An uncertainty principle on compact manifolds

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Breitenberger's uncertainty principle on the torus T and its higher-dimensional analogue on Sᵈ⁻¹ are well understood. We give describe an entire family of uncertainty principles on compact manifolds (M,g), which includes the classical Heisenberg-Weyl uncertainty principle (for M=B(0,1) ⊂ Rᵈ the unit ball with the flat metric) and the Goh-Goodman uncertainty principle (for M=Sᵈ⁻¹ with the canonical metric) as special cases. This raises a new geometric problem related to small-curvature low-distortion embeddings: given a function f:M → R, which uncertainty principle in our family yields the best result? We give a (far from optimal) answer for the torus, discuss disconnected manifolds and state a variety of other open problems.