For compact Riemannian manifolds all of whose geodesics are closed (aka Zoll manifolds) one can define the determinant of a zeroth order pseudodifferential operator by mimicking Szego’s definition of this determinant for the operator: multiplication by a bounded function, on the Hilbert space of square-integrable functions on the circle. In this paper we prove that the non-local contribution to this determinant can be computed in terms of a much simpler “zeta-regularized” determinant.

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If 𝑈:[0,+∞[×𝑀 is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation ∂_𝑡𝑈+𝐻(𝑥,∂_𝑥 𝑈)=0, where 𝑀 is a not necessarily compact manifold, and 𝐻 is a Tonelli Hamiltonian, we prove the set Σ(𝑈), of points in ]0,+∞[×𝑀 where 𝑈 is not differentiable, is locally contractible. Moreover, we study the homotopy type of Σ(𝑈). We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

1 Preliminaries 3 1.1 Linear algebra 3 1.1. 1 Vectors and matrices 3 1.1. 2 Symmetric bilinear forms 5 1.1. 3 Vector subspaces 6 1.1. 4 Linear maps from Rn to Rn 7 1.1. 5 A convention 9 1.2 Vector calculus 9

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