Geodesics and the Hadamard parametrix
Christopher D. Sogge · Princeton University Press eBooks · 2017
This chapter studies the spectrum of Laplace–Beltrami operators on compact manifolds. It begins by defining a metric on an open subset Ω ⊂ Rn, in order to lift their results to corresponding ones on compact manifolds. The chapter then details some elliptic regularity estimates, before embarking on a brief review of geodesics and normal coordinates. The purpose of this review is to show that, with given a particular Laplace–Beltrami operator and any point y0 in Ω, one can choose a natural local coordinate system y = κ(x) vanishing at y0 so that the quadratic form associated with the metric takes a special form. To conclude, the chapter turns to the Hadamard parametrix.