Singularities and the Wave Equation on Conic Spaces

where r is the distance from the point and dω is the round metric on the sphere. If X is an arbitrary manifold with boundary, the class of conic metrics on X is modeled on this special case. Namely, a conic metric is a Riemannian metric on the interior of X such that for some choice of the defining function x of the boundary (x ∈ C∞(X) with ∂X = {x = 0}, x ≥ 0, dx 6= 0 on ∂X), the metric takes the form g = dx + xh on X = X\∂X, near ∂X. Here h is a smooth symmetric 2-cotensor on X such that h0 = h|∂x is a metric on ∂X. In fact a general conic metric can be reduced to a form even closer to (1) in terms of an appropriately chosen product decomposition of X near ∂X, that is, by choice of a smooth diffeomorphism