Proving that a product of Riemannian Manifolds is locally conformally flat

Question

I like to proof an embedding Theorem for LCF-Manifolds. But im stuck on a little detail. The setting is the following. Given a locally conformally flat Manifold with Boundary. I want to attach a Cylinder around the Boundary and proof that this Product is LCF- Compatibel in the sense that $\partial M \times (-1,1)$ is LCF. So the question is if we can locally extend the Metric past the boundary $\partial M $ to a conformally flat metric on $\partial M \times (-1,1) $. I tried using Warped Products for this but it was pointed out to me that if a Warped Product $W= M \times _{f} I , I \epsilon \mathbb {R}$
is LCF than M has to be a Manifold of constant curvature. This however is to restrictive for my purpose. Does any one else have an idea how to Proof that $\partial M \times (-1,1) $ is LCF? Thanks for any Help and sorry for my bad english i hope my question became clear.