Strichartz estimates for non elliptic Schr\"odinger equations on compact manifolds

In this note we consider the Schr"odinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-G'erard-Tzvetkov to the non-elliptic case and improves the result by Salort. We also investigate the optimality of the result for the case on ${\mathbb{S}}^3\times {\mathbb{S}}^3$.