If we consider the AdS-Schwarzschild manifold, defined by $ M^n=[s_0,\infty)\times\mathbb{S}^{n-1}$ equipped with the Riemannian metric $ $ \overline{g}=\frac{1}{1-ms^{2-n}+s^2}ds\otimes ds+s^2g_{\mathbb{S}^{n-1}},$ $ where $ m>0$ is a fixed positive number, $ s_0$ is the unique positive solution of the equation $ 1+s_0^2-ms_0^{2-n}=0$ and $ g_{\mathbb{S}^{n-1}}$ is the standard round metric on the unit sphere $ \mathbb{S}^{n-1}$ . The scalar curvature of $ M$ is equals $ -n(n-1)$ .
My question is:
There exists some spin structure on $ M$ ?
Since the Hyperbolic space is a particular case when $ m\to0$ and the hyperbolic space admits a spin structure, is natural asking for a spin structure on the AdS- Schwarschild space?
