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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?

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The existence or nonexistence of a spin structure on a smooth manifold $M$ is a topological question, in that it does not depend on the choice of a Riemannian metric on $M$. Spin structures are obstructed by the second Stiefel-Whitney class $w_2\in H^2(M;\mathbb Z/2)$, in that $M$ admits a spin structure iff $w_2(TM) = 0$.

Since $M^2 = [s_0,\infty)\times S^{n-1}$, its tangent bundle decomposes as

$$TM^3 \cong \pi_1^*T([s_0,\infty)) \oplus \pi_2^*TS^{n-1},$$

where $\pi_1,\pi_2$ are the projections from $M^n$ onto its two components. Since $[s_0,\infty)$ is contractible, its tangent bundle is trivial. The tangent bundle to $S^{n-1}$ is stably trivial, since embedding $S^{n-1}\hookrightarrow\mathbb R^n$ realizes $\underline{\mathbb R}^n \cong T\mathbb R^n|_{S^{n-1}} = TS^n\oplus\nu$, where $\nu$ is the normal bundle to the embedding, and $\nu$ is a trivial bundle.

Since $w_2$ is a stable characteristic class, it vanishes on stably trivial vector bundles, so $w_2(M^n) = 0$, and therefore $M^n$ admits a spin structure.

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  • $\begingroup$ Your answer was very useful! I wondering now about the existence of killing imaginary spinors in the AdS Schwarschild manifold, there exists some one? $\endgroup$ – Geom math Dec 22 '17 at 18:55
  • $\begingroup$ Unfortunately, I don't know what an imaginary spinor is. $\endgroup$ – Arun Debray Dec 22 '17 at 18:57

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