# Questions tagged [spin-geometry]

For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}(n)$ and $\operatorname{Spin}^c(n)$. This tag should also be used for any questions about the geometry of spin manifolds, including questions involving Dirac operators and the Lichnerowicz formula.

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### Question about the covering space of a spin manifold

I'm a little confused about how to prove that a covering space $\widetilde{M}$ of a spin manifold $M$ is also a spin manifold. If not, is there some counterexample? Could you please give me some help ...
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### Explicit equations describing complex spin groups as affine algebraic groups

Although there are tons of questions about spin groups here on math.SE, I could not find there what I want. What I want is this. Take the complex spin group $\operatorname{Spin}(2n,\mathbb C)$. It has ...
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### Spinor bundle $\mathbb{S}(M)$ for connected sum $\mathbb{S}(M_1 \#M_2)$?

I was just reading This question Wherein the OP wants to know what the tangent bundle looks like in a general connected sum $$T\left(M_{1}\#M_{2}\right)$$ (where the connected sum is along some ...
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### Lie algebra of the $\mathop{Spin}(4)$ group as 2-forms on tangent bundle
While trying to understand connections on spin manifolds, I stumbled over this (apparently obvious) statement. Consider a spin $4$-manifold $M$. I am struggling to understand the intuition of the ...