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Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.

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Induced action by an involution on spinor bundle and Dirac operator

For question 2, the original manifold M will be a double cover of the quotient space since it's an involution, and if you have an operator that commutes with the involution on the covering space, it s …
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