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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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Compact surface with genus$\geq 2$ with Killing field
Let M be a compact Riemannian surface of genus$\geq 2$.
Can M have a globally defined Killing field ?
Can M have a Killing field defined on M-(finite set of points)?