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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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Elliptic theory on compact manifolds

For the specific question of getting rid of the low-order term $\|u\|_p$, what you really need is the compact embedding of $W^{2,p}$ in $L^p$ (Rellich–Kondrachov theorem). Indeed, suppose there is no …
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