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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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When is a Riemannian manifold an open subset of a complete one?
This isn't an answer its a conjecture. Nice question.
Suppose that $M, N$ are a Riemannian manifolds and $M\subset N$ is an open subset and $N$ is complete. Lets assume that $M$ is path connected …