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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 can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?

This question has been definitively answered in the following paper: http://nzjm.math.auckland.ac.nz/images/f/f4/When_is_a_Connection_Metric_Connection%3F.pdf
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