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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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The momentum constraints in the ADM formulation of general relativity

The connection is metric compatible. In particular the covariant derivative of the square root of the determinant of g is zero. So the above equation does coincide.
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