A pseudo-Riemannian manifold is a smooth manifold equipped with a non-degenerate, symmetric bilinear form on the tangent space at each point, which allows for both positive and negative signature metrics. This type of manifold generalizes the concept of Riemannian manifolds by enabling the inclusion of time-like and space-like intervals, making it particularly useful in theories like general relativity. The structure is essential for understanding geodesics, which describe paths of shortest distance or extremal paths in this generalized context, as well as for operations like the Hodge star operator that relate forms on these manifolds.
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