A **Riemannian submersion** is a specific type of mathematical structure that arises in differential geometry. It involves two Riemannian manifolds and a smooth map between them that preserves certain geometric properties. More formally, let \( (M, g_M) \) and \( (N, g_N) \) be two Riemannian manifolds, where \( g_M \) and \( g_N \) are their respective Riemannian metrics.
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A surjective submersion between Riemannian manifolds is Riemannian when its derivative is a linear isometry from the orthogonal complement of its kernel onto the target tangent space. Its vertical space is the kernel and its horizontal space is the orthogonal complement.