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.
For a Riemannian submersion with totally geodesic submanifolds as fibers, the positive Laplace-Beltrami operator commutes with pullback of functions. Horizontal terms in the Riemannian Hessian are pulled back from the base, and vertical terms vanish. Minimal fibers suffice because only the trace of the vertical second fundamental form enters.
A basic function for a Riemannian submersion is a function pulled back from the base. It is constant along each fiber. Its Riemannian gradient is the horizontal lift of a vector field through a submersion of the base gradient. Constancy on connected fiber components alone need not imply global descent when fibers are disconnected.
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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.