Submersion

A smooth map is a submersion when its differential is surjective at every point.

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 surjective Riemannian submersion $\pi :M\to N$, let $H={\sum }_{\alpha }({\nabla }_{V_{\alpha }}{V}_{\alpha }{)}^{\mathrm{hor}}$ be the unnormalised mean curvature vector of its fibres, using an orthonormal vertical frame. With the positive Laplace-Beltrami operator, tracing the Riemannian Hessian in horizontal and vertical directions gives ${\Delta }_M(f\circ \pi )=({\Delta }_{N}f)\circ \pi +d(f\circ \pi )(H)$. Horizontal traces agree with those on the base; a vertical Hessian term is $-d(f\circ \pi )({\nabla }_{V_{\alpha }}{V}_{\alpha })$. Thus minimal fibres, in particular totally geodesic submanifolds as fibres, make pullback an injection from each base eigenspace into the corresponding total-space eigenspace. Its image is exactly the eigenfunctions constant on each whole fibre.

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.

If $F:X^n\to Y^m$ is a submersion at $p$, there are local coordinates in whichEvery fiber of a submersion is consequently an embedded submanifold of codimension $m$.

After choosing a Riemannian metric on the source of a submersion $F:X\to Y$, the orthogonal complements $H_p=(\ker D_{p}F{)}^{\perp }$ map isomorphically to $T_{F(p)}Y$. Therefore every vector field $w$ on $Y$ has the smooth lift

The fibers of a proper submersion over a connected base are mutually diffeomorphic. Lift a compactly supported vector field moving one nearby base point to another; properness makes the lifted field compactly supported, and its time-one flow identifies the fibers.