Shrinking the Fibers of a Submersion Splits the Riemann Tensor

TL;DR

This paper derives a formula describing how the curvature forms of a semi-Riemannian submersion change when fibers are shrunk, revealing their limiting behavior and relation to topological invariants like the Euler characteristic.

Contribution

It provides a concise formula for the curvature forms of a fiber-shrunk submersion using Karcher's formulation, linking geometric deformation to topological limits.

Findings

Curvature forms approach a sum of vertical and base curvature forms as fibers shrink.

The Gauss-Bonnet integrand converges to a wedge product involving the vertical form and base form.

Pushforward of the curvature form approaches the Euler characteristic times the base curvature form.

Abstract

This paper uses Karcher's formulation [Kar99] of the O'Neill tensors [O'N66,Gra67] to derive a concise formula for the family $\Omega^\epsilon$ of curvature forms obtained by shrinking the fibers of a submersion $\pi:M\to B$ of semi-Riemannian manifolds by a factor of $1-\epsilon$. The formula clearly shows that as $\epsilon$ approaches 1, $\Omega^\epsilon$ approaches the sum of the vertical curvature form $\Omega^\mathrm{V}$ and the pullback $\pi^*\Omega^B$ of the curvature form of $B$. The Gauss-Bonnet integrand $\mathrm{Pf}(\Omega^\epsilon)$ therefore approaches the wedge $\mathrm{Pf}(\Omega^\mathrm{V})\wedge\pi^*\mathrm{Pf}(\Omega^B)$. So if $\pi$ has compact fiber $F$, the pushforward $\pi_*\mathrm{Pf}(\Omega^\epsilon)$ approaches $\chi(F)\cdot\mathrm{Pf}(\Omega^B)$.