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This paper uses Karcher's formulation [Kar99] of the O'Neill tensors[O'N66,Gra67] to derive a concise formula for the family $\Omega^\epsilon$ ofcurvature 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 clearlyshows that as $\epsilon$ approaches 1, $\Omega^\epsilon$ approaches the sum ofthe 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 $\chi(F)\cdot\mathrm{Pf}(\Omega^B)$.

Shrinking the Fibers of a Submersion Splits the Riemann Tensor