Shrinking the Fibers of a Submersion Splits the Riemann Tensor

Abstract

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