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 ${\Omega }^{\epsilon }$ 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 ${\pi }^{\ast }{\Omega }^B$ of the curvature form of $B$. The Gauss-Bonnet integrand $\mathrm{Pf}({\Omega }^{\epsilon })$ therefore approaches the wedge $\mathrm{Pf}({\Omega }^{\mathrm{V}})\wedge {\pi }^{\ast }\mathrm{Pf}({\Omega }^B)$. So if $\pi$ has compact fiber $F$, the pushforward ${\pi }_{\ast }\mathrm{Pf}({\Omega }^{\epsilon })$ approaches $\chi (F)\cdot \mathrm{Pf}({\Omega }^B)$.

5 pages, comments welcome

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