A generalization of Thom's transversality theorem

Details A generalization of Thom's transversality theorem A generalization of Thom's transversality theorem

2010-01-13

arxiv.org/abs/1001.2054v1

Arch. Math. (Brno) 44 (2008), no. 5, 523--533

Mathematics

Differential Geometry

Scientific paper

We prove a generalization of Thom's transversality theorem. It gives conditions under which the jet map $f_*|_Y:Y\subseteq J^r(D,M)\ra J^r(D,N)$ is generically (for $f:M\ra N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\ra P$ to the preimage $(j^sf)^{-1}(A)$ of a submanifold $A\subseteq J^s(M,N)$ in terms of transversality properties of the original map $f$. Our main result is that for a reasonable class of submanifolds $A$ and a generic map $f$ the restriction $g|_{(j^sf)^{-1}(A)}$ is also generic. We also present an example of $A$ where the theorem fails.

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