Let $M$ and $N$ be smooth manifolds and let $S$ be a submanifold of $N$ ($\dim S < \dim N$). Let $\mathfrak S$ be a foliation of $S$. We say that a map between $M$ and $N$ is transverse to $\mathfrak S$ if it is transverse to every leaf of $\mathfrak S$.
Now, suppose $f : M \rightarrow N$ be is a smooth map transverse to a foliation $\mathfrak S$ of $S$ at a point $x \in M$ ($f(x) \in S$ and $f$ is transverse to the leaf of $\mathfrak S$ passing through $f(x)$ at the point $x \in M$). Is it possible to find a map $g : M \rightarrow N$, arbitrarily close to $f$ in the strong (Whitney) topology, with $g(x) = f(x)$ and such that $g$ is transverse to $\mathfrak S$ at all points of $M$?
Let $M$ and $N$ be smooth manifolds and let $S$ be a submanifold of $N$. N$($\dim S < \dim N$). Let$\mathfrak S$be a foliation of$S$. We say that a map between$M$and$N$is transverse to$\mathfrak S$if it is transverse to every leaf of$\mathfrak S$. Now, suppose$f : M \rightarrow N$be a smooth map transverse to a foliation$\mathfrak S$of$S$at a point$x \in M$($f(x) \in S$and$f$is transverse to the leaf of$\mathfrak S$passing through$f(x)$at the point$x \in M$). Is it possible to find a map$g : M \rightarrow N$, arbitrarily close to$f$in the strong (Whitney) topology, with$g(x) = f(x)$and such that$g$is transverse to$\mathfrak S$at all points of$M$? 2 added 1 characters in body; edited tags Let$M$and$N$be smooth manifolds and let$S$be a submanifold of$N$. Let$\mathfrak S$be a foliation of$S$. We say that a map between$M$and$N$is transverse to$\mathfrak S$if it is transverse to every leaf of$\mathfrak S$. Now, suppose$f : M \rightarrow N$be a smooth map transverse to a foliation$\mathfrak S$of$S$at a point$x \in M$($f(x) \in S$and$f$is transverse to the leaf of$\mathfrak S$passing through$f(x)$at the point$x \in M$). Is it possible to find a map$g : M \rightarrow N$, arbitrarily close to$f$in the strong (Whitney) topology, with$g(x) = f(x)$and such that$g$is transverse to$\mathfrak S$at all points of$M$.M$?