= Transversality of a map to a submanifold
{title2=$f\pitchfork S$}
= Transversality
{synonym}
A <smooth map> $f:M\to N$ is transverse to an <embedded submanifold> $S\subset N$ when
$$
df_p(T_pM)+T_{f(p)}S=T_{f(p)}N
\quad\text{for every }p\in f^{-1}(S).
$$
This is a sum of <vector subspaces>, not necessarily a <direct sum>. Equivalently the composition of $df_p$ with the quotient to $T_{f(p)}N/T_{f(p)}S$ is surjective. In a <slice chart for an embedded submanifold> of codimension $k$, the $k$ normal coordinate functions composed with $f$ have surjective derivative on their common zero set. If the preimage is empty the condition holds vacuously.
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