I am reading Geometry Theory of Foliations by Camacho and Neto and came upon this definition:
"Let $N$ be a manifold. We say that $g:N \to M$ is transverse to a foliation $F$ when $g$ is transverse to all the leaves of $F$, i.e., if for every $p \in N$ we have $$Dg(p).T_p(N) + T_q(F) = T_q(M), \quad q=g(p)$$ where by $T_q(F)$ we mean the tangent space to the leaf of $F$ which passes trough $q$."
My question is: Does $T_q(F)$ makes sense for all $q$? From what I've understood the sum is made in $T_q(M)$, so $T_q(F)$ only makes sense if his leaf is a submanifold of $M$, and that's not always true (the leaves are always immersed submanifolds though, does that help?)
Thanks in advance!
Locally, a foliation is simple. For every $x\in M$ there exist a chart $f:U\rightarrow\mathbb{R}^n$ such that the leaves of the image of the foliation in $f(U)$ are defined by $x_1=c_1,...,x_p=c_p$ where $c_1,..,c_p$ are constant. $T_xF$ is $df^{-1}_{f(x)}(V)$ where $V$ is the vector space defined by $x_1=...=x_p=0$. $V$ is the direction of the leaves of the image of the foliation in $f(U)$.