Now, I've been reading V.I. Arnol'd, Characteristic Class Entering in Quantization Conditions and I'm stuck on page 10. Here is the link
He tolds that
$uM^{(n)}$ is such that its tangential mapping $\tau$ is transversal to every $\Lambda^k(n) \hookrightarrow \Lambda(n), k=1,2,\dots$.
However, I think the dimension of $\tau^* T_x (uM^{(n)}) + T_{\tau(x)} \Lambda^k(n)$ is strictly less than $T_{\tau(x)} \Lambda(n)$ for some large $k$.
How do I interpret this statement correctly?
EDIT: In here, $uM^{(n)}$ is an $n$-dimensional Lagrangian manifold. $\Lambda(n)$ is an algebraic variety, called Lagrangian grassmanian, which is a collection of all Lagrangian subspaces of $\mathbb{R}^{2n}$. $\Lambda^k(n)$ is a manifold of Lagrangian subspaces of $\mathbb{R}^{2n}$ whose intersection with standard position subspace($q=0$) is $k$-dimensional and it is $\dfrac{k(k+1)}{2}$ codimensional submanifold of $\Lambda(n)$. $\tau$ is a tagential mapping from $M^{(n)}$ to $\Lambda(n)$; mapping a point $x$ to the tangent space $T_xM$ of $M$ at the point, and it is well-defined since $M$ is a Lagrangian manifold in $\mathbb{R}^{2n}$.
The context is, he tried to assert that $uM^{(n)}$ is in general position from showing that each element of $\tau(uM^{(n)})$ is transverse to $\overline{\Lambda^1(n)}$.