Let $(M,g)$ be a riemannian manifold, and $S\subset M$ a submanifold. I would like to know if there is a result which states that, for some hypothesis about the codimension of $S$, the property for a vector field $X\in\mathcal{T}(M)$ to being nowhere tangent to $S$ is generic.
Writing, for $p\in S$, the orthogonal decomposition $T_pM=T_pS\oplus N_pS$, and $f_X:S\to NS$ defined by $f_X(p)=(X_p)^\perp$, the property can be rewrited as $f_X^{-1}(0_{NS})=\varnothing$, the latter being equivalent to $f_X\pitchfork 0_{NS}$ iff $\dim S-\mathrm{codim}\,0_{NS}<0$, i.e. $2\dim S<\dim M$. Since transversality is a generic property, a generic vector field is nowhere tangent to $S$ if $2\dim S<\dim M$.
I would like to know if this kind of reasoning is correct, and if this result is true we can actually find sharper bounds for the inequality. Thank you!