Everyone: I am trying to make sense of these statements (doubts are ones contained within asterisks **); both $M$,$N$ are surfaces.:
Setup: $N$ is a surface trivially-embedded in $X^4$, a 4-manifold. $M$ is a compact surface embedded in $X^4$ , with $\partial M$ contained in $N$, and ** $M$ is normal to $N$ along its boundary *, (1)and, * Interior of $M$ is transverse to $N$ ** (2)
This is what I (think) I understand:
(1) *$M$ is normal to $N$ alongs its bdry *: I guess this means that the curve $\partial M$ is in a position such that at each point $p$ where $\partial M\cap N$ is not empty, $T_p(\partial M)$ is perpendicular to $T_pN$. Is it? I know the normal line is (a) perpendicular to the tangent so I am extrapolating that if two manifolds intersect each other normally, that their respective tangent spaces at points (as local linearizations) of intersection are perpendicular to each other, i.e., the tangent space to one of the manifolds is normal to the tangent space of the other manifold.
(2)*"The interior of $M$ is transverse to $N$ *" Here, by interior of $M$, we mean $M-\partial M$. Since both $M$,$N$ are embedded in $X^4$, and are transverse to each other, they intersect at points (since codim(M)+codim(N)=4), and do not intersect in some other way.
Is this correct? Is there something else I should consider?