I am currently trying to understand the $b$-calculus developed by R. Melrose. an important part of the theory is the stretched product of a manifold $X$ with boundary $\partial X$.
looking at the product $X \times X$ with diagonal $\triangle$, we blow up the intersection $(\partial X \times \partial X) \cap \triangle$ by replacing it with the
"doubly inward - pointing part of its normal bundle".
I am afraid I don't undertand this concept of doubly inward pointing - from Lee's book on smooth manifolds I recall that a vector $N$ in the tangent space $T_pX$ of a boundary point $p \in \partial X$ is said to be inward pointing if $N \notin T_p\partial X$ and for some $\varepsilon > 0$ there exists a smooth curve segment $\gamma \colon [0,\varepsilon] \to X$ such that $\gamma(0) = p$ and $\gamma'(0) = N$.
now, on the above intersection I am looking at (a subset of) the corner $\partial X \times \partial X$ which has codimension $2$, so I guess this is where the "doubly inward pointing" notion comes from but I am not sure about this ..
what do I need to add (or modify) in order to obtain a definition for doubly inward pointing normal vectors ?
many thanks!