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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!

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    Just out of curiosity, what material were you using to study b-calculus?2018-06-28

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Yes, at least for me too, this seems a reasonable meaning for it:

A vector $(N,N_1)\in T_{(p,q)}(X\times X)$ would be defined doubly inward pointing for $(p,q)\in\partial X\times\partial X$, if neither $N$ nor $N_1$ is tangent to $\partial X$, but there is a curve in $X\times X$ with this starting speed. That is, simply if both $N$ and $N_1$ are double inward pointing vectors.

So, most probably this "doubly" just wants to mean that "w.r.t. $X\times X$" and not w.r.t $X$.