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A fibred manifold is a triple $(E,\pi,M)$ where $E$ and $M$ are manifolds and $\pi : E \rightarrow M$ is a surjective submersion. (Saunders, The Geometry of Jet bundles)

A special case of this is the fibre bundle which is a fibred manifold, where for each relatively open subset $U \subset M$ the space $\pi^{-1}(M)$ looks like (i.e. is diffeomorphic to) a product $U \times F$ for some smooth manifold $F$.

Even more special are linear bundles and, of course, the tangential bundle of a smooth manifold.

I do not know a generalization of these constructions to manifolds with boundary. For example, it seems intuitive that at a boundary point of a smooth manifold with boundary, we do not have a tangential space, but rather a tangential cone. Similarly one would adapt the more general classes of fibre bundles.

Unfortunately, this seems to be non-standard in differential geometry and differential topology, and while it seems natural to me to work with cones in that case, reinventing the wheel might be a waste of time. Do you know a good reference that settles the basic constructions and possible pitfalls for "fibred manifolds with boundary"?

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    $\begingroup$ Usually you just demand that $\pi_{|\partial E}$ is a map from $\partial E$ to $M$ of which is a fibering of the boundary. At least, that's the typical usage one sees in most of topology. $\endgroup$
    – Ryan Budney
    Dec 9, 2011 at 0:02
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    $\begingroup$ So this is the "fibers are transverse to the boundary" case. $\endgroup$
    – Ryan Budney
    Dec 9, 2011 at 0:19
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    $\begingroup$ One can take doubling of $M$ and ask that the induced structure in the doubling is a fibred manifold. But in any case the def should depend on the application you have in mind. $\endgroup$
    – Anton Petrunin
    Dec 9, 2011 at 2:13
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    $\begingroup$ Understood as a submanifold, the boundary $\partial M$ of $M$ has co-dimension 1: accordingly, the tangent bundle of the former should be a submanifold of co-dimension 2 of the latter. Hence, the category of manifolds with boundary is not rich enough for your purposes: you may give a look at Michor's "Manifolds of Differential Mappings", which provides a solid framework for this sort of objects. E.g., in ¶2.6 you will find the definition of the tangent bundle to a "manifold with corners". $\endgroup$
    – Giovanni Moreno
    Sep 2, 2013 at 12:19

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