I have a doubt that assails me.
The technique of gluing along edges between manifolds is generally considered in the topological context.
I don't know if there are other gluing techniques.
I was wondering if there were theorems regarding the possibility of obtaining smooth gluing between isometric manifolds to each other.
That is, if there is an isometry between two smooth manifolds (even local, not necessarily global), is there always at least one way to glue them so that the resulting manifold is still smooth?


Could you give me some information about it and possibly point out some references?



Isometry and gluing between smooth manifolds - some references
Let (Mi)i=1,2 be smooth manifolds with boundary, Ni ⊆ ∂Mi unions of connected components of the boundaries of M1 and M2, respectively, and φ : N1 → N2 a diffeomorphism. Then there exists a smooth structure on the space M1 ∪φ M2 that arises by gluing M1 to M2 along N1 ≃ N2. This structure is unique up to a diffeomorphism that leaves all points from the original boundaries of M1 and M2, including N1 ≃ N2, fixed.


This statement is copied from <http://awgd.org/talks/morse/notes2.pdf> , the reference given there is Munkres “Elementary Differential Topology”.


