I would like to know a book or an article where the connected sum of Riemannian manifolds is explained with some detail. I roughly know how to do the construction but I want to be able to check a more detailed explanation. Also, since I'm doing some notes on this I want to be able to cite the reference. Thankk in advance.

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    $\begingroup$ Do you want a reference on connected sum of smooth manifolds? In the Riemannian context, there is no canonical Riemannian metric on the connected sum. $\endgroup$ Jul 17 '14 at 19:11
  • $\begingroup$ Thanks for your comment. I would like to put a Riemannian metric in a way that it "extends" the metrics on the summands. Is this possible? $\endgroup$
    – Sak
    Jul 18 '14 at 6:14
  • $\begingroup$ Yes, it is possible, but not canonically. $\endgroup$ Jul 18 '14 at 21:31
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    $\begingroup$ For instance, embed your manifolds isometrically in some $R^N$ and then do the embedded connected sum as described somewhere as MSE. $\endgroup$ Jul 18 '14 at 21:32
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    $\begingroup$ Here is the MSE reference for the embedded connected sum: math.stackexchange.com/questions/836619/…. I do not know a textbook reference. It might be in the book by Chow and Knopf on Ricci flow (or other books on the Ricci flow), since Ricci flow with surgeries uses the construction. It should be, however, in books/papers on manifolds of positive scalar curvature, since it is the simplest example of Gromov-Lawson surgery (preserving positivity of scalar curvature), see www2.math.umd.edu/~jmr/psc2006.pdf for references. $\endgroup$ Jul 19 '14 at 16:37

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