Connected Sum of Riemannian Manifolds As topological spaces, one can take connected sums of riemannian manifolds. Is there a way to give a Riemannian metric to the connected sum of two Riemannian Manifolds? If so is there a way to take this metric "canonical" in the sense that it extends the metrics of the manifolds used? 
My intuiton tells me that when one takes a ball out of a manifold, the metric is not necessarily complete anymore. Is the metric on the connected sum complete?
 A: Completeness is not a problem. Removing an open ball out of a complete metric space, one obtains another complete metric space (manifold with boundary, in our situation). Then two manifolds with boundary are glued along the boundary. In the topological setting that would be the end of the story: the half-space charts on both sides of the cut stick together well enough for topological structure. 
But if we want smooth gluing, one has to either change the metric on original manifolds or add a cylinder connecting the manifolds. Indeed, imagine glueing two $2$-spheres by removing small disks from each. The total curvature of the two spheres is almost $8\pi$, but the Gauss-Bonnet theorem says the connected sum must have total curvature $4\pi$, since  it is again a sphere. Therefore, negative curvature must appear somewhere. 
One can argue as follows: pick a point $p\in M$ and two small radii $r<R$. The region between the spheres $S(p,r)$ and $S(p,R)$ is diffeomorphic to a cylinder (product of sphere with a line segment). Push forward the metric on $M$ to this cylinder. Then smoothly modify it (using partition of unity) so that near the end that corresponds to $S(p,r)$ it becomes the standard metric of the right circular cylinder. You can do the same on the other manifold $M'$. The ends corresponding to $S(p,r)$ and $S(p',r')$  glue together smoothly.
