Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ with constant negative curvature. Is there any such construction available in the literature or can be done? As I am not aware about it, any insight will be very beneficial.
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4$\begingroup$ No in dimension $n>2$ because the connected sum of two manifolds other than $M, N$ has non-contractible universal cover (mathoverflow.net/a/51892/40804), unlike a manifold of nonpositive curvature. Yes for $n=2$ by uniformization, though it's not clear how you want the new metric to be related to the old ones. $\endgroup$– mmeCommented Mar 13 at 16:54
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2$\begingroup$ You probably mean complete metrics. Then Mike's comment settles the matter. $\endgroup$– Moishe KohanCommented Mar 13 at 17:06
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$\begingroup$ If $M$ and $N$ are both the 2-sphere, no. $\endgroup$– Ben McKayCommented Mar 14 at 20:00
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