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The construction of negatively Ricci curved manifolds. (English) Zbl 0542.53024

We give the affirmative answer to the problem rose by Bourguignon that the connected sum of two Riemannian manifolds with negative Ricci curvature has a metric with negative Ricci curvature. We also proved that the ”connected sum along circles” of two negatively Ricci curved manifolds has a metric with negative Ricci curvature. This ”connected sum” is defined as follows. Let \(X_ 1\) and \(X_ 2\) be two \(C^{\infty}\) manifolds, and \(\gamma_ i\) is a simple closed curve in \(X_ i (i=1,2)\). Take a smooth tubular neighborhood \(V_ i\) of \(\gamma_ i\) in \(X_ i\), let \(\phi:\partial V_ 1\to \partial V_ 2\) be a diffeomorphism of the boundaries. We use \(X_ 1\oplus_{\phi}X_ 2\) to denote the manifold obtained by gluing \(X_ 1-V_ 1\) and \(X_ 2- V_ 2\) along the boundaries by \(\phi\). We call \(X_ 1\oplus_{\phi}X_ 2\) the connected sum along circles of \(X_ 1\) and \(X_ 2\). Particularly in dimension three, we use the above results and some topological constructions to obtain new manifolds with a negative Ricci curvature metric. Given a complete three dimensional Riemannian manifold M with a negative Ricci curvature metric, then there are complete negatively Ricci curved metrics on \(M\oplus S^ 2\times S^ 1\); \(M\oplus S^ 2\times S^ 1\oplus L(p,q)\), and on \(M\oplus S^ 2\times S^ 1\oplus \Sigma \times S^ 1\), where \(\Sigma\) is any oriented Riemannian surface.

MSC:

53C20 Global Riemannian geometry, including pinching
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References:

[1] [H] Hempel, J.: 3-Manifolds. Ann. of Math. Studies 86. Princeton University Press 1976
[2] [BJ] Bourguignon, J.P.: Ricci curvature and Einstein metrics, global differential geometry. Lect. Notes Math.838, pp. 42-63. Berlin, Heidelberg, New York: Springer 1981
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