Kahler manifolds with constant bisectional curvature - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:09:56Z http://mathoverflow.net/feeds/question/105687 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105687/kahler-manifolds-with-constant-bisectional-curvature Kahler manifolds with constant bisectional curvature Reza 2012-08-28T06:53:26Z 2012-08-28T17:44:17Z <p>It is well known that the universal covering of a complete Kahler manifold with constant bisectional curvature is \$\mathbb{C}^n\$, \$\mathbb{B}^n\$ or \$\mathbb{CP}^n\$. I need original paper(s) that prove this theorem.</p> http://mathoverflow.net/questions/105687/kahler-manifolds-with-constant-bisectional-curvature/105732#105732 Answer by Robert Bryant for Kahler manifolds with constant bisectional curvature Robert Bryant 2012-08-28T15:24:45Z 2012-08-28T15:24:45Z <p>You asked for references to original papers. In Kobayashi and Nomizu, Vol. 2, pp. 170–171, they give a proof of this result and then write, "This has been proved independently by Hawley [1] and Igusa [1]." The two papers they cite are</p> <p>Hawley, N.S., Constant holomorphic curvature, Canad. J. Math 5 (1953), 53–56. (MR 14,690)</p> <p>and </p> <p>Igusa, J., On the structure of a certain class of Kähler manifolds, Amer. J. Math. 76 (1954), 669–678. (MR 16,172)</p> <p>Note, however, that the Math Reviews article on the Hawley paper (written by A. G. Walker) attributes the result to Bochner, but doesn't give a reference.</p> <p>I, myself, have never read these papers.</p> http://mathoverflow.net/questions/105687/kahler-manifolds-with-constant-bisectional-curvature/105743#105743 Answer by YangMills for Kahler manifolds with constant bisectional curvature YangMills 2012-08-28T17:36:53Z 2012-08-28T17:44:17Z <p>This is theorem 7.9 in the book of Kobayashi-Nomizu "Foundations of Differential Geometry Vol.II". There the authors attribute it to <a href="http://dx.doi.org/10.4153/CJM-1953-007-1" rel="nofollow">Hawley</a> and <a href="http://dx.doi.org/10.2307/2372709" rel="nofollow">Igusa</a> independently. These are probably the first papers where this result was proved.</p> <p>Of course, as Robert Bryant points out, the correct assumption is "constant holomorphic sectional curvature".</p>