# Kahler manifolds with constant bisectional curvature

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.

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I'm a little puzzled. Do you mean constant holomorphic bisectional curvature (as one would expect, since this is the usual meaning of "bisectional curvature")? As Goldberg and Kobayashi point out in their original 1967 JDG paper, for $n>1$, the holomorphic bisectional curvatures of $\mathbb{B}^n$ and $\mathbb{CP}^n$ are not constant. They vary between $c$ and $c/2$, where $c$ is the holomorphic sectional curvature. –  Robert Bryant Aug 28 '12 at 17:23
This wrong terminology "constant bisectional curvature" comes from the book of Tian "Canonical metrics...", he really means constant holomorphic sectional curvature. –  YangMills Aug 28 '12 at 17:29
@YangMills: Thanks for clearing that up. In fact, I had written the answer below earlier, thinking that the OP wanted 'holomorphic sectional curvature' and then deleted my answer when I realized that he had written 'bisectional curvature'. Now, I've undeleted my answer even though it is essentially the same as yours. –  Robert Bryant Aug 28 '12 at 18:45
@Robert: sorry for the duplicate answer. –  YangMills Aug 28 '12 at 20:04
@YangMills: No need to apologize. I don't think you could have seen my original answer until I undeleted it. The only reason I did so was because of the reference to Bochner in Walker's review, which I didn't have time to follow. Fortunately, 'macbeth' did, and it pretty much confirmed my suspicions that Bochner had known the local uniqueness theorem (which is the interesting part) in 1947, so that paper probably deserves to be credited with the original result. (Bochner was the one who first determined the irreducible decomposition of the curvature tensor of a Kähler manifold, I think.) –  Robert Bryant Aug 28 '12 at 21:49

This is theorem 7.9 in the book of Kobayashi-Nomizu "Foundations of Differential Geometry Vol.II". There the authors attribute it to Hawley and Igusa independently. These are probably the first papers where this result was proved.

Of course, as Robert Bryant points out, the correct assumption is "constant holomorphic sectional curvature".

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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

Hawley, N.S., Constant holomorphic curvature, Canad. J. Math 5 (1953), 53–56. (MR 14,690)

and

Igusa, J., On the structure of a certain class of Kähler manifolds, Amer. J. Math. 76 (1954), 669–678. (MR 16,172)

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.

I, myself, have never read these papers.

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Regarding Walker's comment on Hawley's paper: The Bochner paper which is cited by Hawley is "Curvature in Hermitian metric" (1947). In this paper Bochner proves the local version of the result: that the metric of constant holomorphic bisectional curvature $b$ is unique up to local isometry. Maybe Walker felt that passing to the global version (as done by Hawley/Igusa) was straighforward. –  macbeth Aug 28 '12 at 18:59