Compact K�hler manifolds with nonnegative sectional curvature

Gray, Alfred

doi:10.1007/bf01390163

Cited by 19 publications

(16 citation statements)

References 15 publications

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“…A striking result concerning the curvature of Kähler manifolds is Theorem 4.1 (Gray, 1977) A compact Kähler manifold with nonnegative sectional curvature and constant scalar curvature is locally symmetric.…”

Section: Curvature and Volumementioning

confidence: 99%

Almost parallel structures

Salamon¹

2001

Global Differential Geometry: The Mathematical Legacy of Alfred Gray

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Section: Curvature and Volumementioning

confidence: 99%

Almost parallel structures

Salamon¹

2001

Global Differential Geometry: The Mathematical Legacy of Alfred Gray

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“…In the case of positively curved compact Kfihler-Einstein manifolds Berger [1] (respectively Gray [3]) proved that every compact Kfihler-Einstein manifold with positive (respectively nonnegative) sectional curvature is locally symmetric. A Bochner type argument was used in their proofs.…”

Section: (Kma X (P) _ Kmin (P)) __< Kav (P) __ Kmin (P) ____< ~ (Kma mentioning

confidence: 99%

“…Let 712 and 72 be the Chern-Weil functions of the curvature tensor of M whose integrals over M are the Chern numbers c 2 and c 2 . Consider the function ~b = 4(372-712)- [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] on M, where ~ is the scalar curvature. By a very complicated and delicate analysis of the relationships among the components of the curvature tensor with respect to the frame field where Kmi .…”

Section: (Kma X (P) _ Kmin (P)) __< Kav (P) __ Kmin (P) ____< ~ (Kma mentioning

confidence: 99%

Compact K�hler-Einstein surfaces of nonpositive bisectional curvature

Siu

Yang

1981

Invent Math

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In this paper we consider the question of characterizing compact quotients of the complex 2-ball by curvature conditions. It was conjectured by Frankel [2] that the complex projective space can be characterized as a compact K/ihler manifold with positive holomorphic bisectional curvature. This conjecture was recently proved by Mori [4] and Siu-Yau [6]. The confirmation of the Frankel conjecture leads one to ask whether there is a corresponding curvature characterization for compact quotients of the complex ball which is the noncompact counterpart of the complex projective space. Mostow-Siu [5] constructed a compact Kfihler surface which has negative sectional curvature and yet whose universal covering is not biholomorphic to the complex 2-ball. Because of this example the Frankel conjecture does not have a strictly analogous noncompact counterpart. One has to impose additional conditions. The additional condition we consider in this paper is that the metric whose curvature is negative is Kfihler-Einstein. We will consider only the case of complex dimension two.

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“…Gray [3,4], we prove the following Theorem. Let M be an n-dimensional compact totally real submanifold immersed in an n-dimensional complex space form with parallel mean curvature vector.…”

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confidence: 95%

“…Gray. We use the same notation as in [3,4]. Let M be an «-dimensional Riemannian manifold and denote by R, K and p the curvature tensor, sectional curvature and Ricci tensor of M, respectively.…”

mentioning

confidence: 99%