Holomorphic bisectional curvature

Goldberg, Samuel I.; Kobayashi, Shôshichi

doi:10.4310/jdg/1214428090

Cited by 142 publications

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“…He also proved that in the case of a complete connected Kähler manifold with positive sectional curvature the same conclusion holds if we replace the hypothesis totally geodesic by analytic. Such results were proved by Goldberg and Kobayashi [5] for Kähler manifolds with positive bisectional curvature. Frankel's theorems were extended by Gray [6] to nearly Kähler manifolds, by Marchiafava [10] to quaternionic Kähler manifolds and by Ornea [12] to locally conformal Kähler manifolds and to Sasakian manifolds in the case when the submanifolds N and P are invariant and tangent to the structure vector field of M.…”

Section: Introductionmentioning

confidence: 58%

On the topology of Sasakian manifolds

Pitiş¹

2003

MATH. SCAND.

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Section: Introductionmentioning

confidence: 58%

On the topology of Sasakian manifolds

Pitiş¹

2003

MATH. SCAND.

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“…By Steps 1 and 2, noting that P^C sl^} 1 ) is irreducible and 1-codimensional in P(E*) 9 we obtain: in Gieseker [8; Lemma 1.8] will be given for our later purpose. …”

Section: (S)mentioning

confidence: 99%

C³-Actions and algebraic threefolds with ample tangent bundle

Mabuchi

1978

Nagoya Mathematical Journal

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“…We can choose a local orthonormal frame field Moreover i?2-2 22 = p -Rj\22 = P -(P -<#τnτ) = ^TUT Using these relations and the Bianchi identity, we can express the eigen-values of P easily by the components of curvature tensor and we get If we assume that the bisectional curvatures [7] are nonnegative, we can show that /^0 and hence / = 0 by (4.4). Then M is symmetric and isometric either to P 2 Koszul [8] that T o is negative definite.…”

Section: P:t(g)t-+t(g)tmentioning

confidence: 99%

Remarks on Kähler-Einstein Manifolds

Matsushima

1972

Nagoya Mathematical Journal

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The main purpose of this note is to characterize a compact KahlerEinstein manifold in terms of curvature form. The curvature form Ω is an EndT valued differential form of type (1,1) which represents the curvature class of the manifold. We shall prove that the curvature form of a Kahler metric is the harmonic representative of the curvature class if and only if the Kahler metric is an Einstein metric in the generalized sense {g.s.) y that is, if the Ricci form of the metric is parallel. It is well known that a Kahler metric is an Einstein metric in the g. s. if and only if it is locally product (globally, if the manifold is simply connected and complete) of Kahler-Einstein metrics. We obtain an integral formula, involving the integral of the trace of some operators defined by the curvature tensor, which measures the deviation of a Kahler-Einstein metric from a Hermitian symmetric metric. In the final section we shall prove the uniqueness up to equivalence of Kahler-Einstein metrics in a simply connected compact complex homogeneous space. This result was proved by Berger [3] in the case of a complex projective space and our proof is completely different from Berger's.1. Throughout this paper we shall denote by M a compact Kahler manifold and by T and Γ* the holomorphic tangent bundle and the holomorphic cotangent bundle of M respectively. The real differentiable tangent bundle of M will be denoted by T& The vector space of smooth sections of a vector bundle F will be denoted by Γ(F). A section X of T is a complex vector field of holomorphic type or of type (1,0) and we denote by X the conjugate of X; X is a section of the conjugate bundle T of T. We denote by <, > the Hermitian metric in 7\ that is, if X, Y e Γ(T) f then = g{X, Ϋ),where g denotes the Kahler metric in M. We have then

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Holomorphic bisectional curvature

Cited by 142 publications

References 9 publications

On the topology of Sasakian manifolds

On the topology of Sasakian manifolds

C³-Actions and algebraic threefolds with ample tangent bundle

Remarks on Kähler-Einstein Manifolds

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Holomorphic bisectional curvature

Cited by 142 publications

References 9 publications

On the topology of Sasakian manifolds

On the topology of Sasakian manifolds

C3-Actions and algebraic threefolds with ample tangent bundle

Remarks on Kähler-Einstein Manifolds

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C³-Actions and algebraic threefolds with ample tangent bundle