On a 2-dimensional Einstein Kaehler submanifold of a complex space form

Matsuyama, Yoshio

doi:10.1090/s0002-9939-1985-0810170-0

Cited by 2 publications

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“…M n is complete) (cf. [1], [2], [8], [9], [10], [11], [12], [13], [14], [16] and [17]). In this direction, one of the open problems so far is as follows:…”

Section: Introductionmentioning

confidence: 99%

Curvature Pinching for Kaehler Submanifolds of a Complex Projective Space

Matsuyama¹

2008

Tsukuba J. Math.

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“…M n is complete) (cf. [1], [2], [8], [9], [10], [11], [12], [13], [14], [16] and [17]). In this direction, one of the open problems so far is as follows:…”

Section: Introductionmentioning

confidence: 99%

Curvature Pinching for Kaehler Submanifolds of a Complex Projective Space

Matsuyama¹

2008

Tsukuba J. Math.

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“…(h(e i, el), h(e j, e j)) = ( h(e i, ei), h( g,j, ~j)) = k6ij h(x, ei, e j) = 0 for i +j and any x ~ TpM +(2) (h(ei, el, el), h(ei, el, el)) = (h(ei, ei, el), h(el, el, el)) =3k(2k-cO.Proof. We fix an arbitrary i (1 <i< n).…”

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Einstein K�hler submanifolds with codimension 2 in a complex space form

Tsukada

1986

Math. Ann.

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We denote by Mm(c') an m-dimensional simply connected and complete K~ihler manifold of constant holomorphic sectional curvature 6, which is called a complex space form. Let Pm(C) be an m-dimensional complex projective space endowed with a Fubini-Study metric. Then Pm(C) is a complex space form of positive holomorphic sectional curvature. A complex Euclidean space (12 m endowed with the usual Hermitian metric is a complex space form of zero holomorphic sectional curvature. An open unit ball D m in ~m endowed with the natural complex structure and the Bergmann metric is a complex space form of negative holomorphic sectional curvature. Any m-dimensional complex space form /Qm(c') is (after multiplying the metric by a suitable constant) complex analytically isometric to P~(~), ~'~, or D m according as the holomorphic sectional curvature 6 is positive, zero, or negative.By a Kfihler submanifold we mean a complex submanifold with the induced Kfihler metric. We consider Kfihler submanifolds in a complex space form which are Einsteinian with respect to the induced metric. Einstein K~ihler hypersurfaces in a complex space form are completely classified by Smyth [5] and Chern [1]. That is, an n (>2)-dimensional Einstein Kfihler hypersurface M in a complex space form M" § 1(6) is totally geodesic if 6<0 and is either totally geodesic or locally a complex hyperquadric Q,(~) if 6 > 0. In this note we consider the case of codimension 2 and show the following. Theorem. Let M be an n (> 2)-dimensional Einstein Kdhler submanifold immersed in~,+z(~. If 6<0, then M is totally 9eodesic. If 6>0, then M is either totally 9eodesic or locally a complex hyperquadric Q,(C) in P,+I(~) which is totally geodesic in P, + z(~).Remark 1. When n = 2, the above result is already shown by Matsuyama [2].Remark 2. If we replace our assumption on the Ricci tensor in the theorem by a weaker one that the Ricci tensor is parallel, an open submanifold of PI(C) x P2(C) in Ps(~) is only added in the conclusion of the theorem. The immersion of Pt(~)

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On a 2-dimensional Einstein Kaehler submanifold of a complex space form

Cited by 2 publications

References 8 publications

Curvature Pinching for Kaehler Submanifolds of a Complex Projective Space

Curvature Pinching for Kaehler Submanifolds of a Complex Projective Space

Einstein K�hler submanifolds with codimension 2 in a complex space form

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