Indefinite K�hler manifolds

Barros, Manuel; Romero, Alfonso

doi:10.1007/bf01456410

Cited by 115 publications

(144 citation statements)

References 13 publications

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“…Consider the semi-Euclidean space R("+x) with the indefinite Kahlerian structure (cf. Barros-Romero [1] PROOF. By direct calculations taking account of (1.5), (2.13), (2.14) and (4. [7].…”

Section: (33)mentioning

confidence: 97%

Real hypersurfaces of indefinite Kaehler manifolds

Bejancu

Duggal

1992

International Journal of Mathematics and Mathematical Sciences

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“…Consider the semi-Euclidean space R("+x) with the indefinite Kahlerian structure (cf. Barros-Romero [1] PROOF. By direct calculations taking account of (1.5), (2.13), (2.14) and (4. [7].…”

Section: (33)mentioning

confidence: 97%

Real hypersurfaces of indefinite Kaehler manifolds

Bejancu

Duggal

1992

International Journal of Mathematics and Mathematical Sciences

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“…Barros and Romero [2] showed that locally any complex space formM n s (4ǫ) is isometric holomorphically to C n s , CP n s (4ǫ) CH n s (4ǫ) according to ǫ = 0, ǫ > 0 or ǫ < 0.…”

Section: Lagrangian Submanifolds In Complex Space Formsmentioning

confidence: 99%

Quasi-biharmonic Lagrangian surfaces in Lorentzian complex space forms

Sasahara

2011

Annali di Matematica

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In this paper, we introduce the notion of a quasi-biharmonic submanifold in a pseudo-Riemannian manifold and classify quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms.2000 Mathematics Subject Classification. Primary 53C42; Secondary 53B25.Key words and phrases. Lagrangian surfaces, marginally trapped surfaces, bitension field. IntroductionA submanifold with lightlike mean curvature vector field is called a marginally trapped submanifold. In the theory of cosmic black holes, a marginally trapped surface in a space-time plays an extremely important role. Recently, some classification results on marginally trapped surfaces from the viewpoint of differential geometry have been obtained (see, for instance, [5]).On the other hand, a submanifold is called biharmonic if the bitension field of the isometric immersion defining the submanifold vanishes identically. The theory of biharmonic submanifolds has advanced greatly during this last decade (see, for instance, [1] and [7]). This paper introduces the notion of a quasi-biharmonic submanifold. It is a submanifold such that the bitension field of the isometric immersion defining the submanifold is lightlike at each point.In [9], the author has classified biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms. They exist only in the flat Lorentzian complex plane. In this paper, we classify quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms. We find that the situation in the quasi-biharmonic case is quite different from the biharmonic case. In fact, there exist a lot of quasi-biharmonic marginally trapped Lagrangian surfaces in nonflat Lorentzian complex space forms. Preliminaries 2.1 Lagrangian submanifolds in complex space formsLetM n s (4ǫ) be a complex space form of complex dimension n, complex index s(≥ 0) and constant holomorphic sectional curvature 4ǫ. The complex index is defined as the complex dimension of the largest complex negative definite vector subspace of * Annali di Matematica Pura ed Applicata 192 (2013), 191-201. A section is added at the end.

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“…For more details, we refer the reader to [2]. LetM n s (4c) be an indefinite complex space form of complex dimension n and complex index s. The complex index is defined as the (complex) dimension of the largest complex negative definite vector subspace of the tangent space.…”

Section: Indefinite Complex Space Forms and Their Lagrangian Submanifmentioning

confidence: 99%

“…We refer to [2] for the construction of the standard models of indefinite complex space forms CP n s (4c), whenc > 0, CH n s (4c), whenc < 0 and C n s . For our purposes it is sufficient to know that there exist pseudo-Riemannian submersions, called Hopf fibrations,…”

Section: Indefinite Complex Space Forms and Their Lagrangian Submanifmentioning

confidence: 99%

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Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms

Vrancken

2001

Proc. Amer. Math. Soc.

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Abstract. We study minimal Lagrangian immersions from an indefinite real space form M n s (c) into an indefinite complex space formM n s (4c). Provided that c =c, we show that M n s (c) has to be flat and we obtain an explicit description of the immersion. In the case when the metric is positive definite or Lorentzian, this result was respectively obtained by Ejiri (1982) and by Kriele and the author (1999). In the case that c =c, this theorem is no longer true; see for instance the examples discovered by Chen and the author (accepted for publication in the Tôhoku Mathematical Journal).

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Indefinite K�hler manifolds

Cited by 115 publications

References 13 publications

Real hypersurfaces of indefinite Kaehler manifolds

Real hypersurfaces of indefinite Kaehler manifolds

Quasi-biharmonic Lagrangian surfaces in Lorentzian complex space forms

Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms

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