Kähler manifolds of quasi-constant holomorphic sectional curvatures

Georgi Ganchev; Vesselka Mihova

Open Mathematics (2008)

  • Volume: 6, Issue: 1, page 43-75
  • ISSN: 2391-5455

Abstract

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The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ℂn. Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.

How to cite

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Georgi Ganchev, and Vesselka Mihova. "Kähler manifolds of quasi-constant holomorphic sectional curvatures." Open Mathematics 6.1 (2008): 43-75. <http://eudml.org/doc/269588>.

@article{GeorgiGanchev2008,
abstract = {The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ℂn. Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.},
author = {Georgi Ganchev, Vesselka Mihova},
journal = {Open Mathematics},
keywords = {Kähler manifolds with J-invariant distributions; Kähler manifolds of quasi-constant holomorphic sectional curvatures; biconformal transformations; biconformal invariants; even dimensional rotational hypersurfaces},
language = {eng},
number = {1},
pages = {43-75},
title = {Kähler manifolds of quasi-constant holomorphic sectional curvatures},
url = {http://eudml.org/doc/269588},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Georgi Ganchev
AU - Vesselka Mihova
TI - Kähler manifolds of quasi-constant holomorphic sectional curvatures
JO - Open Mathematics
PY - 2008
VL - 6
IS - 1
SP - 43
EP - 75
AB - The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ℂn. Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.
LA - eng
KW - Kähler manifolds with J-invariant distributions; Kähler manifolds of quasi-constant holomorphic sectional curvatures; biconformal transformations; biconformal invariants; even dimensional rotational hypersurfaces
UR - http://eudml.org/doc/269588
ER -

References

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  9. [9] Kobayashi S., Nomizu K., Foundations of Differential Geometry, Vol. II, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1969 Zbl0175.48504
  10. [10] Tashiro Y., On contact structure of hypersurfaces in complex manifolds I, Tôhoku Math. J. (2), 1963, 15, 62–78 http://dx.doi.org/10.2748/tmj/1178243870 
  11. [11] Tashiro Y., On contact structure of hypersurfaces in complex manifolds II, Tôhoku Math. J. (2), 1963, 15, 167–175 http://dx.doi.org/10.2748/tmj/1178243843 Zbl0126.38003
  12. [12] Tachibana S., Liu R.C., Notes on Kählerian metrics with vanishing Bochner curvature tensor, Kōdai Math. Sem. Rep., 1970, 22, 313–321 http://dx.doi.org/10.2996/kmj/1138846167 Zbl0199.25303
  13. [13] Tricerri F., Vanhecke L., Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc., 1981, 267, 365–397 http://dx.doi.org/10.2307/1998660 Zbl0484.53014

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