Almost complex projective structures and their morphisms

Jaroslav Hrdina

Archivum Mathematicum (2009)

  • Volume: 045, Issue: 4, page 255-264
  • ISSN: 0044-8753

Abstract

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We discuss almost complex projective geometry and the relations to a distinguished class of curves. We present the geometry from the viewpoint of the theory of parabolic geometries and we shall specify the classical generalizations of the concept of the planarity of curves to this case. In particular, we show that the natural class of J-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving of this class turns out to be the necessary and sufficient condition on diffeomorphisms to become homomorphisms or anti-homomorphisms of almost complex projective geometries.

How to cite

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Hrdina, Jaroslav. "Almost complex projective structures and their morphisms." Archivum Mathematicum 045.4 (2009): 255-264. <http://eudml.org/doc/250693>.

@article{Hrdina2009,
abstract = {We discuss almost complex projective geometry and the relations to a distinguished class of curves. We present the geometry from the viewpoint of the theory of parabolic geometries and we shall specify the classical generalizations of the concept of the planarity of curves to this case. In particular, we show that the natural class of J-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving of this class turns out to be the necessary and sufficient condition on diffeomorphisms to become homomorphisms or anti-homomorphisms of almost complex projective geometries.},
author = {Hrdina, Jaroslav},
journal = {Archivum Mathematicum},
keywords = {linear connection; geodetics; $F$-planar; $A$-planar; parabolic geometry; Cartan geometry; almost complex structure; projective structure; linear connection; geodetic; -planar; -planar; parabolic geometry; Cartan geometry; almost complex structure; projective structure},
language = {eng},
number = {4},
pages = {255-264},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Almost complex projective structures and their morphisms},
url = {http://eudml.org/doc/250693},
volume = {045},
year = {2009},
}

TY - JOUR
AU - Hrdina, Jaroslav
TI - Almost complex projective structures and their morphisms
JO - Archivum Mathematicum
PY - 2009
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 045
IS - 4
SP - 255
EP - 264
AB - We discuss almost complex projective geometry and the relations to a distinguished class of curves. We present the geometry from the viewpoint of the theory of parabolic geometries and we shall specify the classical generalizations of the concept of the planarity of curves to this case. In particular, we show that the natural class of J-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving of this class turns out to be the necessary and sufficient condition on diffeomorphisms to become homomorphisms or anti-homomorphisms of almost complex projective geometries.
LA - eng
KW - linear connection; geodetics; $F$-planar; $A$-planar; parabolic geometry; Cartan geometry; almost complex structure; projective structure; linear connection; geodetic; -planar; -planar; parabolic geometry; Cartan geometry; almost complex structure; projective structure
UR - http://eudml.org/doc/250693
ER -

References

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  1. Čap, A., Slovák, J., Weyl structures for parabolic geometries, Math. Scand. 93 (2003), 53–90. (2003) Zbl1076.53029MR1997873
  2. Čap, A., Slovák, J., Parabolic geometries I, Background and general theory, Math. Surveys Monogr., vol. 154, AMS Publishing House, 2009, p. 628. (2009) Zbl1183.53002MR2532439
  3. Hrdina, J., Slovák, J., Generalized planar curves and quaternionic geometry, Global analysis and geometry 29 (2006), 349–360. (2006) Zbl1097.53008MR2251428
  4. Hrdina, J., Slovák, J., Morphisms of almost product projective geometries, Differential Geometry and Applications, World Scientific, 2008, pp. 243–251. (2008) Zbl1168.53013MR2462798
  5. Kobayashi, S., Transformation groups in differential geometry, Springer-Verlag, New York-Heidelberg, 1972. (1972) Zbl0246.53031MR0355886
  6. Mikeš, J., Sinyukov, N. S., On quasiplanar mappings of spaces of affine connection, Soviet Math. 27 (1) (1983), 63–70. (1983) 
  7. Šilhan, J., Algorithmic computations of Lie algebras cohomologies, Rend. Circ. Mat. Palermo (2) Suppl. 71 (2003), 191–197, Proceedings of the 22nd Winter School “Geometry and Physics” (Srní, 2002), www.math.muni.cz/ silhan/lie. (2003) Zbl1032.17037MR1982446
  8. Yano, K., Differential geometry on complex and almost complex spaces, The Macmillan Company NY, 1965. (1965) Zbl0127.12405MR0187181

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