On skew 2-projectable almost complex structures on T M

Anton Dekrét

Archivum Mathematicum (1998)

  • Volume: 034, Issue: 2, page 285-293
  • ISSN: 0044-8753

Abstract

top
We deal with a ( 1 , 1 ) -tensor field α on the tangent bundle T M preserving vertical vectors and such that J α = - α J is a ( 1 , 1 ) -tensor field on M , where J is the canonical almost tangent structure on T M . A connection Γ α on T M is constructed by α . It is shown that if α is a V B -almost complex structure on T M without torsion then Γ α is a unique linear symmetric connection such that α ( Γ α ) = Γ α and ∇ Γ α ( J α ) = 0 .

How to cite

top

Dekrét, Anton. "On skew 2-projectable almost complex structures on $TM$." Archivum Mathematicum 034.2 (1998): 285-293. <http://eudml.org/doc/248214>.

@article{Dekrét1998,
abstract = {We deal with a $(1, 1)$-tensor field $\alpha $ on the tangent bundle $TM$ preserving vertical vectors and such that $J\alpha =-\alpha J$ is a $(1, 1)$-tensor field on $M$, where $J$ is the canonical almost tangent structure on $TM$. A connection $\Gamma _\{\alpha \}$ on $TM$ is constructed by $\alpha $. It is shown that if $\alpha $ is a $VB$-almost complex structure on $TM$ without torsion then $\Gamma _\{\alpha \}$ is a unique linear symmetric connection such that $\alpha (\Gamma _\{\alpha \})=\Gamma _\{\alpha \}$ and $\nabla _\{\Gamma _\{\alpha \}\} (J\alpha ) =0$.},
author = {Dekrét, Anton},
journal = {Archivum Mathematicum},
keywords = {tangent bundle; skew 2-projectable; $(1, 1)$-vector fields; almost complex structure; connection; vertical tangent bundle; skew 2-projectable vector fields; almost complex structure; connection},
language = {eng},
number = {2},
pages = {285-293},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On skew 2-projectable almost complex structures on $TM$},
url = {http://eudml.org/doc/248214},
volume = {034},
year = {1998},
}

TY - JOUR
AU - Dekrét, Anton
TI - On skew 2-projectable almost complex structures on $TM$
JO - Archivum Mathematicum
PY - 1998
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 034
IS - 2
SP - 285
EP - 293
AB - We deal with a $(1, 1)$-tensor field $\alpha $ on the tangent bundle $TM$ preserving vertical vectors and such that $J\alpha =-\alpha J$ is a $(1, 1)$-tensor field on $M$, where $J$ is the canonical almost tangent structure on $TM$. A connection $\Gamma _{\alpha }$ on $TM$ is constructed by $\alpha $. It is shown that if $\alpha $ is a $VB$-almost complex structure on $TM$ without torsion then $\Gamma _{\alpha }$ is a unique linear symmetric connection such that $\alpha (\Gamma _{\alpha })=\Gamma _{\alpha }$ and $\nabla _{\Gamma _{\alpha }} (J\alpha ) =0$.
LA - eng
KW - tangent bundle; skew 2-projectable; $(1, 1)$-vector fields; almost complex structure; connection; vertical tangent bundle; skew 2-projectable vector fields; almost complex structure; connection
UR - http://eudml.org/doc/248214
ER -

References

top
  1. Special tangent valued forms and the Frölicher-Nijenhuis bracket, Arch. Mathematicum (Brno) Tom. 29 (1993), 71–82. (1993) MR1242630
  2. Almost complex structures and connections on T M , Proc. Conf. Differential Geometry and Applications, Masaryk Univ. Brno (1996), 133–140. (1996) MR1406333
  3. Lifts of some tensor fields and connections to product preserving functors, Nagoya Math. Journal 135 (September 1994), 1–41. (September 1994) MR1295815
  4. Structure presque-tangent et connections I., Ann. Inst. Gourier (Grenoble) 22 (1972), 287–334. (1972) MR0336636
  5. Remarks on the Nijenhuis tensor and almost comples connections, Arch. Math. (Brno) 26 No. 4 (1990), 229–240. MR1188975
  6. Foundations of differential geometry II., Interscience publishers, 1969. (1969) 
  7. Natural operations in differential geometry, Springer-Verlag, 1993. (1993) MR1202431
  8. Tangent and cotangent bundles, M. Dekker Inc. New York, 1973. (1973) MR0350650
  9. Differential geometry on complex and almost complex spaces, Pergamon Press, New York, 1965. (1965) Zbl0127.12405MR0187181

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.