| Title:
|
On skew 2-projectable almost complex structures on $TM$ (English) |
| Author:
|
Dekrét, Anton |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
34 |
| Issue:
|
2 |
| Year:
|
1998 |
| Pages:
|
285-293 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\nabla _{\Gamma _{\alpha }} (J\alpha ) =0$. (English) |
| Keyword:
|
tangent bundle |
| Keyword:
|
skew 2-projectable |
| Keyword:
|
$(1, 1)$-vector fields |
| Keyword:
|
almost complex structure |
| Keyword:
|
connection |
| MSC:
|
53C05 |
| MSC:
|
53C15 |
| MSC:
|
58A20 |
| idZBL:
|
Zbl 0910.53020 |
| idMR:
|
MR1645324 |
| . |
| Date available:
|
2009-02-17T10:12:04Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107653 |
| . |
| Reference:
|
[1] Cabras, A., Kolář, I.: Special tangent valued forms and the Frölicher-Nijenhuis bracket.Arch. Mathematicum (Brno) Tom. 29 (1993), 71–82. MR 1242630 |
| Reference:
|
[2] Dekrét, A.: Almost complex structures and connections on $TM$.Proc. Conf. Differential Geometry and Applications, Masaryk Univ. Brno (1996), 133–140. MR 1406333 |
| Reference:
|
[3] Gancarzewicz, J., Mikulski, W., Pogula, Z.: Lifts of some tensor fields and connections to product preserving functors.Nagoya Math. Journal 135 (September 1994), 1–41. MR 1295815 |
| Reference:
|
[4] Griffone, J.: Structure presque-tangent et connections I..Ann. Inst. Gourier (Grenoble) 22 (1972), 287–334. MR 0336636 |
| Reference:
|
[5] Janyška, J.: Remarks on the Nijenhuis tensor and almost comples connections.Arch. Math. (Brno) 26 No. 4 (1990), 229–240. MR 1188975 |
| Reference:
|
[6] Kobayashi, S., Nomizu, K.: Foundations of differential geometry II..Interscience publishers, 1969. |
| Reference:
|
[7] Kolář, I., Michor, P.W., Slovák, J.: Natural operations in differential geometry.Springer-Verlag, 1993. MR 1202431 |
| Reference:
|
[8] Yano, K., Ishihara, S.: Tangent and cotangent bundles.M. Dekker Inc. New York, 1973. MR 0350650 |
| Reference:
|
[9] Yano, K.: Differential geometry on complex and almost complex spaces.Pergamon Press, New York, 1965. Zbl 0127.12405, MR 0187181 |
| . |
