| Title:
|
Riemannian foliations with parallel or harmonic basic forms (English) |
| Author:
|
El Chami, Fida |
| Author:
|
Habib, Georges |
| Author:
|
Nakad, Roger |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
51 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
51-65 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we consider a Riemannian foliation that admits a nontrivial parallel or harmonic basic form. We estimate the norm of the O’Neill tensor in terms of the curvature data of the whole manifold. Some examples are then given. (English) |
| Keyword:
|
Riemannian foliation |
| Keyword:
|
parallel and harmonic basic forms |
| Keyword:
|
O’Neill tensor |
| MSC:
|
53C12 |
| MSC:
|
53C20 |
| MSC:
|
53C24 |
| MSC:
|
57R30 |
| idZBL:
|
Zbl 06487020 |
| idMR:
|
MR3338765 |
| DOI:
|
10.5817/AM2015-1-51 |
| . |
| Date available:
|
2015-04-01T12:52:02Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144233 |
| . |
| Reference:
|
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| Reference:
|
[2] Gromoll, D., Grove, K.: The low dimensional metric foliations of Euclidean spheres.J. Differential Geom. 28 (1988), 143–156. Zbl 0683.53030, MR 0950559 |
| Reference:
|
[3] Grosjean, J.–F.: Minimal submanifolds with a parallel or a harmonic $p$–form.J. Geom. Phys. 51 (2004), 211–228. Zbl 1074.53050, MR 2078671, 10.1016/j.geomphys.2003.10.009 |
| Reference:
|
[4] Habib, G., Richardson, K: Modified differentials and basic cohomology for Riemannian foliations.J. Geom. Anal. 23 (2013), 1314–1342. Zbl 1276.53031, MR 3078356, 10.1007/s12220-011-9289-6 |
| Reference:
|
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| Reference:
|
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| Reference:
|
[7] O’Neill, B.: The fundamental equations of a submersion.Michigan Math. J. 13 (1966), 459–469. MR 0200865, 10.1307/mmj/1028999604 |
| Reference:
|
[8] Ranjan, A.: On a remark of O’Neill.Duke Math. J. 53 (1981), 363–373. MR 0835797 |
| Reference:
|
[9] Reinhart, B.: Foliated manifolds with bundle-like metrics.Ann. of Math. (2) 69 (1959), 119–132. Zbl 0122.16604, MR 0107279, 10.2307/1970097 |
| Reference:
|
[10] Tondeur, P.: Geometry of Foliations.Birkhäuser, Boston, 1997. Zbl 0905.53002, MR 1456994 |
| . |
