no. 2
De Rham decomposition theorems for foliated manifolds
Annales de l'Institut Fourier, Tome 33 (1983) no. 2, pp. 183-198.

Nous démontrons que si M est une variété riemannienne complète simplement connexe et F est un feuilletage totalement géodésique sur M dont le fibré orthogonal est involutif, alors M est topologiquement un produit et les deux feuilletages sont les feuilletages produits. Nous démontrons aussi un théorème de décomposition pour les feuilletages riemanniens et un théorème de structure pour les feuilletages riemanniens à courbure récurrente.

We prove that if M is a complete simply connected Riemannian manifold and F is a totally geodesic foliation of M with integrable normal bundle, then M is topologically a product and the two foliations are the product foliations. We also prove a decomposition theorem for Riemannian foliations and a structure theorem for Riemannian foliations with recurrent curvature.

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     author = {Blumenthal, Robert A. and Hebda, James J.},
     title = {De {Rham} decomposition theorems for foliated manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {183--198},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {33},
     number = {2},
     year = {1983},
     doi = {10.5802/aif.923},
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     mrnumber = {84j:53042},
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}
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Blumenthal, Robert A.; Hebda, James J. De Rham decomposition theorems for foliated manifolds. Annales de l'Institut Fourier, Tome 33 (1983) no. 2, pp. 183-198. doi : 10.5802/aif.923. https://aif.centre-mersenne.org/articles/10.5802/aif.923/

[1] R. A. Blumenthal, Transversely homogeneous foliations, Annales Inst. Fourier, 29 (1979), 143-158. | Numdam | MR | Zbl

[2] R. A. Blumenthal, Riemannian homogeneous foliations without holonomy, Nagoya Math. J., 83 (1981), 197-201. | MR | Zbl

[3] R. A. Blumenthal, Riemannian foliations with parallel curvature, Nagoya Math. J. (to appear). | Zbl

[4] R. Bott, Lectures on characteristic classes and foliations (notes by L. Conlon), Lecture Notes in Math., no. 279, Springer-Verlag, New York, 1972, 1-80. | MR | Zbl

[5] Y. Carriere and E. Ghys, Feuilletages totalement géodésiques, An. Acad. Brasil, Ciênc., 53 (1981), 427-432. | MR | Zbl

[6] A. Haefliger, Variétés feuilletées, Ann. Scuola Norm. Pisa, 16 (1962), 367-397. | Numdam | MR | Zbl

[7] R. Hermann, On the differential geometry of foliations, Annals of Math., 72 (1960), 445-457. | MR | Zbl

[8] R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc., 11 (1960), 236-242. | MR | Zbl

[9] D. Johnson and L. Whitt, Totally geodesic foliations on 3-manifolds, Proc. Amer. Math. Soc., 76 (1979), 355-357. | MR | Zbl

[10] D. Johnson and L. Whitt, Totally geodesic foliations, Journal of Diff. Geom., 15 (1980), 225-235. | MR | Zbl

[11] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. I, Interscience Tracts in Pure and Appl. Math., 15, Interscience, New York, 1963. | Zbl

[12] C. Lazarov and J. Pasternak, Secondary characteristic classes for Riemannian foliations, Journal of Diff. Geom., 11 (1976), 365-385. | MR | Zbl

[13] P. Molino, Etude des feuilletages transversalement complets et applications, Ann. Scient. Ec. Norm. Sup., 10 (1977), 289-307. | Numdam | MR | Zbl

[14] J. F. Plante, Foliations with measure preserving holonomy, Annals of Math., 102 (1975), 327-361. | MR | Zbl

[15] J. F. Plante, Measure preserving pseudogroups and a theorem of Sacksteder, Annales Inst. Fourier, 25, 1 (1975), 237-249. | Numdam | MR | Zbl

[16] B. Reinhart, Foliated manifolds with bundle-like metrics, Annals of Math., 69 (1959), 119-132. | MR | Zbl

[17] G. De Rham, Sur la réductibilité d'un espace de Riemann, Comm. Math. Helv., 26 (1952), 328-344. | MR | Zbl

[18] R. Sacksteder, Foliations and pseudogroups, Amer. J. of Math., 87 (1965), 79-102. | MR | Zbl

[19] D. Tischler, On fibering certain foliated manifolds over S1, Topology, 9 (1970), 153-154. | MR | Zbl

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