Decomposition theorems of Riemannian manifolds

Wang, Pyng

doi:10.1090/S0002-9947-1973-0328824-8

Decomposition theorems of Riemannian manifolds
HTML articles powered by AMS MathViewer

by Pyng Wang PDF

Trans. Amer. Math. Soc. 184 (1973), 327-341 Request permission

Abstract:

Given two complementary orthogonal parallel foliations on a complete connected Riemannian manifold M, a necessary and sufficient condition for the direct product of the two leaves through a point m being a covering manifold of M is obtained. It is shown that the direct product of the two leaves through m of the two foliations is a Riemannian covering of M if the two leaves are regular at m. Moreover, if one of the foliations is regular and the intersection of the two leaves through m contains only the point m, then M is isometric to the direct product of the two leaves.

References

Claude Chevalley, Theory of Lie groups. I, Princeton University Press, Princeton, N. J., 1946 1957. MR 0082628, DOI 10.1515/9781400883851
Sh\B{o}bin Kashiwabara, On the reducibility of an affinely connected manifold, Tohoku Math. J. (2) 8 (1956), 13–28. MR 80958, DOI 10.2748/tmj/1178245005
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
Richard S. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. 22 (1957), iii+123. MR 121424
Georges Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Publ. Inst. Math. Univ. Strasbourg, vol. 11, Hermann & Cie, Paris, 1952 (French). MR 0055692
Georges de Rham, Sur la reductibilité d’un espace de Riemann, Comment. Math. Helv. 26 (1952), 328–344 (French). MR 52177, DOI 10.1007/BF02564308
Bruce L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119–132. MR 107279, DOI 10.2307/1970097
S. A. Robertson, Grid manifolds, J. Differential Geometry 4 (1970), 245–253. MR 268814, DOI 10.4310/jdg/1214429388
Joseph A. Wolf, Locally symmetric homogeneous spaces, Comment. Math. Helv. 37 (1962/63), 65–101. MR 148012, DOI 10.1007/BF02566963
Joseph A. Wolf, On locally symmetric spaces of non-negative curvature and certain other locally homogeneous spaces, Comment. Math. Helv. 37 (1962/63), 266–295. MR 154228, DOI 10.1007/BF02566977
H. Wu, On the de Rham decomposition theorem, Illinois J. Math. 8 (1964), 291–311. MR 161280
H. Wu, Decomposition of Riemannian manifolds, Bull. Amer. Math. Soc. 70 (1964), 610–617. MR 163263, DOI 10.1090/S0002-9904-1964-11212-X