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On the existence of complete parallel vector fields. (English) Zbl 0603.53024

Author’s abstract: ”In 1966, Chern asked which compact orientable manifolds carry a Riemannian metric and a vector field which is parallel under this metric. The author [Ill. J. Math. 30, 9-18 (1986; Zbl 0581.53027)] answered this without assuming orientability. The present paper applies techniques which work without compactness to identify manifolds which can carry complete parallel vector fields.”
Reviewer: H.Karcher

MSC:

53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 0581.53027
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References:

[1] S. S. Chern, The geometry of \?-structures, Bull. Amer. Math. Soc. 72 (1966), 167 – 219. · Zbl 0136.17804
[2] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. · Zbl 0091.34802
[3] Peter Percell, Parallel vector fields on manifolds with boundary, J. Differential Geom. 16 (1981), no. 1, 101 – 104. · Zbl 0451.57012
[4] Polychronis Strantzalos, Dynamische Systeme und topologische Aktionen, Manuscripta Math. 13 (1974), 207 – 211 (German, with English summary). · Zbl 0315.57030
[5] Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Co., Glenview, Ill.-London, 1971. · Zbl 0241.58001
[6] David J. Welsh Jr., Manifolds that admit parallel vector fields, Illinois J. Math. 30 (1986), no. 1, 9 – 18. · Zbl 0581.53027
[7] Shing Tung Yau, Remarks on the group of isometries of a Riemannian manifold, Topology 16 (1977), no. 3, 239 – 247. · Zbl 0372.53020
[8] K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. · Zbl 0051.39402
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