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- Title
On non-negatively curved metrics on open five-dimensional manifolds.
- Authors
Mikael Bengtsson; Valery Marenich
- Abstract
Abstract??LetVnbe an open manifold of non-negative sectional curvature with a soul ? of co-dimension two. The universal cover$$\tilde n$$of the unit normal bundleNof the soul in such a manifold is isometric to the direct productMn-2??R. In the study of the metric structure ofVnan important role plays the vector fieldXwhich belongs to the projection of the vertical planes distribution of the Riemannian submersion$$\pi\!\!:v\to\sigma$$on the factorMin this metric splitting$$\tilde n = m\times r$$. The casen?=?4 was considered in [Gromoll, D., Tapp, K.: Geom. Dedicata99, 127?136 (2003)] where the authors prove thatXis a Killing vector field while the manifoldV4is isometric to the quotient of$$m^2\times (r^2,g_f)\times r$$by the flow along the corresponding Killing field. Following an approach of [Gromoll, D., Tapp, K.: Geom. Dedicata99, 127?136 (2003)] we consider the next casen?=?5 and obtain the same result under the assumption that the set of zeros ofXis not empty. Under this assumption we prove that bothM3and ?3admit an open-book decomposition with a bending which is a closed geodesic and pages which are totally geodesic two-spheres, the vector fieldXis Killing, while the whole manifoldV5is isometric to the quotient of$$m^3\times (r^2,g_f)\times r$$by the flow along corresponding Killing field.
- Subjects
RIEMANNIAN submersions; VECTOR fields; RIEMANNIAN geometry; SUBMERSIONS (Mathematics)
- Publication
Annals of Global Analysis & Geometry, 2007, Vol 31, Issue 2, p213
- ISSN
0232-704X
- Publication type
Article
- DOI
10.1007/s10455-006-9044-6