Geometry of Cyclic Quotients, I: Knotted Totally Geodesic Submanifolds in Positively Curved Spheres

Details Geometry of Cyclic Quotients, I: Knotted Totally Geodesic Submanifolds in Positively Curved Spheres Geometry of Cyclic Quotients, I: Knotted Totally Geodesic Submanifolds in Positively Curved Spheres

1994-10-16

arxiv.org/abs/dg-ga/9410006v1

Mathematics

Differential Geometry

6 pages, plane TEX

Scientific paper

We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a totally geodesic projective plane with Euler number four. Surpisingly, the technique borrows a lot from the Mostow-Siu-Gromov-Thurston constuction of exotic negatively curved manifolds.

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