The Sectional Curvature Remains Positive when Taking Quotients by Certain Nonfree Actions

Details The Sectional Curvature Remains Positive when Taking Quotients by Certain Nonfree Actions The Sectional Curvature Remains Positive when Taking Quotients by Certain Nonfree Actions

2007-10-22

arxiv.org/abs/0710.3912v1

Published in Siberian Advances in Mathematics 18(2008), pages 1-20

Mathematics

Differential Geometry

26 pages, 1 MetaPOST figure, 7 references. To be published in Siberian Advances in Mathematics in the beginning of 2008

Scientific paper

10.3103/S105513440801001X

We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ such that the quotient space can be endowed with a smooth structure using the fibrations $S^3/S^1{\simeq}S^2$ and $S^7/S^3{\simeq}\nobreak S^4$. We prove that the quotient space carries a metric of positive sectional curvature, provided that the original metric has positive sectional curvature on all 2-planes orthogonal to the orbits of the action.

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