Geodesic
The Sectional Curvature Remains Positive When Taking Quotients by Certain Nonfree Actions
Matematičeskie trudy, Tome 10 (2007) no. 2, pp. 62-91

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We study some cases in which 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 $S^3$ for which the quotient space can be endowed with a smooth structure by means of the fibrations $S^3/S^1\simeq S^2$ and $S^7/S^3\simeq S^4$. We prove that the quotient space possesses 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. 
@article{MT_2007_10_2_a2,
     author = {S. V. Dyatlov},
     title = {The {Sectional} {Curvature} {Remains} {Positive} {When} {Taking} {Quotients} by {Certain} {Nonfree} {Actions}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {62--91},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2007_10_2_a2/}
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S. V. Dyatlov. The Sectional Curvature Remains Positive When Taking Quotients by Certain Nonfree Actions. Matematičeskie trudy, Tome 10 (2007) no. 2, pp. 62-91. http://geodesic.mathdoc.fr/item/MT_2007_10_2_a2/