Geodesic
The Sectional Curvature Remains Positive When Taking Quotients by Certain Nonfree Actions
Matematičeskie trudy, Tome 10 (2007) no. 2, pp. 62-91 
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/
@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},
     year = {2007},
     volume = {10},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2007_10_2_a2/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Bazaikin Ya. V., “Ob odnom semeistve 13-mernykh zamknutykh rimanovykh mnogoobrazii polozhitelnoi krivizny”, Sib. mat. zhurn., 37:6 (1996), 1219–1237 | MR | Zbl

[2] Besse A., Mnogoobraziya Einshteina, Mir, M., 1990 | MR | Zbl

[3] Kobayasi Sh., Gruppy preobrazovanii v differentsialnoi geometrii, Nauka, M., 1986 | MR

[4] Gao L. Z., “The construction of negatively Ricci curved manifolds”, Math. Ann., 271:2 (1985), 185–208 | DOI | MR | Zbl

[5] O'Neill B., “The fundamental equations of a submersion”, Michigan Math. J., 13 (1966), 459–469 | DOI | MR

[6] Vilms J., “Totally geodesic maps”, J. Differential Geom., 4:4 (1970), 73–79 | MR | Zbl

[7] Wallach N., “Compact homogeneous Riemannian manifolds with strictly positive curvature”, Ann. of Math., 96 (1972), 277–295 | DOI | MR | Zbl