Geodesic
Nonhomogeneous $G$-spaces of compact Lie groups
Matematičeskie zametki, Tome 13 (1973) no. 4, pp. 523-529.

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A duality theorem for a compact $G$-manifold $M$ of a compact Lie group $G$ is proved. For the case when all the orbits in $M$ are principal, it is proved that the quotient space of the complexification of $M$ by the action of the associated algebraic group $G^c$ is isomorphic to the quotient space of $M$ by the action of $G$. 
@article{MZM_1973_13_4_a5,
     author = {M. Ya. Blinkin},
     title = {Nonhomogeneous $G$-spaces of compact {Lie} groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {523--529},
     publisher = {mathdoc},
     volume = {13},
     number = {4},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_13_4_a5/}
}
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M. Ya. Blinkin. Nonhomogeneous $G$-spaces of compact Lie groups. Matematičeskie zametki, Tome 13 (1973) no. 4, pp. 523-529. http://geodesic.mathdoc.fr/item/MZM_1973_13_4_a5/