Is it known if the total space of an $S^7$-bundles over $S^8$ with structure group $SO(8)$ admits a cohomogeneity one action?



Cohomogeneity one action on $S^7$-bundles over $S^8$
I don't think the argument is written down anywhere, but not all $S^7$-bundles over $S^8$ admit cohomogeneity one actions.


Focusing on those with Euler class $\pm 1$, the resulting total space $E$ is a homotopy sphere. Homotopy spheres admitting cohomogneity one actions have been classified by Straume in



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> Compact connected Lie transformation groups on spheres with low cohomogeneity I. Mem. Amer. Math. Soc., 119(569), 1996.
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In particular, Straume shows that in dimensions *not* of the form $4k+1\geq 5$, that a homotopy sphere admitting a cohomogeneity one action is diffeomorphic to the standard sphere. (In dimension $4k+1$, one can also get Brieskorn varieties.)


So, in dimension $15$, the only homotopy sphere which admits a cohomogeneity one action is the standard sphere. Since there are exotic spheres which are the total space of an $S^7$-bundle over $S^8$ (as shown by Shimada in



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> Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. J. 12 (1957), 59–69. MR0096223 (20 #2715) Zbl 0145.20303,
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the result follows.


