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Certain spheres admit nontrivial fibrations, i.e. the Hopf fibrations and the maps to projective spaces. Also, a product of spheres is a sphere bundle in more than one way.

Are there manifolds which are sphere bundles in more than one way, not factoring through the above examples?

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    $\begingroup$ A compact Lie group $G$ having two subgroups $H$ and $K$ both isomorphic to $SU(2)$ is a $3$-sphere bundle over both $G/H$ and $G/K$. I think this would count as a new class of examples. $\endgroup$
    – Tom Goodwillie
    Commented Oct 26, 2016 at 23:42
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    $\begingroup$ Check out Jason de Vito's answer to this: math.stackexchange.com/questions/129001/… $\endgroup$
    – Igor Rivin
    Commented Oct 27, 2016 at 1:29

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Here is another example, involving $\operatorname{SU}(4)$. I can't produce arrows so I will write the examples as homogeneous spaces. We have $\operatorname{SU}(4)/\operatorname{SU}(3) =S^7$ and $\operatorname{SU}(4)/\operatorname{Sp}(2)=S^5$. I should say that for me the term "sphere bundle" can mean that either the fiber or the base is a sphere, and that may not be common usage.

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