Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric application at the end.1991 Mathematics Subject Classification. Primary 53C05.

We provide geometric realizations of both classical and “exotic” $G$-manifolds such as spheres, Kervaire manifolds, bundles over spheres, homogeneous spaces, and connected sums among them. As an application, we apply the process known as Cheeger deformation to produce new metrics of both positive Ricci and almost non-negative curvature on such objects.

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This paper presents a direct and simple proof of a result concerning the existence of metrics of positive Ricci curvature on the total space of fiber bundles with compact structure groups. In particular, it generalizes and puts in a unified framework the results of Nash [12] and Poor [15]. With the intention of disseminating this result,we apply it to build new examples of manifolds with positive Ricci curvature, including bundles whose base consists of gradient shrinking Ricci solitons.

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