Geodesic
Non-standard 3-spheres locally foliated by elastic helices
Glasgow mathematical journal, Tome 43 (2001) no. 2, pp. 269-274

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In this note we use the Hopf map to construct a family of metrics in the 3-sphere parametrized on the space of positive smooth functions in the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. Also, the fibres are all geodesics if and only if the metric comes from a constant function and so, we have a Berger 3-sphere. Every geodesic in a 3-dimensional Riemannian manifold is a minimum for each elastic energy functional. Therefore, we characterize those functions on the 2-sphere that locally give metrics which have all the fibres being elastica, i.e., critical points of those functionals. Some applications are given including one to the Willmore-Chen variational problem. 
Cabrerizo, José L.; Fernández, Manuel. Non-standard 3-spheres locally foliated by elastic helices. Glasgow mathematical journal, Tome 43 (2001) no. 2, pp. 269-274. doi: 10.1017/S0017089501020109
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     title = {Non-standard 3-spheres locally foliated by elastic helices},
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     year = {2001},
     volume = {43},
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     doi = {10.1017/S0017089501020109},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089501020109/}
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