In the sub-Riemannian framework, we give geometric necessary and sufficient conditions for the existence of abnormal extremals of the Maximum Principle. We give relations between abnormality, -rigidity and length minimizing. In particular, in the case of three dimensional manifolds we show that, if there exist abnormal extremals, generically, they are locally length minimizing and in the case of four dimensional manifolds we exhibit abnormal extremals which are not -rigid and which can be minimizing or non minimizing according to different metrics.
@article{bwmeta1.element.bwnjournal-article-bcpv32z1p301bwm,
author = {Pelletier, F. and Bouche, L.},
title = {Abnormality of trajectory in sub-Riemannian structure},
journal = {Banach Center Publications},
volume = {31},
year = {1995},
pages = {301-317},
zbl = {0836.53025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv32z1p301bwm}
}
Pelletier, F.; Bouche, L. Abnormality of trajectory in sub-Riemannian structure. Banach Center Publications, Tome 31 (1995) pp. 301-317. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv32z1p301bwm/
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