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Abnormal sub-Riemannian geodesics: Morse index and rigidity. (English) Zbl 0866.58023
Summary: Considering a smooth manifold $$M$$ provided with a sub-Riemannian structure, i.e., with Riemannian metric and nonintegrable vector distribution, we set a problem of finding for two given points $$q^0$$, $$q^1\in M$$ a length minimizer among Lipschitzian paths tangent to the vector distribution (admissible) and connecting these points. Extremals of this variational problem are called sub-Riemannian geodesics and we single out the abnormal sub-Riemannian geodesics, which correspond to the vanishing Lagrange multiplier for the length functional. These abnormal geodesics are not related to the Riemannian structure but only to the vector distribution and, in fact, are singular points in the set of admissible paths connecting $$q^0$$ and $$q^1$$. Developing the Legendre-Jacobi-Morse-type theory of second variation for abnormal geodesics we investigate some of their specific properties such as weak minimality and rigidity-isolatedness in the space of admissible paths connecting the two given points.

##### MSC:
 58E30 Variational principles in infinite-dimensional spaces 93B29 Differential-geometric methods in systems theory (MSC2000) 53C22 Geodesics in global differential geometry
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##### References:
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