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On extensions of sub-Riemannian structures on Lie groups. (English) Zbl 1337.53042

Summary: We define the extension of a left-invariant sub-Riemannian structure in terms of an extension of the underlying Lie group and compatibility of the respective distributions and metrics. We show that geodesics of a structure can be lifted to geodesics of any extension of the structure. In the case of central extensions, we show that the normal geodesics of the minimal extension are the projection (in a sense) of the normal geodesics of any other compatible extension. Several illustrative examples are discussed.

MSC:

53C17 Sub-Riemannian geometry
22E30 Analysis on real and complex Lie groups
49J15 Existence theories for optimal control problems involving ordinary differential equations
53C22 Geodesics in global differential geometry
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