Abstract
In this paper, we define and study sub-Riemannian structures on Banach manifolds. We obtain extensions of the Chow–Rashevsky Theorem for exact controllability, and give conditions for the existence of a Hamiltonian geodesic flow despite the lack of a Pontryagin Maximum Principle in the infinite- dimensional setting.
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Notes
Here, we somewhat stretch the definition of distance to allow infinite values for the distance between points that cannot be horizontally connected.
Each \({\tilde{q}}(s,\cdot )\) is the projection to M of a curve \(({\tilde{q}}(s,\cdot ),{\tilde{p}}(s,\cdot ))\) that follows the Hamiltonian flow with initial condition \(({\tilde{q}}_0+s\delta q_0,n({\tilde{q}}_0+s\delta q_0) p_0)\). Then \({\tilde{u}}(s,t)=u({\tilde{q}}(s,t),{\tilde{p}}(s,t))\), which is \(\mathcal {C}^2\) in (t, s). Moreover, \(g_{{\tilde{q}}(s,t)}({\tilde{u}}(s,t),{\tilde{u}}(s,t))=2h({\tilde{q}}(s,t),{\tilde{p}}(s,t))=2h(q_0,p_0)=c^2\).
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Arguillère, S. Sub-Riemannian Geometry and Geodesics in Banach Manifolds. J Geom Anal 30, 2897–2938 (2020). https://doi.org/10.1007/s12220-019-00184-5
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DOI: https://doi.org/10.1007/s12220-019-00184-5