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Abnormal Sub-Riemannian Minimizers
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
Wensheng Liu, Hector J. Sussmann
A C∞Riemannian metric on Ε is a C∞ section p → Gp of the bundle E* ⊗ E* such that for each p ε M the bilinear form TPM × TPM ϶ (v,w) → Gp(v,w) ε ℝ is symmetric and positive definite. A sub-Riemannian structure on a manifold M is a pair (E, G) where Ε is a nonholonomic C∞ subbundle of TM and G is a C∞ Riemannian metric on E. A sub-Riemannian manifold is a triple (M, E, G) such that M is a C∞ manifold and (E, G) is a sub-Riemannian structure on M. One can always construct a Riemannian metric on any subbundle Ε of ΤM by just taking a Riemannian metric on ΤM and restricting it to E. If p ε M, ν ε Ε(p), then the length ‖v‖G of ν is the number Gp(v, v)1/2. The length ‖γ‖g of an .Ε-admissible arc γ : [a, b] → M is the integral ∫ab||γ˙(t)||Gdt. If p, q ε M, then the infimum of the lengths of all the E-admissible curves γ that go from p to q is the distancefrom p to q, and is denoted by dG(p, q). If M is connected and Ε is nonholonomic, then dG(p, q) < ∞ for all p, and dG : M × M → ℝ is a metric whose associated topology is the one of M. An inadmissible curve γ : [a, b] → M such that dG(γ(a), γ(b)) = ‖γ‖ is called a minimizer.
Nonlinear nonhomogeneous Neumann problem on the Heisenberg group
Published in Applicable Analysis, 2022
The Heisenberg group is an example of a sub-Riemannian manifold homeomorphic, but not bi-Lipschitz equivalent to the Euclidean space. The Heisenberg group (and more generally, stratified groups) is a special case of metric measure spaces with doubling measures. Its metric is derived from curves which are only allowed to move in so-called horizontal directions. The applications of the Heisenberg group include several complex variables, subelliptic equations and noncommutative harmonic analysis (see [1,2]).