A Tour of Subriemannian Geometries, Their Geodesics and Applications

By Richard Montgomery. 2006

1,253 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/9889231a10a1b9a39e0549d819f9e04a5a56d4c7

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In Geometry and Topology

Citations in the corpus

Precise estimates for the subelliptic heat kernel on H-type groups (2008, methodology): “This always holds in the Riemannian setting, but is not generally true in subriemannian geometry; the Martinet distribution (see Chapter 3 of [22]) is a counterexample in which some shortest paths do not satisfy Hamilton’s equations.”
Duality on gradient estimates and Wasserstein controls (2009, background): “heir notion of gradient looks diﬀerent from ours. To explain how we deal with it, we will demonstrate a general framework of sub-Riemannian geometry generated by a family of vector ﬁelds. We refer to [16, 28, 33] for details. Throughout this section, we assume X to be a ﬁnite dimensional, σ-compact,”
Euler estimates for rough differential equations (2006, background): “Theorem 7 (Chow, [18]) For every g ∈ G ( R ) there exists an R-valued Lipschitz path x such that SN (x)0,1 = g.”
Sub-Riemannian interpolation inequalities (2017, background): “For a comprehensive introduction, we refer to [3,44,50].”
A Formula for Popp’s Volume in Sub-Riemannian Geometry (2012, background): “oup. A sub-Riemannian structure (M,D,h·,·i) is left invariant if ∀g ∈ M, the left action Lg : M → M is an isometry. As a trivial consequence of Proposition 5 we recover a well-known result (see again [13]). Corollary 8. Let (M,D,h·,·i) be a left-invariant sub-Riemannian structure. Then Popp’s volume”
Shape deformation analysis from the optimal control viewpoint (2014, background): “…generalized in many ways, and a large number of variants or improvements have been made over the past decades, with particular eﬀorts in order to be able to address optimal control problems involving general state/control constraints (see the survey article [23] and the many references therein).”
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, background): “It can be formulated in sub-Riemannian geometry ([125] and the references therein), see also [169].”