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Riemannian and Sub-Riemannian geodesic flow ; Grong, Erlend in Journal of Geometric Analysis (The) (2016) We show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This result ... [more ▼] We show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This result allows us describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion $\pi:M \to B$, we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in $M$ coincide. [less ▲] Detailed reference viewed: 88 (14 UL)Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: Part I Grong, Erlend ; Thalmaier, Anton in Mathematische Zeitschrift (2016), 282(1), 99-130 We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub ... [more ▼] We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in the inequality. Using this inequality, we obtain a lower bound for the eigenvalues of the sub-Laplacian. This inequality also lays the foundation for proving several powerful results in Part II. [less ▲] Detailed reference viewed: 330 (56 UL)Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: Part II Grong, Erlend ; Thalmaier, Anton in Mathematische Zeitschrift (2016), 282(1), 131-164 Using the curvature-dimension inequality proved in Part I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semi-group P_t ... [more ▼] Using the curvature-dimension inequality proved in Part I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semi-group P_t corresponding to the sub-Laplacian. We give bounds for the gradient, entropy, a Poincaré inequality and a Li-Yau type inequality. These results require that the gradient of P_t f remains uniformly bounded whenever the gradient of f is bounded and we give several sufficient conditions for this to hold. [less ▲] Detailed reference viewed: 295 (32 UL)Submersions, Hamiltonian systems and optimal solutions to the rolling manifolds problem Grong, Erlend in SIAM Journal on Control & Optimization (2016), 54(2), 536-566 Given a submersion $\pi:Q \to M$ with an Ehresmann connection~$\calH$, we describe how to solve Hamiltonian systems on $M$ by lifting our problem to $Q$. Furthermore, we show that all solutions of these ... [more ▼] Given a submersion $\pi:Q \to M$ with an Ehresmann connection~$\calH$, we describe how to solve Hamiltonian systems on $M$ by lifting our problem to $Q$. Furthermore, we show that all solutions of these lifted Hamiltonian systems can be described using the original Hamiltonian vector field on $M$ along with a generalization of the magnetic force. This generalized force is described using the curvature of $\calH$ along with a new form of parallel transport of covectors vanishing on $\calH$. Using the Pontryagin Maximum Principle, we apply this theory to optimal control problems $M$ and $Q$ to get results on normal and abnormal extremals. We give a demonstration of our theory by considering the optimal control problem of one Riemannian manifold rolling on another without twisting or slipping along curves of minimal length. [less ▲] Detailed reference viewed: 88 (7 UL)Model spaces in sub-Riemannian geometry Grong, Erlend E-print/Working paper (2016) We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry of the horizontal tangent spaces is realized by a global isometry. We will show that ... [more ▼] We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry of the horizontal tangent spaces is realized by a global isometry. We will show that these spaces have a canonical partial connection defined on their horizontal bundle. However, unlike the Riemannian case, such spaces are not uniquely determined by their curvature and their metric tangent cone. Furthermore, the number of invariants needed to determine model spaces with the same tangent cone is in general greater than one. [less ▲] Detailed reference viewed: 55 (2 UL)Sub-Riemannian Geometry on Infinite-Dimensional Manifolds Grong, Erlend ; ; in Journal of Geometric Analysis (The) (2015), 25(4), 2474-2515 We generalize the concept of sub-Riemannian geometry to infinite- dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold M, the metric is defined only on a sub-bundle H of ... [more ▼] We generalize the concept of sub-Riemannian geometry to infinite- dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold M, the metric is defined only on a sub-bundle H of the tangent bundle T M, called the horizontal distribution. Similarly to the finite-dimensional case, we are able to split possible candidates for minimizing curves into two categories: semi-rigid curves that depend only on H, and normal geodesics that depend both on H itself and on the metric on H. In this sense, semi-rigid curves in the infinite-dimensional case generalize the notion of singular curves for finite dimensions. In particular, we study the case of regular Lie groups with invariant sub-Riemannian structure. As examples, we consider the group of sense-preserving diffeomorphisms Diff S1 of the unit circle and the Virasoro–Bott group with their respective horizontal distributions chosen to be the Ehresmann connections with respect to a projection to the space of normalized univalent functions. In these cases we prove controllability and find formulas for the normal geodesics with respect to the pullback of the invariant Kählerian metric on the class of normalized univalent functions. The geodesic equations are analogues to the Camassa–Holm, Hunter–Saxton, KdV, and other known non-linear PDE. [less ▲] Detailed reference viewed: 101 (8 UL)Horizontal holonomy and foliated manifolds ; Grong, Erlend ; et al E-print/Working paper (2015) We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle D of the tangent bundle. We provide explicit means of computing ... [more ▼] We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle D of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose-Singer’s and Ozeki’s theorems. We then give necessary and sufficient conditions in terms of the horizontal holonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure. The subbundle D plays the role of an orthogonal complement to the leaves of the foliation in case (a) and of a principal connection in case (b). [less ▲] Detailed reference viewed: 98 (3 UL)Geometric conditions for the existence of a rolling without twisting or slipping ; Grong, Erlend in Communications on Pure and Applied Analysis (2014), 13(1), 435-452 We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show ... [more ▼] We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that, up to technical hypotheses, a rolling along these curves exists if and only if the geodesic curvatures of each curve coincide. By using the anti-developments of the curves, which we claim can be seen as a generalization of the geodesic curvatures, we are able to extend the result to arbitrary absolutely continuous curves. For a manifold of constant sectional curvature rolling on itself, two such curves can only differ by an isometry. In the case of surfaces, we give conditions for when loops in the manifolds lift to loops in the configuration space of the rolling. [less ▲] Detailed reference viewed: 122 (5 UL) |
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