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Characterization of pinched Ricci curvature by functional inequalities Cheng, Li Juan ; Thalmaier, Anton in Journal of Geometric Analysis (2018), 28(3), 2312-2345 In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient ... [more ▼] In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, L^p-inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can be used in particular to characterize general geometric flow and Ricci flow by functional inequalities. [less ▲] Detailed reference viewed: 362 (54 UL)Riemannian and Sub-Riemannian geodesic flow ; Grong, Erlend in Journal of Geometric Analysis (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: 99 (15 UL)Compactness of Relatively Isospectral Sets of Surfaces Via Conformal Surgeries Aldana Dominguez, Clara Lucia ; ; in Journal of Geometric Analysis (2015), 25(2), 1185-1210 We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such ... [more ▼] We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such families via a conformal surgery that allows us to reduce to the case of surfaces hyperbolic near infinity recently studied by Borthwick and Perry, or to the closed case by Osgood, Phillips, and Sarnak if there are only cusps. [less ▲] Detailed reference viewed: 117 (5 UL)Sub-Riemannian Geometry on Infinite-Dimensional Manifolds Grong, Erlend ; ; in Journal of Geometric Analysis (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: 119 (9 UL)A characterization of CR quadrics with a symmetry property Altomani, Andrea ; in Journal of Geometric Analysis (2012), 22(3), 892-909 We study CR quadrics satisfying a symmetry property (S~) which is slightly weaker than the symmetry property (S), recently introduced by W. Kaup, which requires the existence of an automorphism reversing ... [more ▼] We study CR quadrics satisfying a symmetry property (S~) which is slightly weaker than the symmetry property (S), recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric. We characterize quadrics satisfying the (S~) property in terms of their Levi–Tanaka algebras. In many cases the (S~) property implies the (S) property; this holds in particular for compact quadrics. We also give a new example of a quadric such that the dimension of the algebra of positive-degree infinitesimal automorphisms is larger than the dimension of the quadric. [less ▲] Detailed reference viewed: 82 (2 UL) |
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