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Distribution (differential geometry)
About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.
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TL;DR: In this article, the spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold are discussed and a parametrization of the LaplAC is given.
Abstract: In this paper, we discuss spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold. First, we give a survey of results on generalized smooth distributions on manifolds, Riemannian structures and associated Laplacians. Then, under the assumption that the singular foliation generated by the distribution is regular, we prove that the Laplacian associated with the distribution defines an unbounded multiplier on the foliation $C^*$-algebra. To this end, we give the construction of a parametrix.
TL;DR: The Kobayashi pseudo-distance is generalized to complex manifolds which admit holomorphic bracket generating distributions based on Chow’s theorem in sub-Riemannian geometry.
Abstract: In this paper, we generalize the Kobayashi pseudo-distance to complex manifolds which admit holomorphic bracket generating distributions. The generalization is based on Chow's theorem in sub-Riemannian geometry. Let G be a linear semisimple Lie group. For a complex $G$-homogeneous manifold M with a G-invariant holomorphic bracket generating distribution D, we prove that (M,D) is Kobayashi hyperbolic if and only if the universal covering of M is a canonical flag domain and the induced distribution is the superhorizontal distribution.
TL;DR: In this paper, the initial configurations that lead to an eventual approach to a given planet, particularly Jupiter, using the invariant manifold of Lyapunov orbits around Lagrangian points L1 or L2.
Abstract:
This study explores the initial configurations that lead to an eventual approach to a given planet, particularly Jupiter, using the invariant manifold of Lyapunov orbits around Lagrangian points L1 or L2. Reachability to the vicinity of planets would provide information on developing a process for capturing irregular satellites, which are small bodies orbiting around a giant planet with a high eccentricity that are considered to have been captured by the mother planet, rather than formed in situ. A region several times the Hill radius is often used for determining reachability, combined with checking the velocity of the planetesimal with respect to the mother planet. This kind of direct computation is inapplicable when the Hill sphere is widely open and its boundary cannot be clearly recognized. Here, we thus employ Lyapunov periodic orbits (LOs) as a formal definition of the vicinity of Jupiter and numerically track the orbital distribution of the invariant manifold of an LO while varying the Jacobi constant, CJ. Numerical tracking of the manifold is carried out directly via repeated Poincaré mapping of points initially allocated densely on a fragment of the manifold near the fixed points, with the assistance of multi-precision arithmetic using the Multiple Precision Floating-Point Reliable Library. The numerical computations show that the distribution of the semi-major axis of points on the manifolds is quite heavily tailed and that its median spans roughly 1–2 times the Jovian orbital radius. The invariant manifold provides a distribution profile of a that is similar to that obtained using a direct method.
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TL;DR: In this article, a Laplacian operator related to the characteristic cohomology of a smooth manifold endowed with a distribution is studied and it is shown that it is not hypoelliptic in general and does not respect the bigrading on forms in a complex setting.
Abstract: We study a Laplacian operator related to the characteristic cohomology of a smooth manifold endowed with a distribution. We prove that this Laplacian does not behave very well: it is not hypoelliptic in general and does not respect the bigrading on forms in a complex setting. We also discuss the consequences of these negative results for a conjecture of P. Griffiths, concerning the characteristic cohomology of period domains.
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TL;DR: In this article, a generalization of Sturm nonoscillation and comparison theorems to general Hamiltonians is presented. But the results are restricted to optical Hamiltonians and cannot be applied to general non-Riemannian Hamiltonians.
Abstract: Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index.
Starting from the celebrated paper of Arnol'd on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.