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.

Twistors, nilpotent orbits and harmonic maps

Abstract: Twistor theory provides a useful tool which has applications in the theory of harmonic maps. A good example is the Calabi-Penrose twistor fibration ℂℙ3 → S 4. All harmonic spheres in S 4 can be obtained from projections of holomorphic horizontal curves in ℂℙ3 (a holomorphic curve is horizontal if it is tangent to the complex contact distribution Η ⊂ Tℂℙ3 which is perpendicular to the fibres of the twistor fibration with respect to the Fubini-Study metric). In order to construct holomorphic curves tangent to the distribution Η one can use the Bryant correspondence which maps ℂℙ3 birationally to ℙT*ℂℙ2 and maps Η to the canonical complex contact distribution on ℙT*ℂℙ2 (see [6] and [27]). The flag manifold F 12(ℂ3) ≃ ℙT*ℂℙ2 is the twistor space of ℂℙ2 and Burstall shows in [11] that in fact all twistor spaces of compact quaternion-Kahler symmetric spaces of the same dimension are birationally equivalent as complex contact manifolds.

An information geometry algorithm for distribution control

Abstract: In this paper, we consider the problem of distribution control from the viewpoint of information geometry. Different from most existing models used in stochastic control, it is assumed that the control input directly affects the distribution of the system output in probability sense. Here, we set up a new manifold (S), meanwhile the B-spline manifold (B) and the system output manifold (M) can be referred to as its submanifolds. We give an information geometrical algorithm which can be called as geodesic-projection algorithm using the properties of manifold. In the geodesic step, we can obtain the geodesic equation from the initial point V0 = (ω10, ω20, ··· , ω(n−1)0) to the specified point Vg = (ω1g, ω2g, ··· , ω(n−1)g) in B. This gives us an optimal trajectory for the points changing along in B. In the projection step, we project the sample points selected from the geodesic onto M. The coordinates of the projections in M give the trajectory of the control input u.

Geometry of lightlike hypersurfaces of a statistical manifold

Abstract: Lightlike hypersurfaces of a statistical manifold are studied. It is shown that a lightlike hypersurface of a statistical manifold is not a statistical manifold with respect to the induced connections, but the screen distribution has a canonical statistical structure. Some relations between induced geometric objects with respect to dual connections in a lightlike hypersurface of a statistical manifold are obtained. An example is presented. Induced Ricci tensors for lightlike hypersurface of a statistical manifold are computed.

Pseudodifferential operators of mixed type adapted to distributions of $k$-planes

Abstract: We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some sub-elliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we introduce is geometrically invariant, and is adapted to a smooth distribution of tangent subspaces of constant rank. We isolate certain ideals in the algebra whose analysis is of particular interest.

Reducible conformal holonomy in any metric signature and application to twistor spinors in low dimension

Abstract: We prove that given a pseudo-Riemannian conformal structure whose conformal holonomy representation fixes a totally isotropic subspace of arbitrary dimension, there is, w.r.t. a local metric in the conformal class defined off a singular set, a parallel, totally isotropic distribution on the tangent bundle which contains the image of the Ricci-tensor. This generalizes results obtained for invariant isotropic lines and planes and closes a gap in the understanding of the geometric meaning of reducibly acting conformal holonomy groups. We show how this result naturally applies to the classification of geometries admitting twistor spinors described in terms of parallel spin tractors using conformal spin tractor calculus. As an example we obtain together with already known results about generic distributions in dimensions 5 and 6 a complete geometric description of local geometries admitting real twistor spinors in signatures ( 3 , 2 ) and ( 3 , 3 ) . In contrast to the generic case where generic geometric distributions play an important role, the underlying geometries in the non-generic case without zeroes turn out to admit integrable distributions.