Showing papers on "Distribution (differential geometry) published in 2001"

Geometric approach to Goursat flags

Abstract: A Goursat flag is a chain Ds⊂Ds−1⊂⋯⊂D1⊂D0=TM of subbundles of the tangent bundle TM such that corankDi=i and Di−1 is generated by the vector fields in Di and their Lie brackets. Engel, Goursat, and Cartan studied these flags and established a normal form for them, valid at generic points of M. Recently Kumpera, Ruiz and Mormul discovered that Goursat flags can have singularities, and that the number of these grows exponentially with the corank s. Our Theorem 1 says that every corank s Goursat germ, including those yet to be discovered, can be found within the s-fold Cartan prolongation of the tangent bundle of a surface. Theorem 2 says that every Goursat singularity is structurally stable, or irremovable, under Goursat perturbations. Theorem 3 establishes the global structural stability of Goursat flags, subject to perturbations which fix a certain canonical foliation. It relies on a generalization of Gray's theorem for deformations of contact structures. Our results are based on a geometric approach, beginning with the construction of an integrable subflag to a Goursat flag, and the sandwich lemma which describes inclusions between the two flags. We show that the problem of local classification of Goursat flags reduces to the problem of counting the fixed points of the circle with respect to certain groups of projective transformations. This yields new general classification results and explains previous classification results in geometric terms. In the last appendix we obtain a corollary to Theorem 1. The problems of locally classifying the distribution which models a truck pulling s trailers and classifying arbitrary Goursat distribution germs of corank s+1 are the same.

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations

Abstract: The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations \(\) are critical points. Later, we prove the instability for these fibrations.

Local Geometry of Fractals Given by Tangent Measure Distributions

Abstract: Let μ be a self-similar-measure and ν an ergodic shift-invariant measure on a self-similar set A. We show that under weak conditions ν-almost all points x in A show the same local structure, that is, the same tangent measure distribution of μ.

Fluid distribution appliance

Abstract: A fluid distribution appliance comprises a manifold unit integrally formed with top, side, front and back walls, and a bottom surface, and a handle receiving section. The walls and bottom surface of the manifold form an enclosed space, through which fluid flows and is delivered through nozzles located at one of the front walls of the manifold. The manifold also supports a squeegee made of a layer of semi-rigid material, which is secured to the manifold by a holddown plate and screws. A triangular brace support is secured at one end to a handle attached to a handle receiving section of the manifold and, at its other two ends, to the manifold. The manifold, with its brace and handle, can be used independently or in combination with a bristle push broom head which is configured to receive the manifold in fitted, surface to surface contact.

On the numerical solution of involutive ordinary differential systems: higher order methods ∗

Abstract: We analyse some Taylor and Runge—Kutta type methods for computing one-dimensional integral manifolds, i.e. solutions to ODEs and DAEs. The distribution defining the solutions is taken to be defined only on the relevant manifold and hence all the intermediate points occuring in the computations are projected orthogonally to the manifold. We analyse the order of such methods, and somewhat surprisingly there does not appear any new order conditions for the Runge—Kutta methods in our context, at least up to order 4. The analysis shows that some terms appearing in the error expansions can be quite naturally expressed in terms of standard notions of Riemannian geometry. The numerical examples show that the methods work reliably and moreover produce qualitatively correct results for Hamiltonian systems although the methods are not symplectic.

Nonlinear methods of analysis of data with gaps

Abstract: Information on most of natural phenomena can be obtained from time series of direct and proxy data. The analysis of time series generated by natural dynamic systems is a key element in interpreting geophysical and climatic information. Unfortunately, most of available time series have gaps. When there are many gaps with irregular distribution, we do not have any statistical tools for repairing the data. We suggest some approach to solve this problem. It is based on modeling the missing data by small-dimensional manifolds and neu- ral network technologies. In this approach we assume that data under consideration are a set of n-dimensional vectors, which are produced by dynamical system. These vectors model n- dimensional attractor in embedding space. Gaps in the vectors are represented as a linear manifold L of some dimension. The method idea is to model L by another small-dimensional manifold, e.g. a curve. Neural networks are used to find this manifold. We verify the method on real time series data: sunspot numbers, the radiocarbon content in tree rings, the 10 Be in

The spectral density function for the Laplacian on high tensor powers of a line bundle

Abstract: For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density function. We give an explicit computation of the spectral density function, by constructing certain quasimodes on the associated principle bundle.

Some conditions on the Weingarten endomorphism of real hypersurfaces in quaternionic space forms

Abstract: We classify the real hypersurfaces in a non-flat quaternionic space form satisfying several conditions on their Weingarten endomorphism Firstly, we study on the maximal quaternionic distribution of the real hypersurface a relationship between a certain metric tensor and the restriction of the metric of the ambient manifold Secondly, we consider some formulae relating the Weingarten endomorphism to the curvature operator

Correspondence spaces and twistor spaces for parabolic geometries

Abstract: For a semisimple Lie group $G$ with parabolic subgroups $Q\subset P\subset G$, we associate to a parabolic geometry of type $(G,P)$ on a smooth manifold $N$ the correspondence space $\Cal CN$, which is the total space of a fiber bundle over $N$ with fiber a generalized flag manifold, and construct a canonical parabolic geometry of type $(G,Q)$ on $\Cal CN$. Conversely, for a parabolic geometry of type $(G,Q)$ on a smooth manifold $M$, we construct a distribution corresponding to $P$, and find the exact conditions for its integrability. If these conditions are satisfied, then we define the twistor space $N$ as a local leaf space of the corresponding foliation. We find equivalent conditions for the existence of a parabolic geometry of type $(G,P)$ on the twistor space $N$ such that $M$ is locally isomorphic to the correspondence space $\Cal CN$, thus obtaining a complete local characterization of correspondence spaces. We show that all these constructions preserve the subclass of normal parabolic geometries (which are determined by some underlying geometric structure) and that in the regular normal case, all characterizations can be expressed in terms of the harmonic curvature of the Cartan connection, which is easier to handle. Several examples and applications are discussed.

Product of Boundary Distributions

Abstract: 1) We identify new parameter branches for the ultra-local boundary Poisson bracket in d spatial dimension with a (d-1)-dimensional spatial boundary. There exist 2^{r(r-1)/2} r-dimensional parameter branches for each d-box, r-row Young tableau. The already known branch (hep-th/9912017) corresponds to a vertical 1-column, d-box Young tableau. 2) We consider a local distribution product among the so-called boundary distributions. The product is required to respect the associativity and the Leibnitz rule. We show that the consistency requirements on this product correspond to the Jacobi identity conditions for the boundary Poisson bracket. In other words, the restrictions on forming a boundary Poisson bracket can be related to the more fundamental distribution product construction. 3) The definition of the higher functional derivatives is made independent of the choice of integral kernel representative for a functional.