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.

Furnace cooling system and method

Abstract: A cooling system for the distribution and collection of a fluid coolant in a metallurgical vessel used in the processing of molten materials. The cooling system comprises a distribution system including an intake manifold, a plurality of headers attached to the intake manifold, and a plurality of distribution dispensers positioned along each header. A collection system, including a collection manifold, is positioned to collect the fluid coolant. The distribution dispensers are positioned to direct the fluid coolant towards the collection manifold and utilize the majority of the kinetic energy contained within the coolant to direct the coolant towards the collection manifold.

The analogs of the Riemann tensor for exceptional structures on supermanifolds

Abstract: H. Hertz called any manifold M with a given nonintegrable distribution {\it nonholonomic}. Vershik and Gershkovich stated and R. Montgomery proved that the space of germs of any nonholonomic distribution on M with an open and dense orbit of the diffeomorphism group is either (1) of codimension one or (2) an Engel distribution. No analog of this statement for supermanifolds is formulated yet, we only have some examples: our list (an analog of E.Cartan's classification) of simple Lie superalgebras of vector fields with polynomial coefficients and a particular (Weisfeiler) grading contains 16 series similar to contact ones and 11 exceptional algebras preserving nonholonomic structures. Here we compute the cohomology corresponding to the analog of the Riemann tensor for the SUPERmanifolds corresponding to the 15 exceptional simple vectorial Lie superalgebras, 11 of which are nonholonomic. The cohomology for analogs of the Riemann tensor for the manifolds with an exceptional Engel manifolds are computed in math.RT/0202213.

The mixed Einstein-Hilbert action and extrinsic geometry of foliated manifolds

Abstract: We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\rm S}_{\,\rm mix}$ is the averaged sectional curvature over all planes that contain vectors from both distributions of an almost-product structure and the variations we consider preserve orthogonality of the distributions. We derive the directional derivative $D J_{\,\rm mix}$ (of the total ${\rm S}_{\,\rm mix}$) for adapted variations of metrics on closed almost-product manifolds and foliations of arbitrary dimension. The obtained Euler-Lagrange equations are presented in two equiva\-lent forms: in terms of extrinsic geometry and intrinsically using the partial Ricci tensor. Certainly, these mixed field equations admit amount of solutions (e.g., twisted products).

The curvature of contact structure on 3-manifolds

Abstract: We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold $M$ there is a metric, such that the sectional curvature of the contact distribution is equal to -1. We also introduce the notion of Gaussian curvature of the plane distribution. For this notion of curvature we get the similar results.

Codimension Two Spacelike Submanifolds Through a Null Hypersurface in a Lorentzian Manifold

Abstract: Most important examples of null hypersurfaces in a Lorentzian manifold admit an integrable screen distribution, which determines a spacelike foliation of the null hypersurface. In this paper, we obtain conditions for a codimension two spacelike submanifold contained in a null hypersurface to be a leaf of the (integrable) screen distribution. For this, we use the rigging technique to endow the null hypersurface with a Riemannian metric, which allows us to apply the classical Eschenburg maximum principle. We apply the obtained results to classical examples as generalized Robertson–Walker spaces and Kruskal space.