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.

The Einstein-Hilbert type action on metric-affine almost-product manifolds

Abstract: We continue our study of the mixed Einstein-Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler-Lagrange equations (which in the case of space-time are analogous to those in Einstein-Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler-Lagrange equations of the mixed Einstein-Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric~connections.

On the geometry of almost $\mathcal{S}$-manifolds

Abstract: An $f$-structure on a manifold $M$ is an endomorphism field $\phi$ satisfying $\phi^3+\phi=0$. We call an $f$-structure {\em regular} if the distribution $T=\ker\phi$ is involutive and regular, in the sense of Palais. We show that when a regular $f$-structure on a compact manifold $M$ is an almost $\S$-structure, as defined by Duggal, Ianus, and Pastore, it determines a torus fibration of $M$ over a symplectic manifold. When $\rank T = 1$, this result reduces to the Boothby-Wang theorem. Unlike similar results due to Blair-Ludden-Yano and Soare, we do not assume that the $f$-structure is normal. We also show that given an almost $\mathcal{S}$-structure, we obtain an associated Jacobi structure, as well as a notion of symplectization.

Properties and applications of Fisher distribution on the rotation group

Abstract: We study properties of Fisher distribution (von Mises-Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011), and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data.

Melt distribution manifold

Abstract: A melt distribution manifold for use with first and second mold portions moveable with respect to each other, the melt distribution manifold including a melt inlet means for receiving melt from an injection molding machine, a first manifold portion connected to the first mold portion, wherein the first manifold portion is stationary with respect to the first mold portion, a second manifold portion connecting the first manifold portion to the hot runner nozzle, wherein the second manifold portion is moveable with respect to the first manifold portion such that when the first mold portion moves with respect to the second mold portion, the second manifold portion remains connected to the first manifold portion and to the hot runner nozzle, a hinged joint connecting the first manifold portion to the second manifold portion. Each of the first manifold portion, the second manifold portion, and the hinged joint having respective melt distribution bores in fluid communication with each other.

The geometry of a 3-quasi-sasakian manifold

Abstract: quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic ge- ometries. In this paper many geometric properties of this class of almost 3-contact metric manifolds are found. In particular, it is proved that the only 3-quasi-Sasakian manifolds of rank 4l+1 are the 3-cosymplectic manifolds and any 3-quasi-Sasakian manifold of maximal rank is necessarily 3-α-Sasakian. Furthermore, the trans- verse geometry of a 3-quasi-Sasakian manifold is studied, proving that any 3-quasi- Sasakian manifold admits a canonical transversal, projectable quaternionic-Kahler structure and a canonical transversal, projectable 3-α-Sasakian structure. The well-known classes of 3-Sasakian and 3-cosymplectic manifolds belong to the wider family of almost 3-contact metric manifolds. Nevertheless, both classes sit also perfectly into the narrower class of 3-quasi-Sasakian manifolds which, as we will see, is a very natural framework for a unified study of the aforementioned geometries. A similar chain of inclusions takes place in the case of a single almost contact metric structure, whereas the class of quasi- Sasakian manifolds encloses both Sasakian and cosymplectic manifolds, but in the setting of 3-structures the interrelations between the triples of tensors produce key additional properties making the choice of the 3-quasi-Sasakian framework still more natural. 3-quasi-Sasakian manifolds were introduced long ago but their first systematic study was carried out by the authors in (5). There, it was proven that in any 3-quasi-Sasakian manifold (M,�α,�α,�α,g) of dimension 4n + 3 the vertical distribution V generated by the three Reeb vector fields is completely integrable determining a canonical totally geodesic and Riemannian foliation. The characteristic vector fields obey the commu- tation relations (� α ,� β ) = c� γ for any even permutation (�,�,) of {1,2,3} and some c ∈ R. Furthermore, it was shown that the ranks of the 1-forms