Narrow Results

Path planning can be subject to different types of optimization. Some years ago a German researcher, U. Leuthäusser, proposed a new variational method for reducing most types of optimization criteria to one and the same: minimization of path length. This can be done by altering the Riemannian metric of the domain, so that optimal paths (with respect to whatever criterion) are simply seen as shortest. This method offers an extra feature, which has not been exploited so far: it admits direction–dependent criteria.

In this paper we make this feature explicit, and apply it to two different anisotropic settings. One is that of different costs for different directions: E.g. the situation of a countryside scene with ploughed fields. The second is dependence on oriented directions, which is called here "strong" anisotropy: the typical scene is that of a hill side. A covering projection solves the additional difficulty. We also provide some experimental results on synthetic data.

Every surface in the Euclidean space ℝ³ admits a canonical Riemannian metric that has constant Gaussian curvature and is conformal to the original metric. Similarly, 3-manifolds can be decomposed into pieces that admit canonical metrics. Such metrics not only have theoretical significance in 3-manifold geometry and topology, but also have potential applications to practical problems in engineering fields such as shape classification.

In this paper we present an algorithm that is based on a discrete curvature flow to compute constant curvature metrics on 3-manifolds that are hyperbolic and have boundaries of a certain type. We also provide an approach to visualize such a metric by embedding the fundamental domain and universal covering in the hyperbolic space ℍ³. Some experimental results are given for both algorithms.

Furthermore, we propose an algorithm to automatically construct truncated tetrahedral meshes for 3-manifolds with boundaries. It can not only generate inputs to the curvature flow algorithm, but could also serve as an automatic tool for geometers and topologists to build simple models for complicated 3-manifolds, and therefore facilitate their research that requires such models.

In this paper, we give an analogue of the Hermitian structure in the almost complex case, on an $n(k+1)$-dimensional manifold endowed with an almost $k$-complex metric. Also, we study the compatibility between Riemannian metric and polarized $k$-symplectic structure. Also, we study some properties of an almost $k$-complex structure. Moreover, we give an equivalence between almost $k$-complex structures, $k$-almost tangent structures and $k$-almost cotangent structures.

We use the general setting for contrast (potential) functions in statistical and information geometry provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. We study the case when the two-form is degenerate and show how in sufficiently regular cases one reduces it to a pseudometric structures. Transversal Levi-Civita connections for Riemannian foliations are generalized to the Lie groupoid/Lie algebroid case.

A new geometric view of homogeneous isotropic turbulence is contemplated employing the two-point velocity correlation tensor of the velocity fluctuations. We show that this correlation tensor generates a family of pseudo-Riemannian metrics. This enables us to specify the geometry of a singled out Eulerian fluid volume in a statistical sense. We expose the relationship of some geometric constructions with statistical quantities arising in turbulence.

Given a closed symplectic manifold (M²ⁿ, ω) of dimension 2n ≥ 4, we consider all Riemannian metrics on M, which are compatible with the symplectic structure ω. For each such metric g, we look at the first eigenvalue λ₁ of the Laplacian associated with it. We show that λ₁ can be made arbitrarily large, when we vary g. This generalizes previous results of Polterovich, and of Mangoubi.

In this paper, as data, ellipsoids in a color coordinate called the Commission Internationale de l’Eclairage (CIE)-Lab system are given as data for 19 colors. Each ellipsoid is a region where all points are visually recognized as the same color at the center of the coordinate system. Our aim here is to predict the shape of an ellipsoid whose center is given by a new color. We proposed two prediction methods of positive definite matrices determining ellipsoids.

The first one is a nonparametric method with Gaussian kernel. The prediction is provided as a weighted sum of positive definite matrices corresponding to 19 ellipsoids in the training data. The second one is to use a matrix-valued regression model applied to a logarithm of positive definite matrices where explanatory variables are three elements of color centers. The best result was obtained by the nonparametric methods with three bandwidth parameters. The log normal regression had a weaker performance, but even so the model estimation was easily carried out.

We develop variational formulae for the total m-th mean curvatures of codimension-one distributions on a compact Riemannian manifold Mⁿ⁺¹. Based on the method of our recent work Rovenski–Walczak (2007) we show that the total generalized mean curvatures over M(c) of constant curvature c do not depend on the choice of k orthonormal vector fields, that for k = 1 was proved by Brito–Langevin–Rosenberg (1981). We calculate the variations of the total generalized mean curvatures for k orthonormal vector fields on a compact Riemannian manifold M.

We prove that the only general natural Kählerian cotangent bundles of quasi-constant holomorphic sectional curvatures, are those of constant holomorphic sectional curvature.

Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology Σ and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of X into Σ. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on Σ, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of Σ.

We survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.