Harmonic Manifolds and Tubes

The authors showed in a preceding paper that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed only for tubes about geodesic segments. As a consequence, we obtain similar characterizations of harmonic manifolds in terms of the total mean curvature and the total scalar curvature of tubular hypersurfaces about curves. We find simple formulae expressing the volume, total mean curvature, and total scalar curvature of tubular hypersurfaces about a curve in a harmonic manifold as a function of the volume density function.Comment: 14 pages, grant numbers are update

On the rigidity of regular bicycle (n,k)-gons

A bicycle (n, k)-gon is an equilateral n-gon whose k-diagonals are equal. In this paper, the order of infinitesimal flexibility of the regular n-gon within the family of bicycle (n, k)-gons is studied. An equation characterizing first order flexible regular bicycle (n, k)-gons were computed by S. Tabachnikov in [7]. This equation was solved by R. Connelly and the author in [4]. S. Tabachnikov has also constructed nontrivial deformations of the regular bicycle (n, k)-gon for certain pairs (n, k). The main result of the paper is that if the regular bicycle (n, k)-gon is first order flexible, but is not among Tabachnikov’s examples of deformable regular bicycle (n, k)-gons, then this bicycle polygon is second order flexible as well, however, it is third order rigid

Harmonic Manifolds and the Volume of Tubes about Curves

H. Hotelling proved that in the n-dimensional Euclidean or spherical space, the volume of a tube of small radius about a curve depends only on the length of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's theorem to rank one symmetric spaces computing the volumes of the tubes explicitly in these spaces. In the present paper, we generalize these results by showing that every harmonic manifold has the above tube property. We compute the volume of tubes in the Damek-Ricci spaces. We show that if a Riemannian manifold has the tube property, then it is a 2-stein D'Atri space. We also prove that a symmetric space has the tube property if and only if it is harmonic. Our results answer some questions posed by L. Vanhecke, T. J. Willmore, and G. Thorbergsson.Comment: 17 pages, no figures. This version is different from the journal versio

Sphere-like isoparametric hypersurfaces in Damek-Ricci spaces

Locally harmonic manifolds are Riemannian manifolds in which small geodesic spheres are isoparametric hypersurfaces, i.e., hypersurfaces whose nearby parallel hypersurfaces are of constant mean curvature. Flat and rank one symmetric spaces are examples of harmonic manifolds. Damek-Ricci spaces are non-compact harmonic manifolds, most of which are non-symmetric. Taking the limit of an "inflating" sphere through a point $p$ in a Damek-Ricci space as the center of the sphere runs out to infinity along a geodesic half-line $\gamma$ starting from $p$, we get a horosphere. Similarly to spheres, horospheres are also isoparametric hypersurfaces. In this paper, we define the sphere-like hypersurfaces obtained by "overinflating the horospheres" by pushing the center of the sphere beyond the point at infinity of $\gamma$ along a virtual prolongation of $\gamma$. They give a new family of isoparametric hypersurfaces in Damek-Ricci spaces connecting geodesic spheres to some of the isoparametric hypersurfaces constructed by J. C. D\'iaz-Ramos and M. Dom\'inguez-V\'azquez [arXiv:1111.0264] in Damek-Ricci spaces. We study the geometric properties of these isoparametric hypersurfaces, in particular their homogeneity and the totally geodesic condition for their focal varieties.Comment: 20 page

Multicriteria cruise control design considering geographic and traffic conditions

The paper presents the design of cruise control systems considering road and traffic information during the design of speed trajectories. Several factors are considered such as road inclinations, traffic lights, preceding vehicles, speed limits, engine emissions and travel times. The purpose of speed design is to reduce longitudinal energy, fuel consumption and engine emissions without a significant increase in travel time. The signals obtained from the road and traffic are handled jointly with the dynamic equations of the vehicle and built into the control design of reference speed. A robust H∞ control is designed to achieve the speed of the cruise control, guaranteeing the robustness of the system against disturbances and uncertainties