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

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

D’Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders

Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about geodesic segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D’Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D’Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D’Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension