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In this paper, we study the structure of truncated connections on the multidimensional Heisenberg group endowed with a left-invariant sub-Riemannian structure. We find that sub-Riemannian geodesic lines are parabolas whose orthogonal projections onto the corresponding contact planes are straight lines. In addition to such parabolas, some straight lines lying in contact planes are also geodesics.

The main goal of this paper is to show that helix surfaces and the Enneper surface are the only surfaces in the 3-dimensional Euclidean space R 3 whose isogonal lines are generalized helices and pseudo-geodesic lines.

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely (M1,g1) and (M2,g2), represented by the Riemann surfaces which intersect along the curve (M1,g1)∩(M2,g2)=ℒ were addressed. Curve ℒ does not contain geodesic lines in either of the manifolds (M1,g1) and (M2,g2). Consider six points located on the ℒ: 1,…6⊂ℒ. The points 1,…6⊂ℒ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold (M1,g1)/red links, and, alternatively, with the geodesic lines belonging to the manifold (M2,g2)/green links. Points 1,…6⊂ℒ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented.

Existing Source-Free Domain Adaptation (SFDA) methods typically adopt the feature distribution alignment paradigm via mining auxiliary information (eg., pseudo-labelling, source domain data generation). However, they are largely limited due to that the auxiliary information is usually error-prone whilst lacking effective error-mitigation mechanisms. To overcome this fundamental limitation, in this paper we propose a novel Target Prediction Distribution Searching (TPDS) paradigm. Theoretically, we prove that in case of sufficient small distribution shift, the domain transfer error could be well bounded. To satisfy this condition, we introduce a flow of proxy distributions that facilitates the bridging of typically large distribution shift from the source domain to the target domain. This results in a progressive searching on the geodesic path where adjacent proxy distributions are regularized to have small shift so that the overall errors can be minimized. To account for the sequential correlation between proxy distributions, we develop a new pairwise alignment with category consistency algorithm for minimizing the adaptation errors. Specifically, a manifold geometry guided cross-distribution neighbour search is designed to detect the data pairs supporting the Wasserstein distance based shift measurement. Mutual information maximization is then adopted over these pairs for shift regularization. Extensive experiments on five challenging SFDA benchmarks show that our TPDS achieves new state-of-the-art performance. The code and datasets are available at https://github.com/tntek/TPDS.

The geodesic model based on the eikonal partial differential equation (PDE) has served as a fundamental tool for the applications of image segmentation and boundary detection in the past two decades. However, the existing approaches commonly only exploit the image edge-based features for computing minimal geodesic paths, potentially limiting their performance in complicated segmentation situations. In this paper, we introduce a new variational image segmentation model based on the minimal geodesic path framework and the eikonal PDE, where the region-based appearance term that defines then regional homogeneity features can be taken into account for estimating the associated minimal geodesic paths. This is done by constructing a Randers geodesic metric interpretation of the region-based active contour energy functional. As a result, the minimization of the active contour energy functional is transformed into finding the solution to the Randers eikonal PDE. We also suggest a practical interactive image segmentation strategy, where the target boundary can be delineated by the concatenation of several piecewise geodesic paths. We invoke the Finsler variant of the fast marching method to estimate the geodesic distance map, yielding an efficient implementation of the proposed region-based Randers geodesic model for image segmentation. Experimental results on both synthetic and real images exhibit that our model indeed achieves encouraging segmentation performance.

The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank n≥2 in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite) n-cycles of rn volume growth; prime examples include n-cycles associated with n-quasiflats. Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper CAT(0) spaces of asymptotic rank n extends to a class of (n-1)-cycles in the Tits boundaries.

Tensile membrane structures (TMS) consist of the membrane fabric, supporting frame, cables and strut elements. The initial shape of a TMS is not known beforehand and has to be found by a process of ‘form-finding’. The majority of form-finding research has focused on determining the initial shape due to a defined prestress for TMS having only membrane and cable elements, while there has been very little research on TMS supported with struts and anchorage cables. This study extends the updated weight method (UWM) for the form-finding of TMS with struts and anchorage supports. The modified approach provides a robust solution in comparison to constrained strut length approaches. Furthermore, the curved form-found shape is not developable and geodesic lines need to be identified on the final form-found shape to provide seam locations for cutting the fabric. To solve this issue, geodesic lines are included in the UWM as constraints, whereby the geodesic pseudo cable lengths are minimised tangent to the surface. A sequential process is developed to ensure both the equilibrium and the constraint conditions are satisfied. The proposed method is successfully tested on a wide variety of TMS shapes along with the patterning of the cut panels. The study provides an integrated solution for the form-finding and identification of geodesic seam line on TMS having different boundary types.

A translation curve in a Thurston space is a curve such that for given unit vector at the origin, the translation of this vector is tangent to the curve in every point of the curve. In most Thurston spaces, translation curves coincide with geodesic lines. However, this does not hold for Thurston spaces equipped with twisted product. In these spaces, translation curves seem more intuitive and simpler than geodesics. In this paper, geodesics and translation curves in Sol04 space are classified and the curvature properties of translation curves are investigated.

The solutions to the Euler–Poisson equations are geodesic lines of SO (3) manifold with the metric determined by inertia tensor. However, the Poisson structure on the corresponding symplectic leaf does not depend on the inertia tensor. We calculate its explicit form and confirm that it differs from the algebra e (3). The obtained Poisson brackets are used to demonstrate the Liouville integrability of a free rigid body. The general solution to the Euler–Poisson equations is written in terms of exponential of the Hamiltonian vector field.

A new concept of surface waves of interference nature is described in detail for the case of creeping waves propagating along a smooth strictly concave surface embedded in 3D Euclidean space. In numerous papers devoted to whispering gallery and to creeping waves, it was assumed that they propagate along boundaries formed by smooth plane curves. However, the process of surface wave propagation along smooth surfaces is much more complicated than along plane curves. Indeed, the surface waves slide along geodesic lines on the surface where they typically form numerous caustics and that, in turn, generates singularities in the wave field asymptotics. In addition, the geodesic lines themselves are not plane curves in 3D and therefore their torsion has to be taken into account. Our approach allows us to resolve both of these specific problems of wave propagation along smooth surfaces embedded in ℝ 3 . It is based on the consideration of the geodesic flow on the surface, which is associated with the surface wave generated by a source. For each geodesic line, we construct an asymptotic solution of the Helmholtz equation localized in a tube vicinity of the geodesic line and having no singularities on caustics. The surface wave under consideration is then presented as a superposition (integral) of the localized solutions.