Special lines on contact manifolds

In a series of two articles Kebekus studied deformation theory of minimal rational curves on contact Fano manifolds. Such curves are called contact lines. Kebekus proved that a contact line through a general point is necessarily smooth and has a fixed standard splitting type of the restricted tangent bundle. In this paper we study singular contact lines and those with special splitting type. We provide restrictions on the families of such lines, and on contact Fano manifolds which have reducible varieties of minimal rational tangents. We also show that the results about singular lines naturally generalise to complex contact manifolds, which are not necessarily Fano, for instance, quasi-projective contact manifolds or compact contact manifolds of Fujiki class C. In particular, in many cases the dimension of a family of singular lines is at most 2 less than the dimension of the contact manifold

Special lines on contact manifolds

In a series of two articles Kebekus studied deformation theory of minimal rational curves on contact Fano manifolds. Such curves are called contact lines. Kebekus proved that a contact line through a general point is necessarily smooth and has a fixed standard splitting type of the restricted tangent bundle. In this paper we study singular contact lines and those with special splitting type. We provide restrictions on the families of such lines, and on contact Fano manifolds which have reducible varieties of minimal rational tangents. We also show that the results about singular lines naturally generalise to complex contact manifolds, which are not necessarily Fano, for instance, quasi-projective contact manifolds or compact contact manifolds of Fujiki class C. In particular, in many cases the dimension of a family of singular lines is at most 2 less than the dimension of the contact manifold

Consumption and Sustainable Development of Polish Metropolitan Cities

The purpose of the study is to determine the level of sustainable development of metropolitan cities in Poland within the basic dimensions of sustainable development. At the same time, the levels of development of cities in terms of consumption were determined, and an attempt was made to identify the relationship of the levels of development of cities and the levels of development of consumption. Three main dimensions of sustainable development were considered: social, economic and environmental-spatial. It was important to find an answer to the question whether consumption can influence the levels of sustainable development achieved by cities, and in what aspects? A taxonomic method was used, which allows organizing the studied territorial units in a hierarchical manner. The levels of their development in each dimension were determined. Identifying the structure of features allows determining at what distance from the ideal structure of features the studied cities are. The research was carried out in dynamic terms by analyzing 2007 and 2020

Morphometry of the heart orifices and morphometry and topography of the coronary ostia in the goat

Background: The coronary vessels have been described in various species of domestic and wild ruminants. However, no studies on the detailed morphology and morphometry of heart orifices and coronary ostia in the Polish goat are available. Materials and methods: The study was carried out on 112 female, adult dairy goats belonging to Polish Fawn Improved and Polish White Improved breed, closely related to French Alpine and Saanen, respectively. Results: In all examined individuals, all heart orifices and heart valves were of normal structure. There was no significant diametrical difference between the aortic and pulmonary orifice. The right atrioventricular opening was significantly wider than the left atrioventricular opening. The dimension of the left coronary ostium ranged from 1.0 to 5.5 mm with the arithmetic mean (±standard deviation [SD]) of 4.3 ± 0.8 mm. The dimension of the right coronary ostium ranged from 0.5 to 5.0 mm with the arithmetic mean (±SD) of 2.8 ± 0.7 mm. Both coronary artery ostia were located under the sinotubular junction. 39/112 examined goats (34.8%) had variations in the structure of the coronary ostia such as the lack of main trunk or the presence of additional coronary ostia. They were observed in one (in 34/39 goats) or both coronary arteries (5/39 goats). Conclusions:  In goats, the dimensions of aortic and pulmonary orifices are similar while the right atrioventricular opening outsizes the left one. On the other hand, the left coronary ostium is wider than the right one. Morphological variations in the coronary ostia occur in approximately one third of goats, more often in the right than in the left ostium