Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.

Integrating control systems defined on the frame bundles of the space forms

This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

NEW CHARACTERIZATIONS OF COMPLETE SPACELIKE SUBMANIFOLDS IN SEMI-RIEMANNIAN SPACE FORMS

In this paper we study n-dimensional complete spacelike submanifolds with constant normalized scalar curvature immersed in semi-Riemannian space forms. By extending Cheng-Yau`s technique to these ambients, we obtain results to such submanifolds satisfying certain conditions on both the squared norm of the second fundamental form and the mean curvature. We also characterize compact non-negatively curved submanifolds in De Sitter space of index p.

The Borsuk-Ulam theorem for homotopy spherical space forms

In this work, we show for which odd-dimensional homotopy spherical space forms the Borsuk-Ulam theorem holds. These spaces are the quotient of a homotopy odd-dimensional sphere by a free action of a finite group. Also, the types of these spaces which admit a free involution are characterized. The case of even-dimensional homotopy spherical space forms is basically known.

The Gauss-Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

LetQ(4)( c) be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss-Kronecker curvature of a complete minimal hypersurface in Q(4)(c), c <= 0, whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M(3) of a space form Q(4)( c) with constant Gauss-Kronecker curvature K. For the case c <= 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of Q(4)( c) with K constant.

Spherical space forms - Homotopy self-equivalences and homotopy types, the case of the groups Z/a x (Z/b x TL(2)(F(p)))

Let G = Z/a x(mu) (Z/b x TL(2)(F(p))) and X(n) be an n-dimensional CW-complex with the homotopy type of the n-sphere. We determine the automorphism group Aut(G) and then compute the number of distinct homotopy types of spherical space forms with respect to free and cellular G-actions on all CW-complexes X(2dn - 1), where 2d is a period of G. Next, the group E(X(2dn - 1)/alpha) of homotopy self-equivalences of spherical space forms X(2dn - 1)/alpha, associated with such G-actions alpha on X(2dn - 1) are studied. Similar results for the rest of finite periodic groups have been obtained recently and they are described in the introduction. (C) 2009 Elsevier B.V. All rights reserved.

Cellular decomposition of quaternionic spherical space forms

We obtain an explicit cellular decomposition of the quaternionic spherical space forms, manifolds of positive constant curvature that are factors of an odd sphere by a free orthogonal action of a generalized quaternionic group. The cellular structure gives and explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group. As an application we compute the Whitehead torsion of these spaces for any representation of the fundamental group. © 2012 Springer Science+Business Media B.V.

Levi-flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms

The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter subgroup of the isometry group of the complex space form, there is an essentially unique example that is invariant under this one-parameter subgroup. On the other hand, when the curvature of the space form is zero, i.e., when the space form is complex 2-space with its standard flat metric, there is an additional `exceptional' example that has no continuous symmetries but is invariant under a lattice of translations. Up to isometry and homothety, this is the unique example with no continuous symmetries.

Oriented vehicles travelling in spherical space

This paper tackles the path planning problem for oriented vehicles travelling in the non-Euclidean 3-Dimensional space; spherical space S3. For such problem, the orientation of the vehicle is naturally represented by orthonormal frame bundle; the rotation group SO(4). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to control systems defined on Lie groups. The oriented vehicles, in this case, are constrained to travel at constant speed in a forward direction and their angular velocities directly controlled. In this paper we identify controls that induce steady motions of these oriented vehicles and yield closed form parametric expressions for these motions. The paths these vehicles trace are defined explicitly in terms of the controls and therefore invariant with respect to the coordinate system used to describe the motion.

On the stability of the L-p-norm of the Riemannian curvature tensor

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M where R(g) and dv (g) denote the corresponding Riemannian curvature tensor and volume form and p a (0, a). First we prove that the Riemannian metrics with non-zero constant sectional curvature are strictly stable for for certain values of p. Then we conclude that they are strict local minimizers for for those values of p. Finally generalizing this result we prove that product of space forms of same type and dimension are strict local minimizer for for certain values of p.

Steady state ion acceleration by a circularly polarized laser pulse

The steady state ion acceleration at the front of a cold solid target by a circularly polarized flat-top laser pulse is studied with one-dimensional particle-in-cell (PIC) simulation. A model that ions are reflected by a steady laser-driven piston is used by comparing with the electrostatic shock acceleration. A stable profile with a double-flat-top structure in phase space forms after ions enter the undisturbed region of the target with a constant velocity. (C) 2007 Elsevier B.V. All rights reserved.

沾化凹陷泥质岩油气藏成藏特征研究

Mudstone reservoir is a subtle reservoir with extremely inhomogeneous, whose formation is greatly related to the existence of fracture. For this kind of reservoir, mudstone is oil source rock, cover rock and reservoir strata, reservoir type is various, attitude of oil layer changes greatly, and the distribution of oil and gas is different from igneous or clastic rock reservoir as well as from carbonate reservoir of self-producing and self-containing of oil and gas. No mature experience has been obtained in the description, exploration and development of the reservoir by far. Taking Zhanhua depression as an example, we studied in this thesis the tectonic evolution, deposit characteristics, diagenesis, hydrocarbon formation, abnormal formation pressure, forming of fissure in mudstone reservoir, etc. on the basis of core analysis, physical simulation, numerical simulation, integrated study of well logging and geophysical data, and systematically analyzed the developing and distributing of mudstone fissure reservoir and set up a geological model for the formation of mudstone fissure reservoir, and predicted possible fractural zone in studied area. Mudstone reservoir mainly distributed on the thrown side of sedimentary fault along the sloping area of the petroleum generatiion depression in Zhanhua depression. Growing fault controlled subsidence and sedimentation. Both the rate of subsidence and thickness of mudstone are great on the thrown side of growing fault, which result in the formation of surpressure in the area. The unlocking of fault which leads to the pressure discharges and the upward conduct of below stratum, also makes for the surpressure in mudstone. In Zhanhua depression, mudstone reservior mainly developed in sub-compacted stratum in the third segment of Shahejie formation, which is the best oil source rock because of its wide spread in distribution, great in thickness, and rich in organic matter, and rock types of which are oil source mudstone and shale of deep water or semi-deep water sediment in lacustrine facies. It revealed from core analysis that the stratum is rich in limestone, and consists of lamina of dark mudstone and that of light grey limestone alternately, such rock assemblage is in favor of high pressure and fracture in the process of hydrocarbon generation. Fracture of mudstone in the third segment of Shahejie formation was divided into structure fracture, hydrocarbon generation fracture and compound fracture and six secondary types of fracture for the fist time according to the cause of their formation in the thesis. Structural fracture is formed by tectonic movement such as fold or fault, which develops mainly near the faults, especially in the protrude area and the edge of faults, such fracture has obvious directivity, and tend to have more width and extension in length and obvious direction, and was developed periodically, discontinuously in time and successively as the result of multi-tectonic movement in studied area. Hydrocarbon generation fracture was formed in the process of hydrocarbon generation, the fracture is numerous in number and extensively in distribution, but the scale of it is always small and belongs to microfracture. The compound fracture is the result of both tectonic movement and hydrocarbon forming process. The combination of above fractures in time and space forms the three dimension reservoir space network of mudstone, which satellites with abnormal pressure zone in plane distribution and relates to sedimentary faces, rock combination, organic content, structural evolution, and high pressure, etc.. In Zhanhua depression, the mudstone of third segment in shahejie formation corresponds with a set of seismic reflection with better continuous. When mudstone containing oil and gas of abnormal high pressure, the seismic waveform would change as a result of absorb of oil and gas to the high-frequency composition of seismic reflection, and decrease of seismic reflection frequency resulted from the breakage of mudstone structure. The author solved the problem of mudstone reservoir predicting to some degree through the use of coherent data analysis in Zhanhua depression. Numerical modeling of basin has been used to simulate the ancient liquid pressure field in Zhanhua depression, to quantitative analysis the main controlling factor (such as uncompaction, tectonic movement, hydrocarbon generation) to surpressure in mudstone. Combined with factual geologic information and references, we analyzed the characteristic of basin evolution and factors influence the pressure field, and employed numerical modeling of liquid pressure evolution in 1-D and 2-D section, modeled and analyzed the forming and evolution of pressure in plane for main position in different periods, and made a conclusion that the main factors for surpressure in studied area are tectonic movement, uncompaction and hydrocarbon generation process. In Zhanhua depression, the valid fracture zone in mudstone was mainly formed in the last stage of Dongying movement, the mudstone in the third segment of Shahejie formation turn into fastigium for oil generation and migration in Guantao stage, and oil and gas were preserved since the end of the stage. Tectonic movement was weak after oil and gas to be preserved, and such made for the preserve of oil and gas. The forming of fractured mudstone reservoir can be divided into four different stages, i.e. deposition of muddy oil source rock, draining off water by compacting to producing hydrocarbon, forming of valid fracture and collecting of oil, forming of fracture reservoir. Combined with other regional geologic information, we predicted four prior mudstone fracture reservoirs, which measured 18km2 in area and 1200 X 104t in geological reserves.

Geomorfologiczne uwarunkowania rozwoju sieci komunikacyjnej na obszarze młodoglacjalnym (zlewnia Perznicy, Pojezierze Drawskie

The aim of the studies in the Perznica River catchment were relief changes caused by the development of transportation infrastructure. This type of transformation is dependable on the state of economy and the settlements. The development of transportation network in the last two hundred years was examined through the analysis of archival cartographic materials – maps from the years 1789, 1855, 1877, 1935 – and the comparison with the situation from mid 1980s. The Perznica River catchment has an area of 249 km2 and it is located in north-western Poland in the central part of the Drawskie Lakeland macroregion, which belongs to the West Pomeranian Lakeland. The heterogeneous Perznica River catchment relief has a denivelation of 159 m and is within 60 and 219 m a.s.l. The study area is within the Parsęta River lobe. A number of subzones, whose morphological diversity and diversity of sediments lithofacies is mainly a reflection of areal deglaciation of the continental ice-sheet marginal zone, has been distinguished and these are: • subzone of the internal kame moraine – the undulated moraine upland, diversified by kame forms and kettle holes, • subzone of ice-free space forms – the uplands of kame plateaux, • subzone of melt-out lake basins – Lake Wielatowo basin with a characteristic collar ridge, • morphological levels of the northern Pomeranian sloping surface – mainly flat moraine uplands and small outwashes. The economic development of the Perznica River catchment advanced in close connection with the physical and geographical context, mainly with the relief, soils and hydrological conditions. As a result, the flat moraine uplands and marginal outflow plains, which were easiest to cultivate, have been developed and populated faster than any other. Since the early medieval period, large, compact villages, often centered around big estates, were emerging in those areas. In areas with a high relief energy–kame-melt moraines, ice-free space forms and ridges around melt-out lake basins–farming entered on a larger scale from the eighteenth century. Scattered settlements in those areas forced the creation of a dense access road network to farms and fields. In the case of anthropogenic forms of transportation with denivelation exceeding 1 m in the study area, road excavations are present for 37.3 km, road undercuttings for 43.8 km and road embankments for 38.7 km in total length. That gives a high ratio of density of such forms, equal to 2.1 km per km–2.

The geometry of optimal control solutions on some six dimensional lie groups

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E-3, the spheres S-3 and the hyperboloids H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3)

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E³, the spheres S³ and the hyperboloids H³ with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions is illustrated.