An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space

We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce principal lines and an invariant moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of spacelike surfaces in the four-dimensional Minkowski space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures: the tangent indicatrix, and the normal curvature ellipse. We apply our theory to a class of spacelike general rotational surfaces.Comment: 23 pages; to appear in Mediterr. J. Math., Vol. 9 (2012

Kaehler Manifolds of Quasi-Constant Holomorphic Sectional Curvatures

The Kaehler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kaehler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kaehler metrics into Kaehler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kaehler metrics is shown to be exactly the class of Kaehler metrics whose potential function is only a function of the distance from the origin in complex Euclidean space. Finally we show that any rotational even dimensional hypersurface carries locally a natural Kaehler structure, which is of quasi-constant holomorphic sectional curvatures.Comment: 36 page

OUR EXPERIENCE IN HEALING COMPOUND FRACTURES OF THE LEG

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Canonical-type connection on almost contact manifolds with B-metric

The canonical-type connection on the almost contact manifolds with B-metric is constructed. It is proved that its torsion is invariant with respect to a subgroup of the general conformal transformations of the almost contact B-metric structure. The basic classes of the considered manifolds are characterized in terms of the torsion of the canonical-type connection.Comment: 11 pages, The final publication is available at http://www.springerlink.co

Timelike surfaces with zero mean curvature in Minkowski 4-space

On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this system determine a unique (up to a motion) timelike surface with zero mean curvature so that the given parameters are canonical. We find all timelike surfaces with zero mean curvature in the class of rotational surfaces of Moore type. These examples give rise to a one-parameter family of solutions to the system of natural partial differential equations describing timelike surfaces with zero mean curvature.Comment: 15 page

Natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric

A natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric is constructed. The class of these manifolds, where the considered connection exists, is determined. Some curvature properties for this connection, when the corresponding curvature tensor has the properties of the curvature tensor for the Levi-Civita connection and the torsion tensor is parallel, are obtained.Comment: 17 page

AdS(3) holography for non-BPS geometries

By using the approach introduced in arXiv:2107.09677 we construct non-BPS solutions of 6D $(1,0)$ supergravity coupled to two tensor multiplets as a perturbation of AdS$_3\times S^3$. These solutions are both regular and asymptotically AdS$_3\times S^3$, so according to the standard holographic framework they must have a dual CFT interpretation as non-supersymmetric heavy operators of the D1-D5 CFT. We provide quantitative evidence that such heavy CFT operators are bound states of a large number of light BPS operators that are mutually non-BPS.Comment: 36 pages, 2 Mathematica files containing data to reproduce our perturbative expansions, 1 readme file summarising how to use the Mathematica file

Preliminary Results of Microwave Non-Contact Detection and Depth Determination of Disbonds in Low Permitivity and Low Loss Thick Sandwich Composites

To ensure manufacturing quality and safe use of thick dielectric composite structures it is essential to utilize a nondestructive testing technique for inspecting their structural integrity. As the thickness of these composite structures increase, most of the nondestructive testing (NDT) techniques become less capable of detecting defects. Microwave signals can penetrate deep inside dielectric materials and interact with their inner structure. They are also sensitive to changes associated with boundary interfaces, which makes them very attractive for disbond detection in composite structures [1,2]. In a thick sandwich composite structure a disbond can occur between any two layers (i.e. in the place of an adhesive fine). The results of an electromagnetic model investigating the potential of a microwave NDT method for detecting disbonds and the potential of determining their depths in a multi-layered sandwich composite is presented. The model describes the interaction of microwave signals, radiating out of an open-ended rectangular waveguide, with a multi-layered composite structure. The composite structure under consideration includes thirteen layers of various materials (three layers of foam with two skin laminates and two similar substrates in between the foam layers) and two layers of air (standoff in front of the sample and free-space backing). Each layer is bound to another by a thin layer of adhesive. A layer representing a disbond is considered to be present in between any given two layers, replacing the adhesive line. The goal of the modeling is to arrive at optimum measurement parameters (frequency and standoff distance) for detecting a disbond and providing information about its depth

Kahler manifolds with quasi-constant holomorphic curvature

The aim of this paper is to classify compact Kahler manifolds with quasi-constant holomorphic sectional curvature.Comment: 18 page