Search

Infinitesimal bending of curves lying with a given precision on ruled surfaces in 3-dimensional Euclidean space is studied. In particular, the bending of curves on the cylinder, the hyperbolic paraboloid and the helicoid are considered and appropriate bending fields are found. Some examples are graphically presented.

In the present paper, we define a new class of generalized para-Kahler spaces by using two different kinds of covariant derivatives. Some properties of the curvature tensors as well as those of the corresponding Ricci tensors of new generalized para-Kahler spaces are pointed out. Also, we consider some tensor fields that are invariant with respect to holomorphically projective mappings of generalized para-Kahler spaces. Finally, we examine some PDE systems for the existence of infinitesimal holomorphically projective transformations of new types of generalized para-Kahler spaces with parallel torsion.

We derive some interesting properties of generalized almost Hermitian spaces. Additionally, we study basic equations of holomorphically projective mappings between generalized Kahler spaces and examine two non-linear PDE systems for the existence of these mappings. In the case of an equitorsion holomorphically projective mapping, we transform those non-linear PDE-systems into linear PDE systems. Some new types of generalized Kahler spaces that admit equitorsion holomorphically projective mappings and transformations are defined as a natural consequence of the linear PDE systems for the existence of these diffeomorphisms. Our consideration provides new results as well as extensions and improvements of some results previously published in several papers.

We investigate a second order infinitesimal bending of curves in a three-dimensional Euclidean space in this paper. We give the necessary and sufficient conditions for the vector fields to be infinitesimal bending fields of the corresponding order, as well as explicit formulas which determine these fields. We examine the first and the second variation of some geometric magnitudes which describe a curve, specially a change of the curvature. Two illustrative examples (a circle and a helix) are studied not only analytically but also by drawing curves using computer program Mathematica.

In this paper, we consider equitorsion third type almost geodesic mappings of non-symmetric affine connection spaces. Applying different computational methods, we find invariants of these mappings obtained from curvature tensors of a corresponding non-symmetric affine connection space. Weyl projective tensor of a symmetric affine connection space is generalized in this way for the case of a third type almost geodesic mapping.

In the present paper we study equitorsion geodesic mappings between two generalized Riemannian spaces f : GR N → G R ‾ N . In this case these spaces have the same torsions at corresponding points. We prove that a generalized Riemannian space GR N admits an equitorsion geodesic mapping onto a generalized Riemannian space G R ‾ N if and only if in GR N certain differential equations of Cauchy type in covariant derivatives of the θ = 1 , … , 4 kinds have a solution with respect to the unknown tensor a i j , the gradient vector λ i ≠ 0 , and the differentiable function μ θ , θ = 1 , … , 4 . In fact we find four systems of PDE all equivalent to the existence of an equitorsion geodesic mapping and discuss the number of linearly independent solutions of this system of PDE. We establish an upper bound for the number of solutions for the geometrical problem under consideration.

We discuss the infinitesimal bending of curves and knots in [Formula: see text]. A brief overview of the results on the infinitesimal bending of curves is outlined. Change of the Willmore energy, as well as of the Möbius energy under infinitesimal bending of knots is considered. Our visualization tool devoted to visual representation of infinitesimal bending of knots is presented.

We relax the conditions $$\underset{1}{\nabla }F=0$$ and $$\underset{2}{\nabla }F=0$$ with respect to the non-symmetric linear connections $$\underset{1}{\nabla }$$ and $$\underset{2}{\nabla }$$ and an almost complex structure F of a generalized Kahler space. In such a way we introduce a wider class of generalized Kahler spaces which admits a holomorphically projective mapping. These generalized Kahler spaces are named generalized Kahler spaces in Eisenhart’s sense, since they are defined as a particular case of Eisenhart’s generalized Riemannian spaces. Curvature tensors of generalized Kahler spaces in Eisenhart’s sense have some interesting properties that have been pointed out in the present paper. Also, we consider equitorsion holomorphically projective mappings and examine some invariant geometric objects with respect to these mappings. Our results are deduced without restrictive conditions that were given in some of the previous papers.

In a space $L_N$ of asymmetric affine connection by equations (1.1) a submanifold $X_M\subset L_N$ is defined. On $X_M$ and on pseudonormal submanifold $X_{N-M}^N$ asymmetric induced connections are defined. Because of asymmetry of induced connection it is possible to define four kinds of covariant derivative. In this work we are considering integrability conditions of derivational equations \cite{4} obtained by help of the $1^{st}$ and the $2^{nd}$ kind of covariant derivative. The corresponding Gauss-Codazzi equations are obtained too.

This paper is devoted to academician Mileva Prvanovic on the occasion of her 85th birthday. We point out her influence on development of geometrical sciences, as the first doctor oh these sciences in Serbia.

The object of the present paper is to introduce spacetimes with semisymmetric energy-momentum tensor. At first we consider the relation R(X,Y)⋅T=0, that is, the energy-momentum tensor T of type (0,2) is semisymmetric. It is shown that in a general relativistic spacetime if the energy-momentum tensor is semisymmetric, then the spacetime is also Ricci semisymmetric and the converse is also true. Next we characterize the perfect fluid spacetime with semisymmetric energy-momentum tensor. Then, we consider conformally flat spacetime with semisymmetric energy-momentum tensor. Finally, we cited some examples of spacetimes admitting semisymmetric energy-momentum tensor.

In view of the meaning of ruled surfaces in aesthetics, statics, scale and manufacturing technologies, we point out the possibility of a mathematical analysis in the case of infinitesimal deformations by considering the rigidity of surfaces. In this work we consider variations of the shape operator, the normal curvature and the principal curvatures of Gaudi surfaces under infinitesimal bending.

The basic information on generalized Riemannian space GR"N and its submanifold X"M and normal bundle X"N"-"M^N are given. As introduced connections are generally asymmetric, it is possible to define four kinds of covariant derivative. In the present work one considers integrability conditions of derivational equations obtained by using the 1st and the 2nd kind of covariant derivative. We obtain also the corresponding Gauss-Codazzi equations. Similar results using simultaneously the both kinds of covariant derivative are obtained.

In this paper, we consider infinitesimal bending of the second-order of curves and knots. The total curvature of the knot during the second-order infinitesimal bending is discussed and expressions for the first and the second variation of the total curvature are given. Some examples aimed to illustrate infinitesimal bending of knots are shown using figures. Colors are used to illustrate curvature values at different points of bent knots and the total curvature is numerically calculated.

In case of bendable surfaces it is useful to discuss the variation of magnitudes such as the shape operator. The shape operator is a good way to measure how a regular surface S bends in R^3 by valuation how the surface normal @n changes from point to point. We considered the variation of shape operator under infinitesimal bending of surface given in an explicit form. We also gave two examples of the variation of shape operator of Gaudi surface and hyperbolic paraboloid.

In the works by Mincic et al. (1998, 1999, 2007) [8,9,11] we have studied the problem (started by M. Prvanovic): Can a generalized Riemannian space GR"N (with non-symmetric basic tensor) contain a Riemannian subspace (possessing a symmetric induced basic tensor)? In the present work the analogous problem is studied for a space L"N with non-symmetric affine connection and its subspace L"M. In the Section 2 it is assumed that the equations defining submanifold are given and that it is valid (2.4), i.e., induced torsion T~=0, and the connection on the manifold M"N is determined so that T 0. At the third section we start from connection on the manifold M"NT 0. From the mentioned condition (2.4) we determine the Eq. (1.1) defining the submanifold M"M@?M"N, so that induced torsion T~=0, i.e., we get L"M^O@?L"N. It is proved that the answer to the question cited above is affirmative and several examples are constructed. The examinations have local character.

Complex freeform structures are one of the most striking trends in contemporary architecture. In a variety of mathematical surfaces, there is a special kind, as known as Gaudi surfaces. Gaudi surfaces are used as elements of architecture and roof constructions. Including aesthetics, statics, scale and manufacturing technologies, it should point out the possibility of the mathematical analysis in the case of infinitesimal deformations as well as considering the rigidity of these surfaces using tensor calculus approach.

In this paper geodesic mappings of equidistant generalized Riemannian spaces are discussed. It is proved that each equidistant generalized Riemannian space of basic type admits non-trivial geodesic mapping with preserved equidistant congruence. Especially, there exists non-trivial geodesic mapping of equidistant generalized Riemannian space onto equidistant Riemannian space. An example of geodesic mapping of an equidistant generalized Riemannian spaces is presented.

Variation of the total mean curvature of piecewise smooth surfaces in Euclidean 3-spaces under infinitesimal bending is discussed and reduced to a sum of line integrals of a rotation vector field.

In this paper we apply tensor calculus and differential geometry to consider shell structure. Using tensor calculus we examine the stationarity of the Willmore energy under infinitesimal deformations of the surfaces in ℜ3. We obtain the class of the surfaces which does not change its Willmore energy under infinitesimal deformations. In particular, a special kind of deformation is considered—infinitesimal bending which preserves the arc length. The change of the Willmore energy under such deformations is determined. Also, we give a new proof of a well-known theorem (that reads that the total mean curvature of a surface is stationary under an infinitesimal bending), applying tensor calculus.

In this paper we consider a class of toroids generated by a meridian in a shape of quadrangle in an arbitrary position to the axis of revolution. We give geometrical meaning of the non-rigidity condition based on the position of the diagonals of meridian. Special classes of quadrangular meridians (trapezium, parallelogram, deltoid, … ) are discussed. The number of flexible toroids is considered. A tool SurfBend is developed for the rigidity and visualization.

In this paper we define generalized Kahlerian spaces of the first kind ( $$ G_1^K N $$ ) given by (2.1)–(2.3). For them we consider hollomorphically projective mappings with invariant complex structure. Also, we consider equitorsion geodesic mapping between these two spaces ( $$ G_1^K N $$ and $$ G_1^{\bar K} N $$ ) and for them we find invariant geometric objects.

Change of the Willmore energy, as the special case of elastic bending energy, under infinitesimal bending of the vesicle membrane is discussed. A membrane is thought of as a smooth surface in R3 because its thickness is much smaller than its lateral dimension. Variation of the Willmore energy at the surface point under infinitesimal bending of that surface, as well as the condition for the stationarity of the Willmore energy are given. Some examples are visualized. Variation of the Willmore energy of a compact path-connected smooth surface in the Euclidean 3-space is reduced to a line integral of a special vector field and it is found that the Willmore energy of a boundary-free surface is stationary under an infinitesimal bending. Also, the Willmore energy of a minimal surface is stationary under its infinitesimal bending.

The object of the present paper is to study almost pseudo-[Formula: see text]-symmetric manifolds [Formula: see text]. Some geometric properties have been studied which recover some known results of pseudo [Formula: see text]-symmetric manifolds. We obtain a necessary and sufficient condition for the [Formula: see text]-curvature tensor to be recurrent in [Formula: see text]. Also, we provide several interesting results. Among others, we prove that a Ricci symmetric [Formula: see text] is an Einstein manifold under certain condition. Moreover we deal with [Formula: see text]-flat perfect fluid, dust fluid and radiation era perfect fluid spacetimes respectively. As a consequence, we obtain some important results. Finally, we consider [Formula: see text]-spacetimes.

Using recent work as a foundation, we have considered the infinitesimal deformations f:x^i->x^i+@ez^i(x^j) of a space L"N with non-symmetric affine connection L"j"k^i. On the basis of the non-symmetry of the connection, we use four kinds of covariant derivatives to express the Lie derivative and the deformations. The rigidities of the geometric objects (connection, tensors, curvature) are defined and analyzed using the Lie derivative.

Within this paper we consider the higher order infinitesimal bending of a surface in E3. The sufficient condition for a toroid generated by a polygonal meridian to be non-rigid of higher order is given. The work contained is an extension of the results of [2] and [13–18]. Examples of non-rigid surfaces, with a visual representation of infinitesimal bending are provided.

In the papers [19], [20] several Ricci type identities are obtained by using non-symmetric affine connection. In these identities appear 12 curvature tensors, 5 of which being independent [21], while the rest can be expressed as linear combinations of the others. In the general case of a geodesic mapping f of two non-symmetric affine connection spaces GAN and GAN it is impossible to obtain a generalization of the Weyl projective curvature tensor. In the present paper we study the case when GAN and GAN have the same torsion in corresponding points. Such a mapping we name ”equitorsion mapping”. With respect to each of mentioned above curvature tensors we have obtained quantities E θ i jmn (θ = 1, · · · , 5), that are generalizations of the Weyl tensor, i.e. they are invariants based on f . Among E θ only E 5 is a tensor. All these quantities are interesting in constructions of new mathematical and physical structures. AMS Subj. Class.: 53B05.

Starting from the definition of generalized Riemannian space (GRN) [1], in which a non-symmetric basic tensor Gij is introduced, in the present paper a generalized K?hlerian space GK2 N of the second kind is defined, as a GRN with almost complex structure Fhi, that is covariantly constant with respect to the second kind of covariant derivative (equation (2.3)). Several theorems are proved. These theorems are generalizations of the corresponding theorems relating to KN. The relations between Fhi and four curvature tensors from GRN are obtained.

An infinitesimal bending of the curve at E3 is considered and the infinitesimal bending field is determined and discussed. CurveBend, tool for graphical presentation of non rigid curves is presented. Influence of infinitesimal bending field on curves is discussed and visualized by the tool.

Minimal surfaces are the surfaces of the smallest area spanned by a given boundary. The equivalent is the definition that it is the surface of vanishing mean curvature. Minimal surface theory is rapidly developed at recent time. Many new examples are constructed and old altered. Minimal area property makes this surface suitable for application in architecture. The main reasons for application are: weight and amount of material are reduced on minimum. Famous architects like Otto Frei created this new trend in architecture. In recent years it becomes possible to enlarge the family of minimal surfaces by constructing new surfaces.

One of the main tasks of the deformation theory is to point out to the rigid and flexible surfaces. In this paper we signify a torus like class of surfaces generated by parallelogram in ?3. It is proved that this class is rigid due to infinitesimal bending. Infinitesimal bending of generated surfaces is considered using Cohn-Vossen's method. .

In this paper we investigate holomorphically projective mappings of generalized Kahlerian spaces. In the case of equitorsion holomorphically projective mappings of generalized Kahlerian spaces we obtain five invariant geometric objects for these mappings.

In tins paper we consider conoid surfaces as frequently used surfaces in building techniques, mainly as daring roof structures. Different types of conoids are presented using the programme package Mathematica. We describe the generation of conoids and by means of parametric representation we get their graphics. The geometric approach offers a wide range of possibilities in the research of complicated spatial surface systems.

The object of the present paper is to study a spacetime admitting conharmonic curvature tensor and some geometric properties related to this spacetime. It is shown that in a conharmonically flat spacetime with cyclic parallel Ricci tensor, the energy–momentum tensor is cyclic parallel and conversely. Finally, we prove that for a radiative perfect fluid spacetime if the energy–momentum tensor satisfying the Einstein’s equations without cosmological constant is generalized recurrent, then the fluid has vanishing vorticity and the integral curves of the vector field [Formula: see text] are geodesics.

Infinitesimal bending of curves in E 3 is considered and an infinitesimal bending field is determined and discussed. A special case of deformation of a plane curve staying plane under infinitesimal bending is analyzed. Also, the variations of the curvature and the torsion are obtained. Infinitesimal bending of an ellipse and Cassini curve are discussed and presented graphically. We present our tool InfBend aimed at infinitesimal deformation and visualization of curves and surfaces. It is written in C++ and uses OpenGL for modelling three-dimensional curves and surfaces.