On the extremal compatible linear connection of a generalized Berwald manifold

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). It is known (Vincze in J AMAPN 21:199–204, 2005) that such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is some strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. The paper presents the idea of the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is uniquely determined because the number of the Lagrange multipliers is equal to the number of the equations for the compatibility of the linear connection with the Finslerian metric. Using the reference element method, the extremal compatible linear connection can be expressed in terms of the canonical data as well. It is an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection.

Base Point Freeness, Uniqueness of Decompositions and Double Points for Veronese and Segre Varieties

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector.

On in-plane drill rotations for Cosserat surfaces

We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a C 2 -smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations are fixed at a portion of the boundary. Put otherwise, if the tangent vectors of the new surface are obtained locally by only rotating the given tangent vectors, and if these rotations have a rotation axis which coincides everywhere with the normal of the initial surface, then the two surfaces are equal provided they coincide at a portion of the boundary. In the language of differential geometry of surfaces, we show that any isometry which leaves normals invariant and which coincides with the given surface at a portion of the boundary is the identity mapping.

The geometry of equations of motion: particles in equivalent universes

The equations of motion for the simplest non-holonomically constrained system of particles are formulated using six methods: Newton–Euler, Lagrange, Maggi, Gibbs–Appell, Kane, and Boltzmann–Hamel. The challenging tasks of exploring and explaining the relationships and equivalences between these formulations is accomplished by constructing a single representative particle for the system of particles. The single particle is constrained to move on a configuration manifold. The explicit construction of sets of tangent vectors to the manifold and their relation to the forces acting on the single particle are used to provide several helpful geometric interpretations of the relationships between the formulations. These interpretations can also be extended to help understand the relationships between different formulations of the equations of motion for more complex systems, including systems of rigid bodies and particles.

NEW APPROACHES ON DUAL SPACE

In this paper, we give how to define the basic concepts of differential geometry on Dual space. For this, dual tangent vectors that have p as dual point of application are defined. Then, the dual analytic functions defined by Dimentberg are examined in detail, and by using the derivative of the these functions, dual directional derivatives and dual tangent maps are introduced.

Modified field equations from a complexified nonlocal metric

We argue that it is possible to obtain higher-derivative Einstein’s field equations by means of an extended complexified backward–forward nonlocal extension of the space–time metric, which depends on space–time vectors. Our approach generalizes the notion of the covariant derivative along tangent vectors of a given manifold, and accordingly many of the differential geometrical operators and symbols used in general relativity. Equations of motion are derived and a nonlocal complexified general relativity theory is formulated. A number of illustrations are proposed and discussed accordingly.

Pipes Construction Based on the Pipes' Centre Curve Shapes

Parts of pipeline need a long piece, a short segment, and various inflate-deflate models. They require as well the thickness and curvature of the pipes. The objective of this paper is to obtain some formulas for modeling the long pipe, the short tube, and various inflate-deflate pipe patches. Relating to the purposes, we use their cross-section, longitudinal section, and center curves of the pipe parts. The methods are, the using of the polar coordinates and of the real functions, to define the cross and longitudinal section of the pipe patches, respectively. Then, we calculate three orthonormal vectors that are determined by the tangent vectors of the pipe center curves and two unit vectors that are perpendicular to the tangent vectors. After that, we evaluate the formulas to model the long pipes and the short pipes, both inflate-deflate and thickness shapes. The results show that, using its center curves of the pipe, it is handy to design the long and short pipes, multiple thicknesses, various volume fluctuations of the pipes, and useful to model the inflate-deflate pipe parts.

Tangent Vectors on Tangent Euclidean Spaces

This is an article on differential geometry that connects tangent vectors and tangent Euclidean spaces.

Manifolds and tensors

The mathematical basis of general relativity is differential geometry. This chapter establishes the starting point of differential geometry: manifolds, tangent vectors, cotangent vectors, tensors, and differential forms. The metric tensor is introduced, and its symmetries (isometries) are described. The importance of diffeomorphism invariance (or “general covariance”) is stressed.

Maps generating normals on a manifold

The first- and second-order canonical forms are considered on a smooth m-manifold. The approach connected with coordinate expressions of basic and fiber forms, and first- and second-order tangent vectors on a manifold is implemented. It is shown that the basic first- and secondorder tangent vectors are first- and second-order Pfaffian (generalized) differentiation operators of functions set on a manifold. Normals on a manifold are considered. Differentials of the basic first-order (second-order) tangent vectors are linear maps from the first-order tangent space to the second-order (third-order) tangent space. Differentials of the basic first-order tangent vectors (basic vectors of the first-order normal) set a map which takes a first-order tangent vector into the second-order normal for the first-order tangent space of (into the third-order normal for second-order tangent space). The map given by differentials of the basic first-order tangent vectors takes all tangent vectors to the vectors of the second-order normal. Such splitting the osculating space in the direct sum of the first-order tangent space and second-order normal defines the simplest (canonical) affine connection which horizontal subspace is the indicated normal. This connection is flat and symmetric. The second-order normal is the horizontal subspace for this connection. Derivatives of some basic vectors in the direction of other basic vectors are equal to values of differentials of the first vectors on the second vectors, and the order of differentials is equal to the order of vectors along which differentiation is made. The maps set by differentials of basic vectors of r-order normal for (r – 1)-order tangent space take the basic first-order tangent vectors to the basic vectors of (r + 1)-order normal for r-order tangent space. Horizontal vectors for the simplest second-order connection are constructed. Second-order curvature and torsion tensors vanish in this connection. For horizontal vectors of the simplest second-order connection there is the decomposition by means of the basic vectors of the secondand third-order normals.