One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula

In [10] one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane \mathbb E²{{\mathbb E}^2} were obtained. Moreover, the relations between the complex velocities of one-parameter motion in the complex plane were provided by [10]. One-parameter planar homothetic motion was defined in the complex plane, [9]. In this paper, analogous to homothetic motion in the complex plane given by [9], one-parameter planar homothetic motion is defined in the hyperbolic plane. Some characteristic properties about the velocity vectors, the acceleration vectors and the pole curves are given. Moreover, in the case of homothetic scale h identically equal to 1, the results given in [15] are obtained as a special case. In addition, three hyperbolic planes, of which two are moving and the other one is fixed, are taken into consideration and a canonical relative system for one-parameter planar hyperbolic homothetic motion is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, is obtained with the help of this relative system.     

Algebras Generated by Scalar <Emphasis Type="Italic">K</Emphasis>-atic Forms and Their Linear Forms

A linear form with an N-elements basis set {e _i ; i = 1,...,N} generates an algebra which is that of multivectors, provided some commutation relation is defined to give a meaning to the outer product of the basis vectors. If, moreover, an inner product of sets of K basis vectors is also introduced, for a mapping producing a 0-form, a geometric algebra is obtained. The algebra has thus two basic numbers to define its dimension: the dimension N of the basis set and the dimension K of the number of elements to be multiplied together to obtain a scalar. If the dimension K refers to the order of the power of [e _i ] ^K to obtain the scalar we will say that we have a K-atic algebra, the best known example is when the scalar form is a quadratic expression; these algebras are said to have a metric which in general is either diagonal or at least symmetric. Otherwise if the dimension K refers again to the number of different basis vectors to be multiplied together in (with j ≠ i and in general all subindexes different) then we obtain a simplectic algebra where the best known case is also when K = 2 and the metric in this case is antisymmetric. In the present paper we define these sets of algebras, give the commutation relations for the algebras with a K-atic scalar form and relate the results to the best known examples of current use in the literature.     

Tensor products and discriminants of unital quadratic forms over commutative rings

A tensor product for unital quadratic forms is introduced which extends the product of separable quadratic algebras and is naturally associative and commutative. It admits a multiplicative functor vdis, the vector discriminant, with values in symmetric bilinear forms. We also compute the usual (signed) discriminant of the tensor product in terms of the discriminants of the factors. The orthogonal group scheme of a nonsingular unital quadratic formQ of even rank is isomorphic toZ ₂×SO(Q ⁰) whereQ ⁰ is the restriction of –Q to the space of trace zero elements. We use cohomology to interpret the action of separable quadratic algebras on unital quadratic forms, and to determine which forms of odd rank can be realized asQ ⁰.     

Divisors of Zero in the Lipschitz Semigroup

The Lipschitz semigroup is generated by all (invertible and noninvertible) Clifford vectors. We show that all solutions of the equation xy = 0 (where x, y are non-zero elements of the Lipschitz semigroup) are of the form x = av₀, y = v₀b where v₀ is an isotropic vector (i.e., v₀² = 0). This problem turns out to be useful in the construction of multisoliton solutions of integrable systems of nonlinear partial differential equtions.     

Representations of Clifford Algebras with Hyperbolic Numbers

The representations of Clifford algebras and their involutions and anti-involutions are fully investigated since decades. However, these representations do sometimes not comply with usual conventions within physics. A few simple examples are presented, which point out that the hyperbolic numbers can close this gap.      

On split products of quaternion algebras with involution in characteristic two

The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution is shown to have an affirmative answer.     

Hyperbolic surfaces in the Grassmannian

In this article we study real 2-dimensional surfaces in the Grassmannian of 2-planes in a 4-dimensional vector space. These surfaces occur naturally as the fibers of jet bundles of partial differential equations.On the Grassmannian there is an invariant conformal quadratic form and we will use the structure induced by this quadratic form to study the surfaces. We give a topological classification of compact hyperbolic surfaces similar to the classification by Gluck and Warner [Duke Math. J. 50 (1) (1983)] of compact elliptic surfaces. In contrast with elliptic surfaces there are several topological possibilities for hyperbolic surfaces. We make a calculation of the differential invariants under the action of the group of conformal isometries. Finally, we analyze a class of surfaces called geometrically flat and show that within this class there exist many examples of non-trivial compact surfaces.     

Über drei zwangläufig gegeneinander bewegte Cayley/Klein-Ebenen

If three Euclidean planes move relatively to each other, the three poles of rotation are either identical or pairwise distinct and collinear. In the second case the distances of the poles are in the ratio of the motions' angular velocities. These known facts of Euclidean kinematics can be generalized in a largely uniform way to plane Cayley/Klein motions with finite poles. For the angular velocities we give a representation which is valid for all considered Cayley/Klein motions. Application of the duality principle of the projective plane yields a proposition about concurrent fixed lines. We also generalize the generation of a pair of envelope curves of a Euclidean motion as paths of a point of a third moved plane.

     

Classification of three-parametric spatial motions with a transitive group of automorphisms and three-parametric robot manipulators

The paper deals with the differential geometry of submanifolds of the kinematical space of Euclidean space kinematics, which is a six-dimensional pseudo-Riemannian symmetric space of signature (3, 3). The main result is in the proof of the classification theorem for three-dimensional Euclidean space motions with a transitive group of automorphisms. All of them are products (in the group multiplication) of homogeneous spaces and their list is provided. All three-parametric robot manipulators with constant invariants are found as an application of the classification theorem.     

Cyclic surfaces in E 5 generated by equiform motions

In this paper, we study cyclic surfaces in E⁵ generated by equiform motions of a circle. The properties of this cyclic surfaces up to the first order are discussed. We prove the following new result: A cyclic 2-surfaces in E⁵ in general are contained in canal hypersurfaces. Finally we give an example.     

Comtrans algebras, Thomas sums, and bilinear forms

A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.Received: 30 March 2004     

Generic adjoints in comtrans algebras of bilinear spaces

As a first step towards a general structure theory for comtrans algebras (modeled loosely on the Cartan theory for Lie algebras), this paper investigates comtrans algebras of bilinear spaces. Attention focuses on invariants associated with comtrans algebras, and the extent to which these invariants may serve to specify the algebras up to isomorphism within certain classes. Over fields whose characteristic differs from two, comtrans algebras of symmetric forms are determined up to isomorphism by the eigenvalues of generic adjoints, while comtrans algebras of symplectic forms are determined by the dimensions of maximal abelian subalgebras. Examples show that the multiplicity of zero as a root of the characteristic polynomial is generally independent of the dimension of a maximal abelian subalgebra.     

Quantum Clifford Hopf gebra for quantum field theory

We give arguments for the necessity to employ quantum Clifford Hopf gebras in quantum field theory. The role of the antipode is examined, Feynman diagrams are re-interpreted as tangles of graphical calculus. Regularization due to the design of convolution Hopf gebras is given as a program for future research.     

Derived brackets and sh Leibniz algebras

We develop a general framework for the construction of various derived brackets. We show that suitably deforming the differential of a graded Leibniz algebra extends the derived bracket construction and leads to the notion of strong homotopy (sh) Leibniz algebra. We discuss the connections among homotopy algebra theory, deformation theory and derived brackets. We prove that the derived bracket construction induces a map from suitably defined deformation theory equivalence classes to the isomorphism classes of sh Leibniz algebras.     

Hyperbolic manifolds are geodesically rigid

We show that if all geodesics of two non-proportional metrics on a closed manifold coincide (as unparameterized curves), then the manifold has a finite fundamental group or admits a local-product structure. This implies that, if the manifold admits a metric of negative sectional curvature, then two metrics on the manifold have the same geodesics if and only if they are proportional. Oblatum 18-IV-2002 & 12-VIII-2002?Published online: 18 December 2002     

Linear Spaces on the Intersection of Two Quadratic Hypersurfaces,and Systems of <Emphasis Type="Italic">p</Emphasis>-adic Quadratic Forms

We obtain new results on linear spaces on the intersection of two quadratic forms defined over a non-dyadic p-adic field . One of our main tools is a recent result of Parimala and Suresh on isotropy of quadratic forms over functions fields over . As a corollary we also get new bounds for the number of variables necessary to always find a non-trivial p-adic zero of a system of quadratic forms.     

Dunford's property (C) and Weyl's theorems

This paper studies Weyl's theorems, and some related results for operators with Dunford's property (C). Weyl's theorem in some classes of operators (e.g.M-hyponormal,p-hyponormal and totally paranormal operators) is considered.     

Constant Mean Curvature Surfaces with Boundaryin Hyperbolic Space

 We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize umbilical surfaces bounded by a circle among all H-discs with small area. Received 27 March 1997; in final form 11 June 1998