Curvature operators which preserve normal spaces

This paper characterizes the invertible algebraic curvature operators that satisfy the Bianchi identity and also preserve normal spaces to the action of SO(n) on the space of bivectors.     

The problem of defining abstract bivectors

In this paper we investigate the problem of defining bivectors on a purely abstract level. This leads to algebras of symbolic vectors and bivectors.     

Stationary Curvatures of Two-Dimensional Totally Geodesic Submanifolds in the Grassmannian of Bivectors

An infinitesimal criterion indicating when a two-dimensional submanifold of a Riemannian symmetric space is totally geodesic is given. As an application, the classification of two-dimensional totally geodesic submanifolds of the Grassmannian of bivectors is given in a new way, and it is proved that the sectional curvature takes stationary values on tangent spaces of such submanifolds. Bibliography: 9 titles.     

One invariant measure and different poisson brackets for two non-holonomic systems

We discuss the non-holonomic Chaplygin and the Borisov-Mamaev-Fedorov systems, for which symplectic forms are different deformations of the square root from the corresponding invariant volume form. In both cases second Poisson bivectors are determined by L-tensors with non-zero torsion on configuration space, in contrast with the well-known Eisenhart-Benenti and Turiel constructions.     

Maxwell equations and the direction of the electromagnetic field

Let M₀ be the Minkowski space, let Λ₂(M₀) be the space of bivectors in M₀, and let G¹ ⊂ Λ₂(M₀) be the manifold of directions of the physical space, consisting of simple bivectors with square −1. A mapping F: U → Λ₂(M₀), U ⊂ ℝ⁴, satisfying the Maxwell equations is regarded as the tensor of an electromagnetic field in vacuum. The field is described on the basis of a special decomposition F = eω + h(*ω), where the mapping ω: U → G¹ is called the direction of the field, and e: U → (0, +∞) and h: U → ℝ are the electric and magnetic coefficients of the field. The Maxwell equations are reformulated in terms of ω, e, and h. Electromagnetic fields whose set of directions is a point or a one-dimensional subset of G¹ are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 118–146.     

Produit de bivecteurs et quaternions complexes

First we calculate the product of two bivectors in vectorial spaceR(p, q) (p andq are integers such thatp+q=n). Second we prove that this product is a quaternion forR(3, 0) and we generalize to finite number of bivectors. Third we prove that this product is a biquaternion forR(1, 3) and we genaralize in the same way. Fourth we prove that some complex quaternions can be connected with real Clifford algebra by choosing correctly the usual imaginary.      

The geometric associative algebras of Euclidean space

We deduce two associative algebra structures arising from the homogeneity and isotropy of three dimensional space with an Euclidean geometry. These are the Clifford algebrasCl(3,0) andCl(0,3). We define a bivector as the geometric product of two vectors, a definition that differs from the usual. There is a choice of whether the bivectors are constructed tail to tail or head to head leading respectively to a positive definite or negative definite Euclidean metric. The origin of the two metric choices is not identified in the usual approach. Thus we arrive at a definite geometric answer to the question of Crumeyrolle, “What is a Bivector?”.     

Classification of Totally Geodesic Surfaces in the Manifold of Directions in Physical Space

The Grassmannian of bivectors over the pseudo-Euclidean Minkowski 4-space is considered and its two-dimensional totally geodesic submanifolds are classified. The family of such surfaces is described in the language of the affine geometry of three-space. Bibliography: 5 titles.     

Compatibilité d'un fibré de Dirac avec une connexion

We extend to an almost Dirac structure and affine connections the notion of compatibility of a bivectors field or a 2-differential form with a pseudo-metric. Compatibility with a symmetric connection implies integrability. We shall be interested especially by such structures on Lie groups or Riemannian homogeneous spaces.     

Geometry of real grassmann manifolds. Parts I, II

The paper deals with the properties of the exterior algebra ℝ(Λⁿ) related to the Euclidean structure on ℝ(Λⁿ) induced by the scalar product in ℝ(Λⁿ). A geometric interpretation of inner multiplication for simple p-vectors is given. An invariant form of the Cartan criterion for the simplicity of a p-vector is given. The Plücker model realizing the real Grassmann manifold as a submanifold of the Euclidean space ℝ(Λⁿ), and an isometry of this submanifold onto the classical Grassmann manifold with SO(n)-invariant metric are described. A canonical decomposition of bivectors is given. Bibliography: 12 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 84–107. Translated by N. Yu: Netsvetaev.     

Inhomogeneous Plane Waves in Monoclinic Crystals subject to a Bias

Here we investigate the conditions of inhomogeneous plane waves propagation in monoclinic crystals subject to initial electromechanical fields. We obtain the components of the electroacoustic tensor for the class 2, resp. m, of the monoclinic system. For particular isotropic directional bivectors we derive the decomposition of the propagation condition, and we show that the specific coefficients are similar to the case of guided wave propagation in monoclinic crystals subject to a bias. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tensors,Spinors and Multivectors in the Petrov Classification

The Petrov classification of the Weyl tensor is revised in its two main approaches: bivector endomorphism and principal null directions. A more transparent presentation is obtained by the use of the real geometric Clifford algebra, where the consideration of bivectors (E) integrated in the full Grassmann space (E) is basic. This language establishes a more close relationship between both approaches and enables the introduction of a new canonical tensorial form for the Weyl tensor which is directly comparable with the spinorial classification. Special care has been given to present properties in its more general form, without specific restriction to a given dimensionality or to a given signature, whenever possible.     

3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$

It is possible to set up a correspondence between 3D space and \({\mathbb{R}^{3,3}}\), interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become represented as rotors in the geometric algebra of \({\mathbb{R}^{3,3}}\). We show explicitly how various primitive projective transformations (translations, rotations, scalings, perspectivities, Lorentz transformations) are represented, in geometrically meaningful parameterizations of the rotors by their bivectors. Odd versors of this representation represent projective correlations, so (oriented) reflections can only be represented in a non-versor manner. Specifically, we show how a new and useful ‘oriented reflection’ can be defined directly on lines. We compare the resulting framework to the unoriented \({\mathbb{R}^{3,3}}\) approach of Klawitter (Adv Appl Clifford Algebra, 24:713–736, 2014), and the \({\mathbb{R}^{4,4}}\) rotor-based approach by Goldman et al. (Adv Appl Clifford Algebra, 25(1):113–149, 2015) in terms of expressiveness and efficiency.     

On the canonical form of the electromagnetic field

Abstract Using the decomposition of an antisymmetric 2-tensor as a sum of two orthogonal bivectors, the various canonical forms of the electromagnetic tensor field are analyzed, recovering known results. However, introducing 1+3 spacetime splitting techniques, the canonical forms are associated to special frames and observers and this helps to clarify the role played by “measurable” quantities (electric and magnetic fields, Poynting vector) in the classification problem itself. Keywords: Electromagnetic field, Canonical form Mathematics Subject Classification (2000): 83A05, 78A25     

The Geometry of the Lie Algebra of the Orthogonal Group O(ℝ4)

In the six-dimensional space of bivectors, a Lie product similar to the standard vector product in is introduced. The Lie algebra constructed is proved to be isomorphic to the Lie algebra of the orthogonal group , and the isomorphism is a canonical isometry between and the space of antisymmetric operators in . Bibliography: 2 titles.     

Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1 in the form of blades that square to –1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research has been done [1] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cℓ ₃ of \mathbb R³{{\mathbb R^3}} . All these roots of –1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations.     

Bi‐Presymplectic Representation of Liouville Integrable Systems and Related Separability Theory

Bi‐presymplectic chains of one‐forms of arbitrary co‐rank are considered. The conditions in which such chains represent some Liouville integrable systems and the conditions in which there exist related bi‐Hamiltonian chains of vector fields are presented. To derived the construction of bi‐presymplectic chains, the notions of dual Poisson‐presymplectic pair, d‐compatibility of presymplectic forms and d‐compatibility of Poisson bivectors is used. The completely algorithmic construction of separation coordinates is demonstrated. It is also proved that Stäckel separable systems have bi‐inverse‐Hamiltonian representation, i.e., are represented by bi‐presymplectic chains of closed one‐forms. The co‐rank of related structures depends on the explicit form of separation relations.     

Applications of Matrix Algebra to Clifford Groups

In this paper, we will show that all of nonzero vectors and nonzero bivectors in the Clifford algebra ${\mathcal{C} \ell_{0,3}}$ are invertible and we will find some conditions for those objects to be element of the Clifford group ??_0,3 using the corresponding properties in the subalgebra L ₈ of the matrix algebra ${M_8 \mathbb{(R)}}$ .     

Sectional Curvatures and the Separation Set of the Complex Projective Space in Its Plücker Model

The Plücker embedding of the complex projective space in the Grassmannian of bivectors is used for proving several theorems on the relationship between the complex structure of and its Riemannian geometry. It is shown that the separation set of in the Plücker model is the face of for a certain calibration. Bibliography: 11 titles.     

On canonical embeddings of complex projective spaces in real projective spaces

In this paper, we construct a natural embedding \(\sigma :\mathbb{C}P_\mathbb{R}^{n} \to \mathbb{R}P^{n^2 + 2n} \) of the complex projective space ?P ⁿ considered as a 2n-dimensional, real-analytic manifold in the real projective space \(\mathbb{R}P^{n^2 + 2n} \). The image of the embedding σ is called the ?P ⁿ-surface. To construct the embedding, we consider two equivalent approaches. The first approach is based on properties of holomorphic bivectors in the realification of a complex vector space. This approach allows one to prove that a ?P-surface is a flat section of a Grassman manifold. In the second approach, we use the adjoint representation of the Lie group U(n + 1) and the canonical decomposition of the Lie algebra u(n). This approach allows one to state a gemetric characterization of the canonical decomposition of the Lie algebra u(n). Moreover, we study properties of the embedding constructed. We prove that this embedding determines the canonical Kähler structure on ?P _? ⁿ . In particular, the Fubini-Study metric is exactly the first fundamental form of the embedding and the complex structure on ?P _? ⁿ is completely defined by its second fundamental form; therefore, this embedding is said to be canonical. Moreover, we describe invariant and anti-invariant completely geodesic submanifolds of the complex projective space.