Parallel displacements on the surface of a projective space

This paper is devoted to studies of parallel displacements of directions and planes in linear and nonlinear (in the narrow sense) connections along lines on a surface of a projective space considered as the point manifold and the manifold of tangential planes. Parallel displacements are described by means of covariant differentials of quasitensors in the case of nonlinear connections and projective-covariant differentials in linear connections. This work concerns researches in the area of differential geometry. The research is based on an application of G. F. Laptev’s method of defining a connection in a principal fiber bundle and his method of continuations and scopes, which generalizes the moving frame method and Cartan’s method of exterior forms; the research depends on calculation of exterior differential forms.     

Finite element differential forms

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing such differential equations by finite element methods, stable discretization depends on the development of spaces of finite element differential forms. As revealed recently through the finite element exterior calculus, for each order of differential form, there are two natural families of finite element subspaces associated to a simplicial triangulation. In the case of forms of order zero, which are simply functions, these two families reduce to one, which is simply the well-known family of Lagrange finite element subspaces of the first order Sobolev space. For forms of degree 1 and of degree n − 1 (where n is the space dimension), we obtain two natural families of finite element subspaces, unifying many of the known mixed finite element spaces developed over the last decades. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the geometry of Kawaguchi spaces

In the paper, with the use of the E. Cartan exterior forms method, the theory of linear and affine connections of the generalized Kawaguchi space of order two is constructed. It is proved that the linear connection of this space incorporates intrinsic antiquaternional structures, the conditions of their complete integrability are found, and the affine connections associated with the above-mentioned structures are constructed. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 321–334, July–September, 2000. Translated by R. Lapinskas     

Differential forms on locally convex spaces and the stokes formula

In this paper we prove a variant of the Stokes formula for differential forms of a finite codimension in a locally convex space (LCS). The main tool used by us for proving the mentioned formula is the surface layer theorem for surfaces of codimension 1 in a locally convex space which was proved earlier by the first author. Moreover, on some subspace of differential forms of the Sobolev type with respect to a differentiable measure we establish a formula expressing the operator adjoint to the exterior differential via standard operations of the calculus of differential forms and the logarithmic derivative. This connection was established earlier under stronger constraints imposed either on the LCS or on the measure, or on differential forms (the smoothness condition).     

Graded Poisson structures on the algebra of differential forms

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field.     

The Grassmann-like Manifold of Centered Planes

Mathematical Notes - Connections associated with the Grassmann-like manifold of centered planes in the multidimensional projective space are studied. A geometric interpretation of these connections...     

Subregular Spreads of (2n+1,q)

In this paper, we develop some of the theory of spreads of projective spaces with an eye towards generalizing the results of R. H. Bruck (1969,in“Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514, Univ. of North Carolina Press, Chapel Hill). In particular, we wish to generalize the notion of asubregularspread to the higher dimensional case. Most of the theory here was anticipated by Bruck in later papers; however, he never provided a detailed formulation. We fill this gap here by developing the connections between a regular spread of (2n+1)-dimensional projective space and ann-dimensional circle geometry, which is the appropriate generalization of the Miquelian inversive plane. After developing this theory, we provide a fairly general method for constructing subregular spreads of (5,q). Finally, we explore a special case of this construction, which yields several examples of three-dimensional subregular translation planes which are not André planes.     

A recursive basis for primitive forms in symplectic spaces and applications to Heisenberg groups

This paper is divided in two parts: in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space(V~(2n), ω). Successively, in Section 3, we apply our construction in the setting of Heisenberg groups H~n, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin's complex of differential forms in H~n.     

Hodge–Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media

We study in detail Hodge–Helmholtz decompositions in nonsmooth exterior domains Ω??^N filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms of rank q belonging to the weighted L²‐space L_s^{2, q}(Ω), s∈?, into irrotational and solenoidal q‐forms. These decompositions are essential tools, for example, in electro‐magnetic theory for exterior domains. To the best of our knowledge, these decompositions in exterior domains with nonsmooth boundaries and inhomogeneous and anisotropic media are fully new results. In the Appendix, we translate our results to the classical framework of vector analysis N=3 and q=1, 2. Copyright © 2008 John Wiley & Sons, Ltd.     

Geradenkomplexe im Flaggenraum I

In this paper we consider complexes of lines in an isotropic space of degree two, i. e. a three dimensional real affine space with the metricds ²=dx ². Using the method of differential forms we study the local differential geometry of first order and the theory of complex curves. Finally we give some applications in the theory of linear complexes.     

Harmonic Forms on Conformal Euclidean Manifolds: The Clifford Multivector Approach

In this paper we express the theory of harmonic differential forms on conformal Euclidean manifolds in terms of the so called Clifford multivector fields. The aim is to give good definitions for d and d* operators in Clifford multivector case. Using these definitions we derive a formula for the Laplace operator. Three fundamental examples are included in the end of the paper and connections to existing theory is discussed.     

Algebra Forms with $$d^{N} = 0$$ on Quantum Plane. Generalized Clifford Algebra Approach

We construct a q-analog of exterior calculus with a differential d satisfying d ^N = 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found.     

Exterior products of forms and the cohomology ring of a complex space

In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kählerian hypercoverings, in particular, projective algebraic varieties. We define an exterior product on these forms, which induces the cup product on the cohomology of analytic spaces. The main difficulty is to prove that this exterior product is functorial with respect to morphisms of analytic spaces. This exterior product can be used to prove that the cup product is compatible with the mixed Hodge structure on the cohomology.     

Compensated compactness for differential forms in Carnot groups and applications

In this paper we prove a compensated compactness theorem for differential forms of the intrinsic complex of a Carnot group. The proof relies on an Ls-Hodge decomposition for these forms. Because of the lack of homogeneity of the intrinsic exterior differential, Hodge decomposition is proved using the parametrix of a suitable 0-order Laplacian on forms.     

Erratum to: Discrete Lie Advection of Differential Forms

In this paper, we present a numerical technique for performing Lie advection of arbitrary differential forms. Leveraging advances in high-resolution finite-volume methods for scalar hyperbolic conservation laws, we first discretize the interior product (also called contraction) through integrals over Eulerian approximations of extrusions. This, along with Cartan’s homotopy formula and a discrete exterior derivative, can then be used to derive a discrete Lie derivative. The usefulness of this operator is demonstrated through the numerical advection of scalar fields and 1-forms on regular grids.     

Tensor of nonabsolute displacements on Grassman-like manifold of centered planes

For the Grassman-like manifold of centered planes in a projective space, we construct an object, which is the tensor of nonabsolute displacements. If this tensor vanishes, then parallel displacements of clothing planes are absolute.     

Nonholonomic torses of the first kind

In the three-dimensional Euclidean space, we study two-dimensional nonholonomic distributions with zero total curvature of the first kind, called nonholonomic torses of the first kind. The two cases are considered: 1) one of the principal curvatures of the first kind differs from zero (the general case), 2) both of the principal curvatures of the first kind equal zero (a nonholonomic plane). The result obtained in the second case is of the general form. In the study we use the canonical moving frame and apply Cartan’s exterior forms method described by by S. P. Finikov in the book Cartan’s Exterior Forms Method in Differential Geometry (GITTL, Moscow-Leningrad, 1948).     

Sobolev inequalities for differential forms andL q,p -cohomology

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold (M, g) and the L_q,p-cohomology of that manifold. The L_q,p-cohomology of (M,g) is defined to be the quotient of the space of closed differential forms in L^p(M) modulo the exact forms which are exterior differentials of forms in L^q (M).     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications

Integral section formulae for totally geodesic submanifolds (planes) intersecting a compact submanifold in a space form are available from appropriate representations of the motion invariant density (measure) of these planes. Here we present a new decomposition of the invariant density of planes in space forms. We apply the new decomposition to rewrite Santaló's sectioning formula and thereby to obtain new mean values for lines meeting a convex body. In particular we extend to space forms a recently published stereological formula valid for isotropic plane sections through a fixed point of a convex body in R³.