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Higher order contact of real curves in a real hyperquadric. II

Yuli Villarroel (1998)

Archivum Mathematicum

Let Φ be an Hermitian quadratic form, of maximal rank and index ( n , 1 ) , defined over a complex ( n + 1 ) vector space V . Consider the real hyperquadric defined in the complex projective space P n V by Q = { [ ς ] ∈ P n V , Φ ( ς ) = 0 } . Let G be the subgroup of the special linear group which leaves Q invariant and D the ( 2 n ) - distribution defined by the Cauchy Riemann structure induced over Q . We study the real regular curves of constant type in Q , tangent to D , finding a complete system of analytic invariants for two curves to be locally equivalent...

Hodge-Bott-Chern decompositions of mixed type forms on foliated Kähler manifolds

Cristian Ida (2014)

Colloquium Mathematicae

The Bott-Chern cohomology groups and the Bott-Chern Laplacian on differential forms of mixed type on a compact foliated Kähler manifold are defined and studied. Also, a Hodge decomposition theorem of Bott-Chern type for differential forms of mixed type is proved. Finally, the case of projectivized tangent bundle of a complex Finsler manifold is discussed.

Holomorphically projective mappings of compact semisymmetric manifolds

Raad J. K. al Lami (2010)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In this paper we consider holomorphically projective mappings from the compact semisymmetric spaces A n onto (pseudo-) Kählerian spaces K ¯ n . We proved that in this case space A n is holomorphically projective flat and K ¯ n is space with constant holomorphic curvature. These results are the generalization of results by T. Sakaguchi, J. Mikeš, V. V. Domashev, N. S. Sinyukov, E. N. Sinyukova, M. Škodová, which were done for holomorphically projective mappings of symmetric, recurrent and semisymmetric Kählerian...

Homogénéité locale pour les métriques riemanniennes holomorphes en dimension 3

Sorin Dumitrescu (2007)

Annales de l’institut Fourier

Une métrique riemannienne holomorphe sur une variété complexe M est une section holomorphe q du fibré S 2 ( T * M ) des formes quadratiques complexes sur l’espace tangent holomorphe à M telle que, en tout point m de M , la forme quadratique complexe q ( m ) est non dégénérée (de rang maximal, égal à la dimension complexe de M ). Il s’agit de l’analogue, dans le contexte holomorphe, d’une métrique riemannienne (réelle). Contrairement au cas réel, l’existence d’une telle métrique sur une variété complexe compacte n’est...

Homogeneous Cartan geometries

Matthias Hammerl (2007)

Archivum Mathematicum

We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely algorithmic. As an example of an homogeneous parabolic geometry we treat a conformal structure on the product of two spheres.

Homogeneous Geodesics in 3-dimensional Homogeneous Affine Manifolds

Zdeněk Dušek, Oldřich Kowalski, Zdeněk Vlášek (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

For studying homogeneous geodesics in Riemannian and pseudo-Riemannian geometry (on reductive homogeneous spaces) there is a simple algebraic formula which works, at least potentially, in every given case. In the affine differential geometry, there is not such a universal formula. In the previous work, we proposed a simple method of investigation of homogeneous geodesics in homogeneous affine manifolds in dimension 2. In the present paper, we use this method on certain classes of homogeneous connections...

Homogeneous systems of higher-order ordinary differential equations

Mike Crampin (2010)

Communications in Mathematics

The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation...

Horizontal lift of symmetric connections to the bundle of volume forms ν

Anna Gąsior (2010)

Annales UMCS, Mathematica

In this paper we present the horizontal lift of a symmetric affine connection with respect to another affine connection to the bundle of volume forms ν and give formulas for its curvature tensor, Ricci tensor and the scalar curvature. Next, we give some properties of the horizontally lifted vector fields and certain infinitesimal transformations. At the end, we consider some substructures of a F(3, 1)-structure on ν.

How many are affine connections with torsion

Zdeněk Dušek, Oldřich Kowalski (2014)

Archivum Mathematicum

The question how many real analytic affine connections exist locally on a smooth manifold M of dimension n is studied. The families of general affine connections with torsion and with skew-symmetric Ricci tensor, or symmetric Ricci tensor, respectively, are described in terms of the number of arbitrary functions of n variables.

Hypercomplex Algebras and Geometry of Spaces with Fundamental Formof an Arbitrary Order

Mikhail P. Burlakov, Igor M. Burlakov, Marek Jukl (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The article is devoted to a generalization of Clifford and Grassmann algebras for the case of vector spaces over the field of complex numbers. The geometric interpretation of such generalizations are presented. Multieuclidean geometry is considered as well as the importance of it in physics.

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