scholarly journals Generalized Affine Connections Associated with the Space of Centered Planes

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 782
Author(s):  
Olga Belova
Keyword(s):  
Projective Space ◽  
Affine Connection ◽  
Affine Connections ◽  
Curvature Tensors

Our purpose is to study a space Π of centered m-planes in n-projective space. Generalized fiberings (with semi-gluing) are investigated. Planar and normal affine connections associated with the space Π are set in the generalized fiberings. Fields of these affine connection objects define torsion and curvature tensors. The canonical cases of planar and normal generalized affine connections are considered.

About differential equations of the curvature tensors of a fundamental group and affine connections

2019 ◽  
pp. 133-140
Author(s):  
N. Ryazanov
Keyword(s):  
Differential Equations ◽  
Fundamental Group ◽  
Curvature Tensor ◽  
Principal Bundle ◽  
Group Structure ◽  
Affine Connection ◽  
Affine Connections ◽  
First Order ◽  
Structure Equations ◽  
Curvature Tensors

The principal bundle is considered, the base of which is an n-dimensional smooth manifold, and the typical fiber is an r-fold Lie group. Structure equations for the forms of the fundamental group and affine connections are given, each of which contains the corresponding components of the curvature tensor. For each connection, an approach is shown that allows to find the differential equations for the components of the curvature tensor of the corresponding connection in a faster way than by differentiating the expressions of these objects in terms of the connection objects and their Pfaffian derivatives. The method consists in successively solving cubic equations, first by Laptev’s lemma, then by Cartan’s lemma. Taking into account the comparisons modulo basic forms, we obtain already known results (see [3]). Thus, differential equations are derived for the components of the curvature tensor of the first-order fundamentalgroup connection, as well as for the components of the curvature tensor of the affine connection.

Fields of geometric objects associated with compiled hyperplane-distribution in affine space

2020 ◽  
pp. 103-115
Author(s):  
Yu. I. Popov
Keyword(s):  
Projective Space ◽  
Tangent Bundle ◽  
Affine Space ◽  
Affine Connection ◽  
Distribution Characteristic ◽  
Normal Bundles ◽  
Geometric Objects ◽  
Dimensional Projective Space ◽  
Curvature Tensors ◽  
Torsion Free

A compiled hyperplane distribution is considered in an n-dimensional projective space . We will briefly call it a -distribution. Note that the plane L(A) is the distribution characteristic obtained by displacement in the center belonging to the L-subbundle. The following results were obtained: a) The existence theorem is proved: -distribution exists with arbitrary (3n – 5) functions of n arguments. b) A focal manifold is constructed in the normal plane of the 1st kind of L-subbundle. It was obtained by shifting the cen­ter A along the curves belonging to the L-distribution. A focal manifold is also given, which is an analog of the Koenigs plane for the distribution pair (L, L). c) It is shown that a framed -distribution in the 1st kind normal field of H-distribution induces tangent and normal bundles. d) Six connection theorems induced by a framed -distri­bu­tion in these bundles are proved. In each of the bundles , the framed -distribution induces an intrin­sic torsion-free affine connection in the tangent bundle and a centro-affine connection in the corresponding normal bundle. e) In each of the bundles (d) in the differential neighborhood of the 2nd order, the covers of 2-forms of curvature and curvature tensors of the corresponding connections are constructed.

scholarly journals ON METRIZABILITY OF INVARIANT AFFINE CONNECTIONS

2012 ◽  
Vol 09 (01) ◽  
pp. 1250014 ◽  
Author(s):  
ERICO TANAKA ◽  
DEMETER KRUPKA
Keyword(s):  
Affine Connection ◽  
Rotation Group ◽  
Explicit Description ◽  
Invariant Metric ◽  
Euclidean Spaces ◽  
Affine Connections ◽  
Group Of Diffeomorphisms ◽  
Civita Connection ◽  
All Solutions ◽  
Levi Civita Connection

The metrizability problem for a symmetric affine connection on a manifold, invariant with respect to a group of diffeomorphisms G, is considered. We say that the connection is G-metrizable, if it is expressible as the Levi-Civita connection of a G-invariant metric field. In this paper we analyze the G-metrizability equations for the rotation group G = SO (3), acting canonically on three- and four-dimensional Euclidean spaces. We show that the property of the connection to be SO(3)-invariant allows us to find complete explicit description of all solutions of the SO(3)-metrizability equations.

scholarly journals Some relations in non-symmetric affine connection spaces with regard to a special almost geodesic mappings of the third type

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1941-1951 ◽  
Author(s):  
Mica Stankovic ◽  
Nenad Vesic
Keyword(s):  
Affine Connection ◽  
The Third ◽  
Curvature Tensors

We investigate two kinds of special almost geodesic mappings of the third type in this paper. We also find some relations for curvature tensors of almost geodesic mappings of the third type.

scholarly journals Almost geodesic mappings of type π1* of spaces with affine connection

2021 ◽  
Vol 52 ◽  
pp. 30-36
Author(s):  
Volodymyr Evgenyevich Berezovskii ◽  
Josef Mikeš ◽  
Željko Radulović
Keyword(s):  
Constant Curvature ◽  
Affine Connection ◽  
Affine Connections ◽  
Fundamental Equations ◽  
Special Case ◽  
Constant Curvature Spaces

We consider almost geodesic mappings π1* of spaces with affine connections. This mappings are a special case of first type almost geodesic mappings. We have found the objects which are invariants of the mappings π1*. The fundamental equations of these mappings are in Cauchy form. We study π1* mappings of constant curvature spaces.

scholarly journals On the projective geometry of paths

1937 ◽  
Vol 5 (2) ◽  
pp. 103-115 ◽  
Author(s):  
J. Haantjes
Keyword(s):  
Differential Equation ◽  
Projective Geometry ◽  
Affine Connection ◽  
Dimensional Manifold ◽  
Affine Connections ◽  
Linear Fractional Transformations ◽  
Image Position ◽  
Normal Parameter

An affine connection in an n-dimensional manifold Xn defines a system of paths, but conversely a connection is not defined uniquely by a system of paths. It was shown by H. Weyl that any two affine connections whose components are related by an equation of the formwhere is the unit affinor, give the same system of paths. In the geometry of a system of paths, a particular parameter on the paths, called the projective normal parameter, plays an important part. This parameter, which is invariant under a transformation of connection (1), was introduced by J. H. .C. Whitehead. It can be defined by means of a Schwarzian differential equation and it is determined up to linear fractional transformations. In § 1 this method is briefly discussed.

scholarly journals Geodesic Mappings of Spaces with Affine Connections onto Generalized Symmetric and Ricci-Symmetric Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1560
Author(s):  
Volodymyr Berezovski ◽  
Yevhen Cherevko ◽  
Irena Hinterleitner ◽  
Patrik Peška
Keyword(s):  
Symmetric Spaces ◽  
Affine Connection ◽  
Cauchy Type ◽  
Theory Of Relativity ◽  
Covariant Form ◽  
General Theory Of Relativity ◽  
Nontrivial Solutions ◽  
Affine Connections ◽  
Systems Of Pdes ◽  
Mixed Systems

In the paper, we consider geodesic mappings of spaces with an affine connections onto generalized symmetric and Ricci-symmetric spaces. In particular, we studied in detail geodesic mappings of spaces with an affine connections onto 2-, 3-, and m- (Ricci-) symmetric spaces. These spaces play an important role in the General Theory of Relativity. The main results we obtained were generalized to a case of geodesic mappings of spaces with an affine connection onto (Ricci-) symmetric spaces. The main equations of the mappings were obtained as closed mixed systems of PDEs of the Cauchy type in covariant form. For the systems, we have found the maximum number of essential parameters which the solutions depend on. Any m- (Ricci-) symmetric spaces (m≥1) are geodesically mapped onto many spaces with an affine connection. We can call these spaces projectivelly m- (Ricci-) symmetric spaces and for them there exist above-mentioned nontrivial solutions.

The metrizability problem for Lorentz-invariant affine connections

2016 ◽  
Vol 13 (08) ◽  
pp. 1650110
Author(s):  
Zbyněk Urban ◽  
Jana Volná
Keyword(s):  
Euclidean Space ◽  
Lie Groups ◽  
Affine Connection ◽  
Explicit Solutions ◽  
Invariant Metric ◽  
Affine Connections ◽  
Civita Connection ◽  
Levi Civita Connection

The invariant metrizability problem for affine connections on a manifold, formulated by Tanaka and Krupka for connected Lie groups actions, is considered in the particular cases of Lorentz and Poincaré (inhomogeneous Lorentz) groups. Conditions under which an affine connection on the open submanifold [Formula: see text] of the Euclidean space [Formula: see text] coincides with the Levi-Civita connection of some [Formula: see text], respectively [Formula: see text]-invariant metric field are studied. We give complete description of metrizable Lorentz-invariant connections. Explicit solutions (metric fields) of the invariant metrizability equations are found and their properties are discussed.

scholarly journals Some properties of ET-projective tensors obtained from Weyl projective tensor

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 573-584
Author(s):  
Mica Stankovic ◽  
Milan Zlatanovic ◽  
Nenad Vesic
Keyword(s):  
Curvature Tensor ◽  
Affine Connection ◽  
Connection Coefficients ◽  
Linearly Independent ◽  
Projective Curvature ◽  
Projective Tensor ◽  
Connection Space ◽  
Curvature Tensors

Vanishing of linearly independent curvature tensors of a non-symmetric affine connection space as functions of vanished curvature tensor of the associated space of this one are analyzed in the first part of this paper. Projective curvature tensors of a non-symmetric affine connection space are expressed as functions of the affine connection coefficients and Weyl projective tensor of the corresponding associated affine connection space in the second part of this paper.

scholarly journals Tensor Fields and Their Parallelism

1961 ◽  
Vol 18 ◽  
pp. 133-151 ◽  
Author(s):  
Minoru Kurita
Keyword(s):  
Tensor Field ◽  
Complex Structure ◽  
Affine Connection ◽  
Tensor Fields ◽  
Fundamental Tensor ◽  
Affine Connections ◽  
Complex Tensor ◽  
Group Manifold ◽  
Almost Complex ◽  
Riemannian Connection

Much has been studied about an almost complex structure these ten years. One of the problems about the structure is to find an affine connection which makes a given almost complex tensor field parallel. A Riemannian connection is a one without torsion for which the fundamental tensor field of a Riemannian manifold is parallel. Affine connections on the group manifold were investigated fully by E. Cartan in [1]. In this paper we treat in general some tensor fields and affine connections which make the fields parallel. Moreover some studies about certain tensor fields are given.

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