749–754 DOI: 10.18514/MMN.2018.2575 AN ANALOG OF NEIFELD’S CONNECTION INDUCED ON THE SPACE OF CENTRED PLANES OLGA BELOVA Received 26 March, 2018 Abstract

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 2, pp. 749–754 DOI: 10.18514/MMN.2018.2575

AN ANALOG OF NEIFELD’S CONNECTION INDUCED ON THE SPACE OF CENTRED PLANES

OLGA BELOVA Received 26 March, 2018

Abstract. The paper concerns to investigations in the field of differential geometry. It is realized by a method of prolongations and scopes of G.F. Laptev which generalizes a moving frame method and Cartan’s exterior forms method and bases on calculation of exterior differential forms. The space˘ of centredm–planes is considered in projective spacePn. Principal fiber bundle is arised above˘. An analog of Neifeld’s connection is given in this principal fibering.

It is proved that the analog of the Norden normalization of the space of centred planes induces Neifeld’s connection. The torsion of Neifeld’s connection analog is introduced. It is shown, that this object is a tensor.

2010Mathematics Subject Classification: 53A20; 53B25; 53B15

Keywords: projective space, space of centred planes, principal fiber bundle, Neifeld’s connection

1. INTRODUCTION

The object of research of this paper is an analog of Neifeld’s connection on the space of centred planes.

We consider the space of centred planes as space of all m–dimensional centred planes in n–dimensional projective space. The non-classical analytical method is applied in this paper. It has advantage at allocation of subgroups and factor groups of a projective group. The connections in the fiberings associated with Grassmann manifold and the space of centred planes were investigated in [1] by such method. We show that an analog of Neifeld’s connection is induced on the space of centred planes.

By methods of tensor analysis E.G. Neifeld considered two dual linear connections (compare [3]), associated with normalized Grassmann manifold.

We put projective spacePnto the moving framefA; AIg.I; J; K; :::D1; n/with the derivation formulae

dADAC!^IAI; dAI DAIC!_I^JAJC!IA: (1.1) The Pfaffian forms!^I,!_J^I,!I satisfy the Cartan structure equations of the projective groupGP .n/(see, eg., [5]):

D!^I D!^J^!_J^I; D!I D!_I^J^!J;

c 2018 Miskolc University Press

D!_J^I D!_J^K^!_K^I Cı_J^I!K^!^KC!J^!^I: (1.2) 2. THE SPACE OF CENTRED PLANES

The space˘ [1] of all centred m–dimensional planes L_m is considered in Pn. Let’s produce a specialization of the moving framefA; A_a; A_˛g.a; D1; mI ˛; D mC1; n/putting the topAin the centre ofm-dimensional planeLmand putting the topsAa on the planeLm.

From the derivation formulae (1.1) it follows that the equations!^˛D0, !^aD0,

!_a^˛D0are stationarity conditions of the centred planes L_m, i.e. forms!^˛, !^a,

!_a^˛are principal forms for the space˘. The basic forms of the space˘ satisfy the equations following from the structure Cartan equations (1.2)

D!^˛D!^a^˝_a^˛C!^ˇ^˝_ˇ^˛;

D!^aD!^b^˝_b^aC!^˛^˝_˛^a; (2.1) D!_a^˛D!_b^ˇ^˝_{ˇ a}^˛bC!^˛^˝a;

where

˝_ˇ^˛D!_ˇ^˛; ˝_b^aD!_b^a; ˝_˛^aD!_˛^a; ˝aD !a; ˝_{ˇ a}^˛bDı_a^b˝_ˇ^˛ ı_ˇ^˛˝_a^b: (2.2) Remark1. The forms˝_a^˛D!_a^˛are the basic-fibre forms.

Differentiating the forms (2.2) we obtain

D˝_b^aD˝_b^c^˝_c^aC!^˛^˝_b˛^a C!^c^˝_bc^a C!_b^˛^˝_˛^a; (2.3) D˝_ˇ^˛D˝_ˇ^˝^˛C!^˝_ˇ ^˛ C!^a^˝_{ˇ a}^˛ !_a^˛^˝_ˇ^a; (2.4) D˝_˛^aD˝_b˛^{ˇ a}^˝_ˇ^bC!^a^˝˛; (2.5) D˝aD˝_a^b^˝_bC!_a^˛^˝˛; (2.6) where

˝_ˇ ^˛ D ı_ˇ^˛! ı^˛!_ˇ; ˝_{ˇ a}^˛ D ı_ˇ^˛!a;

˝_bc^a D ı_b^a!_c ı^a_c!_b; ˝_b˛^a D ı^a_b!_˛; ˝_˛D !_˛:

The principal fiber bundleL.˘ /is constructed over the space˘ of centred planes and the Lie groupLis the typical fiber. This group acts in the tangent space [2] to the space˘.

Theorem 1. The principal fiber bundleL.˘ /contains the following factorfiber- ings:

(1) factorfibering of linear frames belonging to the centred planeL_mwhich typ- ical fiber is linear factorgroup acting in a bunch of lines belonging to a plane L_m, with the structure equations (2.1) and (2.3);

(2) factorfibering of normal linear frames with the structure equations (2.1) and (2.4);

(3) factorfibering of coaffine frames belonging to the plane L_m which typical fiber is coaffine factorgroup acting in the centred plane with the structure equations (2.1), (2.3) and (2.6);

(4) maximal affine factorfibering which typical fiber is factorgroup acting in a bunch of lines with centreA; the factorfibering has structure equations (2.1) and (2.3–2.5).

3. AN ANALOG OFNEIFELD’S CONNECTION

In the principal fiber bundle we set an analog of Neifeld’s connection [3] by the Laptev — Lumiste way.

Entering new forms

˝Q_b^aD˝_b^a _b˛^a !^˛ _bc^a!^c _b˛^ac!_c^˛; ˝Q_ˇ^˛D˝_ˇ^˛ _ˇ ^˛ ! _{ˇ a}^˛!^a _ˇ ^˛a!_a;

˝Q_˛^aD˝_˛^a _˛ˇ^a !^ˇ _˛b^a !^b _˛ˇ^ab!_b^ˇ; ˝QaD˝a a˛!^˛ ab!^b L^b_a˛!^˛_b and finding their exterior differentials we get that connection in the principal fiber bundleL.˘ /is set with the help of a field of connection object

D f b˛^a; _bc^a; _b˛^ac; _ˇ ^˛ ; _{ˇ a}^˛ ; _ˇ ^˛a; _˛ˇ^a ; _˛b^a ; _˛ˇ^ab; a˛; _ab; L^b_a˛g

on the base˘. We found differential equations of the connection object components:

_b˛^a _bc^a˝_˛^c _b˛^ac˝cCı_b^a˝˛D _b˛^a jˇ !^ˇC _b˛^a jc!^cC _b˛^a j^c_ˇ !_c^ˇ; _bc^a Cı_b^a˝cCı_c^a˝_bD bc^a j^˛!^˛C bc^a j^e!^eC bc^a j^e˛!_e^˛;

_b˛^acCı_b^c˝_˛^aD b˛^acjˇ !^ˇC b˛^acj^e!^eC b˛^acj^eˇ !_e^ˇ;

_ˇ ^˛ _{ˇ a}^˛˝^a _ˇ ^˛a˝aC˝_ˇ ^˛ D ˇ ^˛ j!C ˇ ^˛ j^a!^aC ˇ ^˛ j^a!_a; _{ˇ a}^˛ Cı_ˇ^˛˝aD ˇ a^˛ j!C ˇ a^˛ jb!^bC ˇ a^˛ j^b!_b;

_ˇ ^˛a ı^˛˝_ˇ^aD ˇ ^˛aj!C ˇ ^˛ajb!^bC ˇ ^˛aj^b!_b; (3.1) _˛ˇ^a _˛b^a˝_ˇ^b _˛ˇ^ab˝_{b bˇ}^a ˝_˛^bC _˛ˇ ˝^aD ˛ˇ^a j!C ˛ˇ^a jb!^bC ˛ˇ^a j^b!_b;

_˛b^a _cb^a˝_˛^cC _˛b^ˇ ˝_ˇ^aCı_b^a˝˛D _˛b^a jˇ !^ˇC _˛b^a jc!^cC _˛b^a j^c_ˇ !_c^ˇ; _˛ˇ^abC _˛ˇ^b˝^a _cˇ^ab˝_˛^cD ˛ˇ^ab j !C ˛ˇ^abj^c!^cC ˛ˇ^ab j^c !_c; a˛ ab˝_˛^bC. _a˛^b L^b_a˛/˝_bD ^a˛jˇ !^ˇC ^a˛jb!^bC ^a˛j^bˇ !_b^ˇ;

abC _ab^c ˝cD ab j˛!^˛C abjc !^cC abj^c˛!_c^˛; L^b_a˛C a˛^cb˝cCı_a^b˝˛DL^b_a˛jˇ !^ˇCL^b_a˛jc!^cCL^b_a˛j^c_ˇ !_c^ˇ: There are Pfaffian derivatives in the right side of the equations (3.1).

Theorem 2. The object of group connection contains four simple geometric subobjects 1D f _b˛^ac; _bc^a; _b˛^ag, 2D f _ˇ ^˛a; _{ˇ a}^˛ ; _ˇ ^˛ g, 3D f ¹; a˛; _ab;Ł^b_a˛g,

4D f ¹; 2; _˛b^a ; _˛ˇ^ab, _˛ˇ^a ggiving the connection in the corresponding factorfiber- ings.

We realize an analogue of the strong Norden normalization [4] of the manifold by fields of the following geometrical images:.n m 1/-dimensional planePn m 1, not having the common points with the planeL_m, and.m 1/-dimensional plane Pm 1, belonging to the planeL_mand not passing through its centreA.

We define the planesPn m 1andPm 1by the points, respectively B˛DA˛C^a_˛AaC˛A; BaDAaCaA:

Demanding a relative invariancy of the clothing planes we obtain the differential equations for the components of clothing geometrical object

^a_˛C˝_˛^a0; ˛ ^a_˛˝a !˛0; a ˝a0: (3.2) The analog of the strong Norden normalization [4] of the spaces˘ allows to cover the components of the connection object

ac

b˛ Dı_b^ca_˛; _bc^a D ı_b^ac ı_c^a_b; _b˛^a D ı_b^a˛C^a_˛b;

˛a

ˇ D ı^˛a_ˇ; _{ˇ a}^˛ D ı^˛_ˇa; _ˇ ^˛ D ı^˛_ˇ ı^˛_ˇ;

ab

˛ˇ D ^b_˛^a_ˇ; _˛b^a D ı_b^a˛; _˛ˇ^a D _b^b_˛^a_ˇ; (3.3) L^b_a˛D ı_a^b˛; _abDa_b; a˛D a_b^b_˛;

where˛D˛ ^a_˛a.

The functions (3.3) by virtue of the comparisons (3.2) satisfy the differential equa- tions (3.1) for the components of the connection object .

Thus, we have

Theorem 3. The analog of the strong Norden normalization of the space˘ in- duces analog of Neifeld’s connection in the associated fiberingL.˘ /.

4. ATORSION OFNEIFELD’S CONNECTION ANALOG ON THE SPACE OF CENTRED PLANES

Putting into the structure equations (2.1) of the basic forms!^˛,!^aand!_a^˛of the space˘ the connection forms˝Q_b^a,˝Q_ˇ^˛,˝Q_˛^a,˝Qa we obtain the equations:

D!^˛D!^ˇ^ Q˝_ˇ^˛C!^a^!_a^˛CS_ˇ ^˛ !^ˇ^!CS_{ˇ a}^˛ !^ˇ^!^aCS_ˇ ^˛a!^ˇ^!_a; (4.1) D!_a^˛D!_b^ˇ^.ı_a^b˝Q_ˇ^˛ ı_ˇ^˛˝Q_a^b/C!^˛^ Q˝aCS_aˇ ^˛ !^ˇ^!CS_{aˇ b}^˛ !^ˇ^!^bC

S_aˇ ^˛b !^ˇ^!_bCS_{aˇ c}^˛b !^c^!_b^ˇCS_aˇ ^˛bc!_b^ˇ^!_c;

D!^aD!^b^ Q˝_b^aC!^˛^ Q˝_˛^aCS_˛ˇ^a !^˛^!^ˇCS_˛b^a !^˛^!^bCS_˛ˇ^ab!^˛^!_b^ˇC

S_bc^a !^b^!^cCS_b˛^ac!^b^!_c^˛; where

S_ˇ ^˛ D Œˇ ^˛ ; S_{ˇ a}^˛ D ˇ a^˛; S_ˇ ^˛aD ˇ ^˛a;

S_aˇ ^˛ Dı^˛_Œˇ a ; S_{aˇ b}^˛ Dı_ˇ^˛ ab; S_aˇ ^˛b Dı_ˇ^˛L^b_aCı^˛ _aˇ^b ı^b_{a ˇ}^˛; S_{aˇ c}^˛b Dı_ˇ^˛ _ac^b ı_a^b _{ˇ c}^˛; S_{aˇ c}^˛bcDı^d_a^b _b^˛c_ˇ  ı_b^˛_{ˇ a}^dbc;

S_˛ˇ^a D Œ˛ˇ ^a ; S_˛b^a D ˛b^a a

b˛; S_˛ˇ^abD ˛ˇ^ab; S_bc^a D Œbc^a ; S_b˛^acD b˛^ac: Here the square brackets mean an alternation with the last indices and pairs of indices.

We represent the underlined term in formula (4.1) as

!^a^!_a^˛Dı_ˇ^˛ı^a_b!^b^!_a^ˇ: Then the equation (4.1) takes the form

D!^˛D!^ˇ^ Q˝_ˇ^˛CS_{ˇ b}^˛a!^b^!_a^ˇCS_ˇ ^˛ !^ˇ^!CS_{ˇ a}^˛ !^ˇ^!^aCS_ˇ ^˛a!^ˇ^!_a; where we use the new componentsS_{ˇ b}^˛aDı_ˇ^˛ı_b^aof the torsionS.

The components of the torsion objectSsatisfy the differential comparisons mod- ulo the basic forms!^˛,!^a,!_a^˛:

S_ˇ ^˛ S_{Œˇ a}^˛ ! _^a CS_{Œˇ }^˛a !a0; S_{ˇ a}^˛ S_{ˇ a}^˛b!_b0; S_ˇ ^˛a S_b^˛a!_ˇ^b0;

S_{ˇ b}^˛a0; S_aˇ ^˛ S_{aŒˇ b}^˛ ! _^b CS_{aŒˇ }^˛b !_b S_ˇ ^˛ !a0; S_{aˇ b}^˛ S_{aˇ b}^˛c !c S_{ˇ b}^˛ !a0;

S_aˇ ^˛b S_ac^˛b !_ˇ^cC2S_aˇ ^˛cb!c S_ˇ ^˛b!a0; S_{aˇ c}^˛b S_{ˇ c}^˛b!_b0; S_aˇ ^˛bc0;

S_˛ˇ^a S_Œ˛b^a !_{ˇ }^b CS_{Œ˛ˇ }^ab !bCS_˛ˇ !^a0; S_˛b^a C2S_bc^a !_˛^cCS_˛b^ˇ !_ˇ^a S_b˛^ac!c0;

S_˛ˇ^ab S_cˇ^ab!_˛^cCS_˛ˇ^b!^a0; S_bc^a 0; S_b˛^acCS_˛b^{ˇ c}!_ˇ^a0:

From these comparisons we obtain the following theorem.

Theorem 4. The torsion object S D fS_{ˇ b}^˛a Dı_ˇ^˛ı^a_b; S_ˇ ^˛ ; S_{ˇ a}^˛ ; S_ˇ ^˛a; S_aˇ ^˛ ; S_{aˇ b}^˛ , S_aˇ ^˛b ; S_{aˇ c}^˛b ; S_aˇ ^˛bc; S_˛ˇ^a ; S_˛b^a ; S_˛ˇ^ab; S_bc^a ; S_b˛^acgof the connection is a tensor. It con- tains three elementary subtensors and four simple subtensors.

Remark2. Since the torsion subobjectS_{ˇ b}^˛aDı^˛_ˇı_b^a is a nonzero tensor, the con- nection is always a connection with torsion.

REFERENCES

[1] O. Belova, “Connections in fiberings associated with the grassmann manifold and the space of centered planes,”J. Math. Sci. (New York), vol. 162, no. 5, pp. 605–632, 2009, doi:10.1007/s10958- 009-9649-y.

[2] J. Mikeˇ s and et al.,Differential geometry of special mappings. Palack´y University Olomouc, Faculty of Science, Olomouc, 2015.

[3] E. G. Neifeld, “Affine connections on the normalized manifolds of planes in the projective space,”

News of High schools. Math., vol. 11, pp. 48–55, 1976.

[4] A. P. Norden,Spaces with an affine connection. Moscow: Nauka, 1976.

[5] Y. I. Shevchenko,Equipments of centreprojective manifolds, Kaliningrad, 2000.

Author’s address

Olga Belova

Immanuel Kant Baltic Federal University, Institute of Physics, Mathematics and IT, 14 A. Nevskogo St., 236041 Kaliningrad, Russia

E-mail address:olgaobelova@mail.ru