mathematics Article Generalized Affine Connections Associated with the Space of Centered Planes Olga Belova Institute of Physical and Mathematical Sciences and IT, Immanuel Kant Baltic Federal University, A. Nevsky Str. 14, 236016 Kaliningrad, Russia; [email protected]; Tel.: +7-921-610-5949 Abstract: Our purpose is to study a space P of centered m-planes in n-projective space. Generalized fiberings (with semi-gluing) are investigated. Planar and normal affine connections associated with the space P 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. Keywords: projective space; space of centered planes; planar generalized affine connection; normal generalized affine connection 1. Introduction The theory of connections occupies an important place in differential geometry. The concept of connection was introduced by T. Levi-Civita [1] as a parallel displacement of tangent vectors of a manifold. Weyl introduced the concept of a space of the affine connection [2]. This concept was continued in the works of Cartan [3] and Ehresmann [4]. Citation: Belova, O. Generalized Different ways to determine the connection were inputted by Nomizu [5], Vagner [6], and Affine Connections Associated with Laptev [7]. Veblen and Whitehead [8] introduced a connection in a composite manifold. the Space of Centered Planes. In [9], Laptev gave an invariant definition of connections as a certain law defining the Mathematics 2021, 9, 782. https:// mapping of infinitely close fibres. He also provided a well-known theoretical-group method doi.org/10.3390/math9070782 on the basis of the Cartan calculation. Rashevskii [10], Nomizu [11], and Norden [12] studied affine connections. Generalized Academic Editor: Marian Ioan affine connections were considered in book [13], where a relation was shown between a Munteanu generalized affine connection and a linear connection (see also [14,15]). In [16], all invariant affine connections on three-dimensional symmetric homogeneous spaces admitting a nor- Received: 12 March 2021 mal connection were described; canonical connections and natural torsion-free connections Accepted: 3 April 2021 were also considered. Published: 5 April 2021 The theory of affine connections is widely used in physics [17–21] and by studying geodesics [22] (see, e.g., [23–28]). Publisher’s Note: MDPI stays neutral This paper is the result of the author’s research [29,30], where the concepts of gen- with regard to jurisdictional claims in eralized bundles are used (see, e.g., [31]). Principal fibering with gluing to the base was published maps and institutional affil- introduced in the paper [32] and generalized principal fibering (a principal fibering with iations. semi-gluing to the base) was introduced in the paper [33]. Connections in the fiberings associated with the space of centered planes were studied in [34,35]. In the present paper, generalized affine connections associated with this space are considered. The notion of vector, or specifically, principal bundles over a smooth manifold is one Copyright: © 2021 by the authors. of the central notions in modern mathematics and its applications to mathematical and Licensee MDPI, Basel, Switzerland. theoretical physics. In particular, all known types of physical interactions (gravitational, This article is an open access article electromagnetic, etc.) are described in terms of connections and other geometric structures distributed under the terms and on vector/principal bundles on underlying manifolds. The properties of physical fields conditions of the Creative Commons can be formulated in terms of geometric invariants of connections such as curvature and Attribution (CC BY) license (https:// characteristic classes of corresponding vector/principal bundles [36]. creativecommons.org/licenses/by/ 4.0/). Mathematics 2021, 9, 782. https://doi.org/10.3390/math9070782 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 782 2 of 9 There are affine connections which arise in different contexts. These connections are used in geometry for illustration of parallel transports, and they can be applied to mechanics. Therefore, I hope this article will be of interest to geometers and physicists. The important role of the affine connections and the previous author’s work in the study of manifolds of planes was the motivation for writing this paper. This paper will be arranged in the following way. Section2 will describe the analytical apparatus and the space of centered planes as the object of research. In Sections3 and4, a review of planar and normal generalized affine connections will be given, respectively. A brief summary of the paper is given in Section5. 2. Analytical Apparatus and Object of Research It is a well-known fact that projective space Pn can be represented as a quotient space Ln+1/ ∼ of a linear space Ln+1 with respect to equivalence (collinearity) ∼ of non-zero vectors, that is, Pn = (Ln+1 n f0g)/ ∼. Projective frame in the space Pn is a system formed 0 by points AI0 , I = 0, ..., n, and a unit point E (see [37]). In linear space Ln+1, linearly n independent vectors eI0 correspond to the points AI0 , and a vector e = ∑ eI0 corresponds I0=0 to the point E. Moreover, these vectors are determined in the space Ln+1 with accuracy up to a common factor. The unit point is specified together with the basic points, although you do not have to mention it every time. In the present paper, we will use the method of a moving frame (see, e.g., [38]) fA, AI g, I, ... = 1, ..., n, the derivation formulae of the vertices of which are I J dA = qA + w AI, dAI = qAI + wI AJ + wI A, I I where the form q acts as a proportionality factor, and the structure forms w , wJ , wI of the projective group GP(n), effectively acting on the space Pn, and satisfying the Cartan equations (see [39], cf. [40]) I J I I K I I K I J Dw = w ^ wJ , DwJ = wJ ^ wK + dJ wK ^ w + wJ ^ w , DwI = wI ^ wJ. Throughout this paper, d is the symbol of ordinary differentiation in the space Pn, and D is the symbol of exterior differentiation. This apparatus is successfully used by geometers in Kaliningrad (see, e.g., [41]). In the projective space Pn, a space P (see [34,42,43]) of all centered m-dimensional planes is considered (cf. [44–47]). Vertices A and Aa, a, ... = 1, ..., m, of the moving frame a a a are placed on the centered plane, with vertex A fixed as a centre. The forms w , w , wa (a, ... = m + 1, ..., n) are the basic forms of the space P. Remark 1. The space P is a differentiable manifold whose points are m-dimensional centered planes. The technique used in the present paper was based on the Laptev–Lumiste method. This, in turn, requires knowledge of calculating external differential forms. 3. Planar Generalized Affine Connection Now we introduce the following definition. Definition 1. A smooth manifold with structure equations a b a a a a b a a a Dw = w ^ wb + w ^ wa, Dw = w ^ wb + w ^ wa , a b b a a b a (1) Dwa = wb ^ (da wb − dbwa ) − w ^ wa, a c a c a a a a a a Dwb = wb ^ wc − w ^ (dc wb + db wc) − db w ^ wa + wb ^ wa is called a generalized bundle of planar affine frames [48] and is denoted by Am2+[m](P). Mathematics 2021, 9, 782 3 of 9 Remark 2. The symbol m is enclosed in square brackets, since m forms wa are included in both basic and fibre forms. We will call the forms wa basic-fibre. To define an affine connection in the generalized bundle Am2+[m](P), we extend to it the Laptev–Lumiste method of defining group connections in principal bundles. We a a transform the basic-fibre forms w and fibre forms wb of the fibering Am2+[m](P), using a a a linear combinations of the basic forms w , w , wa [32] as follows: a a a b a a ab a a a a c a a ac a w˜ = w − Cb w − Caw − Ca wb , w˜ b = wb − Gbcw − Gbaw − Gbawc . (2) Finding the exterior differentials of forms (2) with the help of structure equations (1) and applying the Cartan–Laptev theorem in this generalized case, we obtain a = a c + a a + a,c a DCb Cb,cw Cb,aw Cb,awc , a a b ab a a b a b a,b b DCa − Cb wa + Ca wb + wa = Ca,bw + Ca,bw + Ca,bwb , ab ab c ab b ab,c b DCa = Ca,cw + Ca bw + C wc , , a,b (3) a − a − a = a d + a a + a,d a DGbc db wc dc wb Gbc,dw Gbc,aw Gbc,awd , a − a c + ac − a = a c + a b + a,c b DGba Gbcwa Gbawc db wa Gba,cw Gba,bw Gba,bwc , ac + c a = ac d + ac b + ac,d b DGba dbwa Gba,dw Gba,bw Gba,bwd , a where Cb,c, ... are Pfaffian derivatives [32], and D is a tensor differential operator acting a a e a a e according to the law DCb = dCb + Cbwe − Ce wb (see, e.g., [49]). We will use the following terminology (see [50]): A substructure of a structure S is called simple if it is not a union of two substructures of the structure S. A simple substructure is called the simplest if it, in turn, does not have a substructure. P a a ab a a Statement 1. The object of a planar generalized affine connection G= fCb, Ca, Ca , Gbc, Gba, ac a ab Gbag associated with the space P of centered planes contains two simplest tensors Cb, Ca in a a ab a simple quasi-tensor of a connection C = fCb, Ca, Ca g and two simplest subquasi-tensors a ac a a ac Gbc, Gba of a simple quasi-tensor of a planar linear connection fGbc, Gba, Gbag.