Parallel displacements of directions on the Grassmann-like manifold of centered planes Olga Belova Abstract. The Grassmann-like manifold Gr (m, n) of centered m-planes L m (it has the same dimension as the space of m-planes) is considered in projective space. The analog of the strong Norden normalization of the Grassmann-like manifold is realized and induced connections are considered. Parallel displacements of directions on this manifold are studied. M.S.C. 2: 53A2, 53B15, 53B25. Key words: projective space; Grassmann-like manifold; connection; covariant differential; parallel displacement. 1 Introduction Extensive theory of Grassmann submanifolds has been developed by school of Baltic geometers, basically, works of Yu.G. Lumiste [1], V.I. Bliznikas [2], I.V. Bliznikene [3], etc., but here essentially new results are obtained and approaches to researches differ from earlier used. The object of research of the present paper are connections in the fiberings associated with Grassmann-like manifold of centered planes. The work concerns to researches in the area of differential geometry. The research is based on an application of the G.F. Laptev s method [4] of defining a connection in a principal fiber bundle and his method of continuations and scopes, which generalizes the moving frame method and Cartan exterior forms method. The research depends on calculation of exterior differential forms. Grassmann manifold and the space of centered planes in the projective space were already studied by autor in this direction [5]. This paper is devoted to studies of parallel displacements of directions in connections along lines on Grassmann-like manifold of centered planes. BSG Proceedings 21. The International Conference Differential Geometry - Dynamical Systems DGDS-23, October 1-13, 23, Bucharest-Romania, pp. 9-13. c Balkan Society of Geometers, Geometry Balkan Press 24.
1 Olga Belova 2 Grassmann-like manifold and parallel displacements In n-dimensional projective space P n we consider the moving frame {A, A I } (I, = 1, n) with derivation formulas da = θa + ω I A I, da I = θa I + ω J I A J + ω I A, where the form θ plays the role of a proportionality factor and the structure forms ω I, ω I J, ω I of the projective group GP (n), which acts effectively on P n, satisfy the Cartan equations Dω I = ω J ω I J, Dω I J = ω K J ω I K + δ I Jω K ω K + ω J ω I, Dω I = ω J I ω J. The Grassmann-like manifold Gr (m, n) of the centered m-planes L m is considered in P n. Let s produce a specialization of the moving frame {A, A a, A α } (a, = 1, m; α, = m + 1, n) putting the tops A and A a on the plane L m = [A, A a ] and fixing the center A. The Grassmann-like manifold Gr (m, n) [6] is given by the equations where Λ a α, Λ ab α ω a = Λ a αω α + Λ ab α ω α b, are functions; ω α, ω α a are basic forms of this manifold, dimgr (m, n) = (n m)(m + 1). The components of the fundamental object Λ = {Λ a α, Λ ab α } satisfy the differential comparisons modulo the basic forms ω α, ω α a : Λ a α + Λ ab α ω b + ω a α, Λ ab α. The principal fiber bundle G (Gr (m, n)) is constructed over the manifold Gr (m, n) and the stationarity subgroup G of the centered plane L m is the typical fiber. In the principal fiber bundle the fundamental-group connection is given in G. F. Laptev s method: ω a b = ω a b Γ a bαω α L ac bαω α c, ω α β = ω α β Γ α βγω γ L αa βγω γ a, ω a α = ω a α Γ a αβω β L ab αβω β b, ω a = ω a L aα ω α Π b aαω α b, ω α = ω α L αβ ω β Π a αβω β a. The connection in the associative fibering G (Gr (m, n)) is defined by the field of the connection object Γ = {Γ a bα, Lac bα, Γα βγ, Lαa βγ, L aα, Π b aα, Γ a αβ, Lab αβ, L αβ, Π a αβ } on the base Gr (m, n): Γ a bα + L ac bαω c ωbα a, L ac bα ωbα ac, Γ α βγ + L αa βγω a ω α βγ, Γ a αβ + L ab αβω b Γ a bβω b α + Γ γ αβ ωa γ ω a αβ, L aα + (Π b aα + Γ b aα)ω b, L αa βγ ω αa βγ, L ab αβ + L γb αβ ωa γ L ab cβω c α ω ab αβ, Π b aα + L cb aαω c + δ b aω α
Parallel displacements of directions on the Grassmann-like manifold 11 L αβ +(Π a αβ +Γ a αβ)ω a L aβ ω a α +Γ γ αβ ω γ, where Π a αβ Π a bβω b α +L ba αβω b +L γa αβ ω γ, ω a bα = Λ a αω b + δ a b Λ c αω c + δ a b ω α, ω α βγ = δ α β Λ a γω a + δ α β ω γ + δ α γ ω β, ω a αβ = Λ a βω α, ω ac bα = Λ ac α ω b + δ a b Λ ec α ω e δ c bω a α, ω αa βγ = δ α γ ω a β + δ α β Λ ba γ ω b. ω ab αβ = Λ ab β ω α. An analog of the strong Norden normalization [7] for this manifold is carried out. It consists of the fields of the planes C n m 1 and N m 1 : L m C n m 1 =, A / N m 1 L m. The planes C n m 1 and N m 1 we shall set by the points B α = A α + λ a αa a + λ α A and B a = A a + λ a A, respectively. The differential equations for the components of the clothing quasitensor λ = {λ a α, λ α, λ a } are of the form [8]: λ a + ω a = λ aα ω α + λ b aαω α b, We have (see [6]) λ a α + ω a α = λ a αβω β + λ ab αβω β b, λ α + λ a αω a + ω α = λ αβ ω β + χ a αβω β a. Γ a bα = δ a b λ α + µ a αλ b + δ a b µ c αλ c, L ac bα = δ c bλ a α (δ a b Λ ec α + δ e bλ ac α )λ e, Γ α βγ = δ α γ λ β δ α β λ γ + δ α β µ a γλ a, Γ a αβ = Λ a βλ α µ a βλ b λ b α, L aα = µ b αλ a λ b, L αβ = λ α λ β + λ a λ α µ a β λ a λ b λ b αµ a β, L αa βγ = δ α γ λ a β δ α β Λ ba γ λ b, L ab αβ = Λ ab β λ α + Λ ab β λ c αλ c λ a βλ b α, Π b aα = δ b aλ α Λ cb α λ a λ c, Π a αβ = Λ ba β λ b λ α λ β λ a α + Λ ba β λ c αλ b λ c. Theorem 2.1. The analog of the strong Norden normalization of the Grassmann-like manifold Gr (m, n) induces a connection in the associated fibering. We ll consider the straight line in the plane L m and passing through a point A. It crosses an analogue of a normal of the 2nd type N m 1 in the point B = µ a B a = µ a (A a + λ a A). We have db = [θ + ( λ α + µ b αλ b )ω α Λ cb α λ c ω α b ]B+ + µ a B a + µ a [λ a ω α + ω α a ]B α + +µ a [(λ aα + λ a λ b µ b α λ a λ α )ω α + (λ b aα λ a λ c Λ cb α δ b aµ α )ω α b ]A, where µ a α = λ a α Λ a α, µ α = λ α λ a λ a α and µ a = dµ a + µ b ω a b. If the covariant differential of the geometrical object µ vanishes we get parallel displacement and we have
12 Olga Belova Theorem 2.2. The straight line AB L m determined by a point B N m 1 (A / N m 1 L m) is parallel displaced in the plane linear connection Γ 1 = {Γ a bα, Lac bα } iff the point B is displaced in a plane P n m+1 = N n m + B, where N n m is an analogue of a normal of the 1st type (N n m = A + C n m 1 ). We consider a normal straight line AC intersecting an analog of the Cartan plane C n m 1 N n m in the point C = µ α B α = µ α (A α + λ a αa a + λ α A). We find the differential of the point C dc = [θ (Λ a βλ a + µ β )ω β Λ ba β λ b ω β a ]C + µ α B α + +µ α [(λ a αβ λ α µ a β)ω β + (λ ab αβ + λ α Λ ab β λ a βλ b α)ω β b ]B a+ +µ α [(λ αβ λ a λ a αβ + λ a λ α µ a β λ α λ β )ω β + (χαβ a λ b λ ba αβ λ b λ α Λ ba β λ a αµ β )ωa β ]A, where µ α = dµ α + µ β ω α β. Theorem 2.3. The straight line AC N n m determined by a point C C n m 1 (C n m 1 is an analogue of the Cartan plane) is parallel displaced in the normal linear connection Γ 2 = {Γ α βγ, Lαa βγ } iff the point C is displaced in a plane P m+1 = L m + C. We consider general parallel displacements. We consider the point M L n 1 = N m 1 + C n m 1. M = η a B a + η α B α. We find the differential of the point M dm = θm + [ η a + ( η a λ α + η a µ b αλ b + η β λ a βα η β λ β µ a α)ω α + +(η c λ c Λ ab α + η β λ ab βα + η β λ β Λ ab α η β λ a αλ b β η a Λ cb α λ c η c Λ ab α λ c )ωb α ]B a + +[ η α + (δ α β η a λ a η α λ β + η α µ a βλ a )ω β + (δ α β η a η α Λ ba β λ b )ω β a ]B α + +[(η a λ aα + η a λ a λ b µ b α η a λ a λ α η β λ a λ a βα + η β λ βα η β λ a λ β Λ a α η β λ β µ α )ω α + +(η a λ b aα η a λ a λ c Λ cb α η b µ α η β λ a λ ab βα + η β χ b βα η β λ a λ β Λ ab α η β λ b βµ α )ωb α ]A, where η a = dη a + η b ω a b, ηα = dη α + η β ω α β. Theorem 2.4. The straight line AM is parallely displaced in the compound connection Γ 1 Γ 2 by any displacement of the point M, i.e. parallel displacement in this connection is degenerated. References [1] Yu. G. Lumiste, Induced connection in the embeded projective and affine fiberings (in Russian), Acad. Proc. of Tartu University, Tartu, 177 (1965) 6 42. [2] V. I. Bliznikas, Some questions of geometry of hypercomplexes of lines (in Russian), The Proc. of the Geom. Seminar, Moscow, 6 (1974) 43 111. [3] I. V. Bliznikene, About geometry of hemiholonomic congruence of the first kind, The Proc. of the Geom. Seminar, Moscow, 3 (1971) 125 148.
Parallel displacements of directions on the Grassmann-like manifold 13 [4] G. F. Laptev, Differential geometry of the embeded manifolds (in Russian), The Proc. of Moscow Math. Society, Moscow, 2 (1953) 275 383. [5] O. O. Belova, Connections in fiberings associated with the Grassman manifold and the space of centered planes, J. Math. Sci, Springer, 162, 5 (29), 65 632. [6] O. O. Belova, The connection in the fibering associated with Grassmann-like manifold of centered planes (in Russian), Vestnik CHGPU Cheboksary 68 (26), 18 2. [7] A. P. Norden, Spaces with an Affine Connection (in Russian), Nauka, Moscow 1976. [8] O. O. Belova, Connections of the 2nd type in the fibering associated with Grassmann-like manifold of centered planes (in Russian), Diff. Geom. of Figure Manifolds, Kaliningrad. 38 (27), 6 12. Author s address: Olga Belova Institute of Applied Mathematics and Information Technologies, Immanuel Kant Baltic Federal University, A. Nevsky Street, 23641, Kaliningrad, Russia. E-mail: olgaobelova@mail.ru