Grasmmann manifolds

Grasmmann manifolds

O. O. Belova

Abstract. The centred Grassman manifold (a family of planes, passing through one point) is considered in the projective space. The principal fiber bundle which arises has as typical fiber the stationarity subgroup of a centred plane. A fundamental–group connection is given in this fiber- ing. The curvature object of the connection is introduced. It is shown, that this object is a tensor containing certain subtensors. It is proved, that Bortolotti’s equipment [3] of the centred Grassman manifold induces connections of 3–types in associated fibering. The conditions of their co- incidence and their interdependence are obtained.

M.S.C. 2000: 53A20, 51M35, 53C05.

Key words: projective space, centred Grassman manifold, principal fiber bundle, Bortolotti’s equipment, fundamental group connection, curvature.

Let Pn be the n–dimensional real projective space referred to the mobile frame {A, AI} (I, J, K, . . . = 1, n), with infinitesimal displacements defined by:

dA = θA + ω^IAI, dAI = θAI + ω^J_IAJ+ ωIA. (1) The forms ω^I, ω_J^I, ωI satisfy the structural equations of the projective group GP (n):

Dω^I = ω^J∧ ω_J^I, Dω_J^I = ω_J^K∧ ω_K^I + δ_J^IωK∧ ω^K+ ωJ∧ ω^I, DωI = ω^J_I ∧ ωJ. (2) In the projective space Pn we shall consider the centred Grassman manifold V^∗ = Gr^∗(m, n), i.e. the family of all m–dimensional planes Lm, passing through a fixed point. We specialize a mobile frame {A, Aa, Aα} (a, b, c, . . . = 1, m; α, β, γ, . . . = m + 1, n), putting the top A in the given point, and tops Aa — on the centred plane L^∗_m= [A, Aa]. While fixing a point A, we obtain the identity ω^I = 0. From (1) follows, that the equations ω^α_a = 0 are conditions of stationarity of the plane L^∗_m, i.e. the forms ω^α_a are main, and for the centred Grassman manifold — basic (dim V^∗= m(n − m)).

From (2) it follows that these satisfy the structural equations

Dω^α_a = ω_b^β∧ (δ^b_aω_β^α− δ_β^αω^b_a). (3) The exterior differentials of the fibre forms look as follows:

Dω_b^a= ω_b^c∧ ω^a_c+ ω^α_b ∧ ω^a_α, Dω_β^α= ω_β^γ∧ ω^α_γ+ ω^a_β∧ ω^α_a, Dω^a_α= ω_α^b∧ ω_b^a+ ω^β_α∧ ω^a_β; (4)

D

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° Balkan Society of Geometers, Geometry Balkan Press 2006.

Dωa= ω_a^b∧ ωb+ ω_a^α∧ ωα, Dωα= ω_α^β∧ ωβ+ ω_α^β∧ ωβ.

A principal fiber bundle G(V^∗) is constructed over the manifold V^∗ whose fiber is the subgroup G of stationary centred planes L^∗_m∈ V^∗. The fibering G(V^∗) contains the affine factor fibering H(V^∗) with the structural equations (3), (4). In the prin- cipal fiber bundle G(V^∗) we shall set the fundamental-group connection by means of Laptev’s method [4]. We shall consider the transformation of secondary forms with the help of the basic forms of the manifold V^∗:

˜

ω^a_b = ω^a_b − Γ^ac_bαω_c^α, ˜ω_β^α= ω_β^α− Γ^αa_βγω_a^γ, ˜ω^a_α= ω_α^a− Γ^ab_αβω^β_b,

˜

ωa = ωa− Γ^b_aαω_b^α, ˜ωα= ωα− Γ^a_αβω_a^β.

A connection in the principal fiber bundle is given with the help of a field of connection object Γ on base V^∗ whose components should satisfy the equations:

∆Γ^ac_bα+ δ^c_bω^a_α= Γ^ace_bαβω^β_e, ∆Γ^αa_βγ− δ^α_γω_β^a= Γ^αab_βγµω_b^µ,

∆Γ^ab_αβ− Γ^ab_cβω_α^c + Γ^γb_αβω_γ^a= Γ^abc_αβγω_c^γ, ∆Γ^b_aα+ Γ^cb_aαωc+ δ_a^bωα= Γ^bc_aαβω^β_c, (5)

∆Γ^a_αβ+ Γ^ba_αβωb− Γ^a_bβω^b_α+ Γ^γa_αβωγ = Γ^ab_αβγω_b^γ, where the operator ∆ acts in usual manner (see, e.g., [1]).

The structural equations of the connection forms are

D ˜ω_b^a= ˜ω_b^c∧ ˜ω_c^a+ R^acd_bαβω^α_c ∧ ω^β_d, D ˜ω_β^α= ˜ω_β^γ∧ ˜ω_γ^α+ R^αab_βγµω^γ_a∧ ω^µ_b, D ˜ω_α^a = ˜ω^b_α∧ ˜ω_b^a+ ˜ω^β_α∧ ˜ω_β^a+ R^abc_αβγω^β_b ∧ ω_c^γ, D ˜ωa = ˜ω^b_a∧ ˜ωb+ R_aαβ^bc ω_b^α∧ ω_c^β,

D ˜ωα= ˜ω^a_α∧ ˜ωa+ ˜ω^β_α∧ ˜ωβ+ R^ab_αβγω_a^β∧ ω^γ_b,

where the components of the curvature object R of the group connection Γ are ex- pressed by the formulas:

R^acd_bαβ = Γ^a_b[^cd_αβ] − Γ^e_b[^c_αΓ^ad_eβ], R^αab_βγµ= Γ^α_β[^ab_γµ] − Γ^η_β[^a_γΓ^αb_ηµ], R^abc_αβγ = Γ^a_α[^bc_βγ] + Γ^a_e[^b_βΓ^ec_αγ] − Γ^µ_α[^b_βΓ^ac_µγ],

R^bc_aαβ= Γa[^bc_αβ] − Γ^e_a[^b_αΓ^c_eβ], R^ab_αβγ = Γα[^ab_βγ] − Γ^c_α[^a_βΓ^b_cγ] − Γ^µ_α[^a_βΓ^b_µγ].

They satisfy the following relations on the module of basic forms ω^α_a:

∆R^acd_bαβ ≡ 0, ∆R^αab_βγµ ≡ 0, ∆R^abc_αβγ− R^abc_eβγω^e_α+ R^µbc_αβγω^a_µ≡ 0,

∆R^bc_aαβ+ R^ebc_aαβωe≡ 0, ∆R^ab_αβγ+ R^µab_αβγωµ+ R^cab_αβγωc− R^ab_cβγω_α^c ≡ 0.

Theorem 1. The object of curvature R is a tensor, containing 2 elementary subtensors ([5]): R^acd_bαβ, R_βγµ^αab.

The Bortolotti’s equipment of the centred Grassman manifold V^∗is set. It consists of addition to each centred plane L^∗_mof an (n − m − 1)–dimensional plane Pn−m−1,

which has no common points with the plane L^∗_m. The equipping of the plane Pn−m−1

is defined by the sum total of points Bα = Aα+ λ^a_αAa+ λαA. The conditions of stationarity of the equipping plane Pn−m−1are:

∆λ^a_α+ ω^a_α= λ^ab_αβω^β_b, ∆λα+ λ^a_αωa+ ωα= λ^a_αβω^β_a. (6) Theorem 2. The Bortolotti’s equipment setting by a field of quasitensor λ = {λ^a_α, λα} on the manifold V^∗, induces the 1–st type connection

01Γ= {

Γ0^ac_bα,Γ^0αa_βγ,

0

Γ^b_aα,

01

Γ^ab_αβ,Γ^01a_αβ} in the fibering G(V^∗).

The first encompassing of the components of the connection object by the com- ponents of equipping quasitensor is made with the help of the hypotheses and looks as follows:

Γ0^ac_bα= δ_b^cλ^a_α,Γ^0αa_βγ= −δ_γ^αλ^a_β,

0

Γ^b_aα= δ_a^bλα, (7)

01

Γ^ab_αβ= −λ^a_βλ^b_α,Γ^01a_αβ= −λ^a_αλβ. (8) With the help of the object of the group connection Γ, we shall define the concept of covariant differential and of covariant derivative of the quasitensor λ regarding to this connection. By defining the connection forms ˜ω in (6), we obtain the covariant differentials of the components of the object λ

∇λ^a_α= dλ^a_α+ λ^b_αω˜_b^a− λ^a_βω˜_α^β+ ˜ω_α^a, ∇λα= dλα+ λ^a_αω˜a− λβω˜^β_α+ ˜ωα (9) and the covariant derivatives

∇^b_βλ^a_α= λ^ab_αβ− λ^c_αΓ^ab_cβ+ λ^a_γΓ^γb_αβ− Γ^ab_αβ, ∇^a_βλα= λ^a_αβ− λ^b_αΓ^a_bβ+ λγΓ^γa_αβ− Γ^a_αβ. (10) Theorem 3. The covariant derivatives (10) of the equipping quasitensor λ in the group connection Γ are tensors, containing subtensors.

Proof. We examine the components of the covariant derivatives, which have the form:

∆∇^b_βλ^a_α≡ 0, ∆∇^a_βλα+ ∇^a_βλ^b_αωb≡ 0.

With the help of the structural equations, we shall find the exterior differentials from the covariant differentials (9) of the components of equipping quasitensor λ

D∇λ^a_α= ∇λ^b_α∧ ˜ω_b^a− ∇λ^a_β∧ ˜ω_α^β+ T_αβγ^abcω^β_b ∧ ω_c^γ, D∇λα= ∇λ^a_α∧ ˜ωa− ∇λβ∧ ˜ω_α^β+ T_αβγ^ab ω^β_a ∧ ω_b^γ, where

T_αβγ^abc = R^abc_αβγ+ λ^d_αR^abc_dβγ− λ^a_µR^µbc_αβγ, T_αβγ^ab = R^ab_αβγ+ λ^c_αR^ab_cβγ− λµR^µab_αβγ. Thus, the exterior differentiation of the covariant differential contains an object of relative motion ([6]) whose components are combinations of the components of cur- vature tensor of the group connection having as coefficients the components of the

equipping quasitensor. The given object T = {T_αβγ^abc, T_αβγ^ab } is a tensor, including the subtensor T_αβγ^abc, with the following differential relations on the components:

∆T_αβγ^abc ≡ 0, ∆T_αβγ^ab + T_αβγ^cabωc≡ 0.

The condition of vanishing of the covariant derivatives (10) of the components of equipping quasitensor λ is invariant (see Theorem 3). Setting them to zero, we shall find the expressions of the components Γ^a_αβ, Γ^ab_αβ of the connection object Γ through the components Γ^ac_bα, Γ^αa_βγ, Γ^b_aα. We get a bunch of connections of 2–nd type. Taking into account the encompassing (7), we have

Γ02^a_αβ= λ^a_αβ− 2λ^a_αλβ,

02

Γ^ab_αβ= λ^ab_αβ− 2λ^a_βλ^b_α. (11)

Theorem 4. The Bortolotti’s equipment of the centred Grassman manifold V^∗ induces a setof connections of 2–nd type in the associated fibering G(V^∗) from which a unique connection of 2–nd type is infered

02Γ= {Γ^0ac_bα,Γ^0αa_βγ,

0

Γ^b_aα,

02

Γ^ab_αβ,Γ^02a_αβ}.

We further construct the third encompassing of the components of connection object Γ. We shall take into account the continued differential relations of the components of equipping quasitensor λ, the differential equations (5) which are satisfied by the components of connection object Γ, and the encompassing (7). Then the following formulas hold true

03

Γ^ab_αβ= −λ^ab_αβ,Γ^03a_αβ= −λ^a_αβ. (12)

Theorem 5. Bortolotti’s equipment induces a connection of the third type

03Γ= {Γ^0ac_bα,Γ^0αa_βγ,

0

Γ^b_aα,

03

Γ^ab_αβ,Γ^03a_αβ}.

A condition of coincidence of the three constructed encompassings is the equality λ^ab_αβ= λ^a_βλ^b_α, λ^a_αβ= λ^a_αλβ.

Theorem 6. The connection of the 1–st type is the average ([5]) of the connections of 2–nd and 3–d types, i.e. ⁰¹Γ= ¹₂(⁰²Γ +⁰³Γ).

Proof. Straightforward, using the formulas (8), (11), (12), which express the connection components.

Remark. The similar theorem is proved in [2] for the noncentred Grassman manifold.

References

[1] O. O. Belova, The connection in the fibering associated with the Grassman mani- fold (in Russian), Diff. geom. of manifolds of the figures. Kaliningrad, 31 (2000), 8–11.

[2] O. O. Belova, The connections of three types in the fibering associated with the Grassman manifold(in Russian), The problem of math. and phys. sciences. Kalin- ingrad, 2001, 3–5.

[3] E. Bortolotti, Connessioni nelle varieta luogo di spazi, Rend. Semin. Fac. Sci.

Univ. Cagliari 3 (1933), 81–89.

[4] L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, A. P. Shirokov, The differential- geometrical structures on manifolds (in Russian), Probl. of geom. VINITI. I., 9 (1979), 5 – 247.

[5] A. P. Norden, Spaces of affine connection (in Russian), Nauka, 1976.

[6] Ju. I. Shevchenko, Equipment of the centerprojective manifolds (in Russian), Kaliningrad, KSU, 2000.

Author’s address:

Olga Belova

I. Kant State University of Russia,

The University Department of Higher Algebra and Geometry 14 A.Nevsky Str., 236041, Kaliningrad, Russia

email: [email protected]