Grassmannian structures on manifolds

P.F. Dhooghe

Abstract Grassmannian structures on manifolds are introduced as subbundles of the second order framebundle. The structure group is the isotropy group of a Grassmannian. It is shown that such a structure is the prolongation of a subbundle of the first order framebundle. A canonical normal connection is constructed from a Cartan connection on the bundle and a Grassmannian curvature tensor for the structure is derived.

1 Introduction

The theory of Cartan connections has lead S. Kobayashi and T. Nagano, in 1963, to present a rigourous construction of projective connections [3]. Their construction, relating the work of Eisenhart, Veblen, Thomas a.o. to the work of E. Cartan, has a universal character which we intend to use in the construction of Grassmannian- like structures on manifolds. The principal aim is to generalise Grassmannians in a similar way. By doing so we very closely follow their construction of a Cartan con- nection on a principal bundle subjected to curvature conditions and the derivation of a normal connection on the manifold.

The action of the projective group Pl(no) on a Grassmannian G(lo,no)oflo- no no planes in IR is induced from the natural action of Gl(no)onIR .LetH be the isotropy group of this action at a fixed point e of G(lo,no). The generalisation will consist in the construction of a bundle P with structure group H and base manifold

Received by the editors November 1993 Communicated by A. Warrinnier AMS Mathematics Subject Classification : 53C15, 53B15, 53C10 Keywords : Grassmannian structures, Non linear connections, Higher order structures, Second order connections.

Bull. Belg. Math. Soc. 1 (1994), 597–622 598 P.F. Dhooghe

M of dimension mo = loko with ko = no − lo. The bundle P will be equipped with a Cartan connection with values in the Lie algebra of the projective group, which makes the bundle P completely parallelisable. We will show that such a connection exists and is unique if certain curvature conditions are imposed. The Cartan connection identifies the tangent space Tx(M)foreachx ∈ M with the vectorspace L(IR lo , IR ko ). Identifying L(IR lo , IR ko )withV = IR mo , the group H × τ −1 acts on V to the first order as Go = Gl(lo) Gl(ko) / exp tIno properly embedded 0 in Gl(mo). Let g˜ denote the Lie algebra of this group, which is seen as a subspace ∗ of V ⊗ V . We prove that if ko ≥ 2andlo ≥ 2, the Lie algebra h of H, as subspace of V ⊗ V ∗, is the first prolongation of the Lie algebra g˜0. Moreover the second prolongation equals zero. The action of H on V allows to define a homomorphism of P into the second order 2 framebundle F (M). The image, Gr(ko,lo)(M), is called a Grassmannian structure on M. From the previous algebraic considerations it follows that a Grassmannian structure on a manifold is equivalent with a reduction of the framebundle F 1(M)to (ko,lo) a subbundle B (M) with the structure group Go. A Grassmannian connection from this point of view, is an equivalence class of symmetric affine connections, all 1 of which are adapted to a subbundle of F (M) with structure group Go. The action 1 of Go in each fibre is defined by a local section σ : x ∈ M → F (M)(x) together with an identification of Tx(M)withM(ko,lo). This result explains in terms of G-structures the well known fact that the structure group of the tangent bundle on a Grassmannian, G(lo,no), reduces to Gl(ko) × Gl(lo) [6]. The consequences for the geometry and tensoralgebra are partly examined in the last paragraph, but will be studied in a future publication. We remark that as a consequence of the algebraic structure the above defined structure is called Grassmannian if ko ≥ 2andlo ≥ 2. Otherwise the structure is a projective structure. Hence the manifolds have dimension mo = kolo,withko,lo ≥ 2. α ··· IR mo i ··· Let (¯x ), α =1, ,mo be coordinates on ,and(ea), a =1, ,ko; i = ··· a 1, ,lo, the natural basis on M(ko,lo). (xi ) are the corresponding coordinates on 1 M(ko,lo). We will identify both spaces by α =(a−1)lo+i.Letσ : U⊂M → F (M) be a local section andσ ¯ be the associated map identifying the tangent space Tx(M) (x ∈U)withM(ko,lo). An adapted local frame with respect to some coordinates U α −1 i iα ∂ ∇ ∇˜ ( , (x )) is given asσ ¯ (x)(ea)=Ea ∂xα (x). If and are two adapted symmetric (ko,lo) linear connections on B (M), then there exists a map µ : U→M(lo,ko)such that for X, Y ∈X(M):

−1 ∇˜ XY = ∇XY +¯σ [(µ.σ¯(X)) . σ¯(Y )+(µ.σ¯(Y )) . σ¯(X)].

Because µ ∈ M(lo,ko)and¯σ(X)(x) ∈ M(ko,lo), for X ∈X(U), the term (µ.σ¯(X)(x)), as composition of matices, is an element of M(lo,lo) which acts on σ¯(Y )(x) giving thus an element of M(ko,lo). Analogous to the projective case we will construct a canonical normal Grass- mannian connection and calculate the expression of the coefficients with respect to an adapted frame. The curvature of the Grassmannian structure is given by the i a i forms Ωj, Ωb , Ωa, with respect to a Lie algebra decomposition of h. We prove that ≥ ≥ i a if lo 3orko 3 the vanishing of Ωj or Ωb is necessary and sufficient for the Grassmannian structures on manifolds 599

i a local flatness of the bundle P . The two curvature forms Ωj and Ωb are basic forms 2 ⊂ 2 → 1 on the quotient π1 : Gr(ko,lo)(M) F (M) F (M) and hence determine the Grassmannian curvature tensor, whose local components are given by

α i a jα b a iα Kβγσ = KjγσFiβEa + KaγσFiβEb ,

i i α ⊗ β a α ⊗ β jα a with Ωj = Kjαβdx dx and Ωbαβdx dx . Eb is an adapted frame and Fiβ the corresponding coframe. It follows that the vanishing of the Grassmannian cur- vature tensor is a necessary and sufficient condition for the local flatness of the Grassmannian structure for any lo ≥ 2andko ≥ 2. We assume all manifolds to be connected, paracompact and of class C ∞.All ∞ no maps are of class C as well. Gl(no) denotes the general linear group on IR and gl(no) its Lie algebra. We will use the summation convention over repeated in- dices. The indices take values as follows : α, β, ···=1, ···,mo = kolo; a, b, c ··· = 1, ···,ko; i, j, k, ··· =1, ···,lo. Cross references are indicated by [(.)] while refer- ences to the bibliography by [.].

2 Grassmannians

A. Projective Group Actions

no Let G(lo,no) be the Grassmannian of the lo-dimensional subspaces in IR .Dim no G(lo,no)=loko, no = lo + ko.LetS be a ko-dimensional subspace of IR .An associated big cell U(S)toS in G(lo,no) is determined by all transversal subspaces no to S of dimension lo in IR . One observes that

G(lo,no)=∪I U(SI ) where I is any subset of length ko of {1, 2, ···,no} and SI the subspace of dimension I no ko spanned by the coordinates (x )inIR . Let (x1 ···,xlo ,xlo+1, ···,xno ) be the natural coordinates on IR no . For simplic- ity we will choose a rearrangement of the coordinates such that S is given by the condition x1 = x2 = ···= xlo =0. Let M(no,lo)bethespaceof(no × lo) matrices (no rows and lo columns). Any no element may be considered as lo linearly independent vectors in IR . Hence each no y ∈ M(no,lo) determines an lo-plane in IR . We get a natural projection

π : M(no,lo) →U(S), (1) which is a principal fibration over U(S) with structure group Gl(lo). Representing the coordinate system on M(no,lo) by a matrix Z, the big cell U(S) is coordinatised as follows. If Z ∈ M(no,lo), we will write ! Z Z = 0 , Z1 with Z0 an lo × lo matrix and Z1 an ko × lo matrix, no = ko + lo. 600 P.F. Dhooghe

The coordinates are obtained by ˜ −1 Z = Z1.Z0 , whereweassumedZ0 to be of maximal rank. In terms of its elements we get ! i zj Z = a , zj i, j =1, ···,lo and a =1, ···,ko, to which we refer as the homogeneous coordinates. i i Denoting by wj the inverse of zj,weobtain

˜ a a j Z =(xi )=(zj wi ). which are the local coordinates on the cell. In the sequel we will identify the cel with M(ko,lo). no The action of the group Gl(no)onIR induces a transitive action of Pl(no)on G(lo,no). On a big cell the action of Pl(no) is induced from the action of Gl(no)on Z on the left. Let β be in Gl(no). In matrix representation we write β as : ! β β β = 00 01 , (2) β10 β11 with β00 ∈ M(lo,lo), β11 ∈ M(ko,ko), β10 ∈ M(ko,lo), β01 ∈ M(lo,ko). The local action of an open neighbourhood of the identity in the subset of Gl(no) defined by det β00 =0on6 M(ko,lo) is given in fractional form by

−1 φβ : x 7→ (β10 + β11 x)(β00 + β01 x) (3) for β ∈ Gl(no) as in [(2)] and x ∈ M(ko,lo). Because the elements of the center of Gl(no) are in the kernel of φβ this action induces an action of an open neighbourhood of the identity in Pl(no). In terms of the coordinates and using the notation

i i a a −1 i β00 =(βj),β01 =(βa),β10 =(βi ),β11 =(βb )andβ00 =(γj), we find the Taylor expression a a k a − a k j b m x¯l = βk γl +(βb βk γj βb )xmγl − a c k m b n a k m c j n e r ··· βc xkγmβb xnγl + βk γmβc xjγnβe xrγl + . (4) Consequences :

(a) The orbit of the origin of the coordinates in M(ko,lo), is locally given by (0) 7→ −1 β10β00 . (b) The isotropy group H at 0 ∈ M(lo,ko) is the group ! { β00 β01 } H : β = / exp t.Ino , (5) 0 β11 Grassmannian structures on manifolds 601

with β00 ∈ Gl(lo)andβ11 ∈ Gl(ko). The subgroup H in Taylor form is given by

1 x¯a = βaγmxb − [βaγi βkγl + βaγl βkγi]xbxc + ···. (6) j b j m 2 b k c j c k b j i l

B. The Maurer Cartan Equations i i a a ··· ··· Let (ua,uj,ub ,ui ), with i, j =1, ,lo , a, b =1, ,ko, be local coordinates at i i a a the identity on Gl(no) according to the decomposition [(2)] and (¯ωa, ω¯j , ω¯b , ω¯i )the i i a a left invariant forms co¨ınciding with (dua,duj,dub ,dui ) at the identity. The Maurer Cartan equations are

a − a ∧ k − a ∧ b (1) dω¯j = ω¯k ω¯j ω¯b ω¯j i − i ∧ k − i ∧ b (2) dω¯j = ω¯k ω¯j ω¯b ω¯j a − a ∧ k − a ∧ c (3) dω¯b = ω¯k ω¯b ω¯c ω¯b i − i ∧ k − i ∧ b (4) dω¯a = ω¯k ω¯a ω¯b ω¯a.

i a Letω ¯1 =¯ωi andω ¯2 =¯ωa. We define

i i − 1 i a a − 1 a 1 − 1 ωj =¯ωj δj ω¯1,ωb =¯ωb δb ω¯2,ω∗ = ω¯1 ω¯2. (7) lo ko lo ko

Passing to the quotient Gl(no)/ exp t.Ino we find the Maurer Cartan equations on Pl(no).

Proposition 2.1 The Maurer Cartan equations on Pl(no) are

a − a ∧ k − a ∧ b − a ∧ (1) dωj = ωk ωj ωb ωj ωi ω∗ 1 (2) dωi = −ωi ∧ ωk − ωi ∧ ωb + δi ωk ∧ ωc j k j b j l j c k 1 (3) dωa = −ωa ∧ ωk − ωa ∧ ωc + δa ωc ∧ ωk (8) b k b c b k b k c i − i ∧ k − i ∧ b i ∧ (4) dωa = ωk ωa ωb ωa + ωa ω∗ k + l o o a ∧ i (5) dω∗ = ωi ωa. kolo

i a Remark that ωi = ωa =0.

The Lie algebra of Pl(no), g, in this representation is found by taking the tangent k l space at the identity, W , to the submanifold in Gl(no) defined by (det β00) (det β11) = 1. The quotient of the algebra of left invariant vectorfields, originated from W , by the vectorfield expt.Ino determines the Lie algebra structure. The vectorspace for this Lie algebra is formed by the direct sum

g = g−1 ⊕ g0 ⊕ g1, (9) 602 P.F. Dhooghe where

g−1 = L(IR lo, IR ko ) 0 g = {(u, v) ∈ gl(lo) ⊕ gl(ko); k.Tr(u)+l.Tr(v)=0} g1 = L(IR ko , IR lo ). (10)

Let x ∈ g−1, ∗y ∈ g1 and (u, v) ∈ g0, the induced brackets on this vector space are : [u, x]=x.u ;[v,x]=v.x ;[u, ∗y]=u. ∗y ; ∗ ∗ ∗ ∗ [v, y]= y.v ;[x1,x2]=0;[y1, y2]=0;

[u1 + v1,u2 + v2]=[u1,u2]+[v1,v2] ; (11) ∗ ∗ ∗ − ∗ − − Tr(x. y) [x, y]=x y y.x (lo ko) .Idno . 2kolo

IR lo ⊕ IR ko Idno denotes the identity on .

C. Representations and prolongation We will use the following identifications :

κ ko×lo ς mo M(ko,lo) = IR = IR a κ ai ς α (12) xi = x = x

× IR ko lo IR lo ×···×IR lo − ··· where stands for | {z } ; α =(a 1)lo+i, mo = kolo ; α =1, ,mo

ko times

; a =1, ···,ko and i =1, ···,lo.

We introduce the following two subgroups.

(1) The subgroup Go of Gl(lo) × Gl(ko):

ko lo Go = {(A, B) ∈ Gl(lo) × Gl(ko) | (det(A)) .(det(B)) =1}. (13)

0 0 ko lo Let (A, B)and(A ,B)beelementsinGo. Then from (det(A)) (det(B)) =1 0 0 0 0 and (det(A ))ko (det(B ))lo = 1 it follows that (det(AA ))ko (det(BB ))lo =1.We also remark that Go is isomorphic to the subgroup defined by β01 = β10 =0in

Gl(lo + ko)/ exp t.Ino . There indeed always exists an α such that

(det αA)ko .(det αB)lo = αk+l(det(A))ko .(det(B))lo =1.

(2) The subgroup G˜o of Gl(mo) defined by

− {Aα δ(a 1)lo+i δβ = Ai Aa | (Ai ) ∈ Gl(l ), (Aa) ∈ Gl(k )}. (14) β α (b−1)lo+j j b j o b o Multiplication in the group yields

− AαAγ δ(a 1)lo+i δβ = Ai AkAaBc. γ β α (b−1)lo+j k j c b Grassmannian structures on manifolds 603

We will intoduce the following notations

α β α ↔ς i a bj ai ↔κ a b τ j a Aβx =˜x AjAb x =˜x Ab xj Ai =˜xi , (15) which we will use throughout this paper. We also will use κ for κoς.

Let (A1,B1)and(A2,B2)beinGo.Wethenhave

τ −1 τ −1 τ −1 (A1.A2,B1.B2) 7→ ( (A1.A2) ,B1.B2)=( (A1) . (A2) ,B1.B2), which proves the following proposition.

Proposition 2.2 The morphism

τ : Go → G˜o (A, B) 7→ ( τA−1,B) (16) is a group isomorphism sending left invariant vectorfields into left invariant vector- fields.

˜ × Proposition 2.3 The Lie algebra, g˜o of Go is given by the subalgebra of the (mo mo) matrices which are defined by

α κ j a a j zβ =˜ui δb +˜ub δi (17)

− − i ∈ a ∈ with α =(a 1)lo + i, β =(b 1)lo + j, (˜uj) gl(lo), (˜ub ) gl(ko).

It is a direct consequence of proposition [(2.2)] that this Lie algebra, g˜0, is isomorphic 0 τ − i a to g . The isomorphism is induced from u = (˜uj),v=(˜ub ). Let V be the real vectorspace isomorphic to IR mo . The algebra g˜0 is a subalgebra of V ⊗V ∗. The first prologation g˜(1) is defined as the vectorspace V ∗ ⊗g˜0∩S2(V ∗)⊗V and the kth prolongation likewise as the vectorspace [1] [10]

(k) ∗ ⊗···⊗ ∗ ⊗ 0 ∩ k+1 ∗ ⊗ g˜ = V| {z V } g˜ S (V ) V. ktimes

A subspace of V ∗ ⊗ V is called of finite type if g˜(k) = 0 for some (and hence all larger) k and otherwise of infinite type. We refer to [10] [1] [8] for the details. We then have the following theorem.

0 Theorem 2.1 The algebra V ⊕g˜ is of infinite type if ko or lo equals 1. If ko and lo are both different from 1 the algebra is of finite type. Moreover in this case g˜(2) =0 and the algebra V ⊕ g˜0 ⊕ g˜(1) is isomorphic to the algebra g−1 ⊕ g0 ⊕ g1.

In order to prove the theorem we will make use of the representation of g˜0 into the linear polynomial vectorfields on V .Let(xα) be the coordinates on V . Define the subalgebra g0 as the set of vectorfields

∂ u˜αxβ withu ˜α =˜κ ujδa +˜vaδj. (18) β ∂xα β i b b i 604 P.F. Dhooghe

−1 0 1 If ko =1orlo = 1, the algebra g ⊕ g ⊕ g is the algebra of projective mo 0 0 transformations on IR [11]. Hence g = g˜ = gl(mo), from which it follows that the algebra V ⊕ g˜0 is of infinite type. We assume from now on ko and lo to be different from 1. The second prolongation g˜(2) is zero as a consequence of a classification theorem by Matsushima [7] [8] or by a direct calculation from g˜(1) once this is derived. Before proving the theorem we will prove the following lemmas.

κ ilc a d ∂ Lemma 2.1 Any second order vectorfield X = Tadkxi xl c , such that ∂xk

∂ b ∈ 0 [[ ul b ,X]] g˜ ∂xl is of the form ilc i l c l i c Tadk = uaδkδd + udδkδa.

Proof b ∂ ∈ For any uj b V the bracket with any homogeneous second order vectorfield ∂xj ijc a b ∂ 0 Tabkxi xj c taking values in g˜ satisfies the equation ∂xk

b ∂ ijc a b ∂ l c c l d ∂ [[ uj b ,Tabkxi xj c ]] = [ Akδd + Bdδk]xl c , ∂xj ∂xk ∂xk

l c for some constants Ak and Bd. This equation becomes a ilc l c c l 2ui Tadk = Akδd + Bdδk. ilc lic Which together with the symmetry Tadk = Tdak proves the lemma. @A

Call W be the vector space of the second order vectorfields of the form X =κ ilc a d ∂ ∈ ∈ 0 ∈ Tadkxi xl c .LetX V , Y g˜ and Z W . Because the set of all formal ∂xk vectorfields on V is a Lie algebra, we can consider the Jacobi identity

[[ [[ X, Y ]] ,Z]] + [[ [[ Y,Z]] ,X]] + [[ [[ Z, X]] ,Y]] = 0 .

Lemma 2.2 Let Y ∈ g˜0 and Z ∈ W .Then:

[[ Y,Z]] ∈ g˜0.

Proof Because [[ X, Y ]] ∈ V the first term takes values in g˜0. The thirth term also takes values in g˜0 by the construction of W . Hence the second term [[ [[ Y,Z]] ,X]] t a k e s values in g˜0. But this imples that [[ Y,Z]]takesvaluesinW by the former lemma. @A Grassmannian structures on manifolds 605

As a consequence of both lemmas we are able to write the algebra V ⊕ g˜0 ⊕ g˜(1) as the vectorspace L spanned by the vectorfields

a ∂ j a a j b ∂ k a j j a k b c ∂ (˜ui a , (˜ui δb +˜ub δi )xj a , (˜ub δc δi +˜ucδb δi )xjxk a ) ∂xi ∂xi ∂xi

a ∂ j a ∂ a b ∂ k a c ∂ =(˜ui a , u˜i xj a +˜ub xj a , u˜c xkxi a ). (19) ∂xi ∂xi ∂xj ∂xi We find the following proposition.

Proposition 2.4 Both Lie algebras L and g are isomorphic. The isomorphism

τ : g = g−1 ⊕ g0 ⊕ g1 →L is induced from

a a i − i a a i i τ(ui )=˜ui ,τ(uj)= u˜j,τ(ub )=˜ub ,τ(ua)=˜ua, (20)

a i a i ∈ with (ui ,uj,ub ,ua) g. This proposition together with both lemmas proves the theorem.

3 The Cartan connections

A. The structure equations 2 − Let P be a principal bundle, of dimension no 1(no = ko +lo), over M with fibre group H, the isotropy group [(5)]. We then have dim P/H = kolo.Therightaction of H on P is denoted as Ra,fora ∈ H, while ad stands for the adjoint representation of H on the Lie algebra g = pl(no). Every A ∈ h induces in a natural manner a vectorfield A?, called fundamental vectorfield, on P as a consequence of the action of H on P . The vectorfield A? obviously is a vertical vectorfield on P . A Cartan connection on P is a 1-form ω on P , with values in the Lie algebra g, such that :

(1) ω(A?)=A, ∀ A ∈ h ? −1 ∈ (2) Ra ω = ad(a ) ω, a H (3) ω(X) =06 , ∀X ∈X(P )withX =06 . (21)

The form ω defines for each x ∈ P an isomorphism of TxP with g. Hence the space P is globally parallelisable. In terms of the natural basis in matrix representation of pl(no) as given in [(10)] a i a i i a and [(11)], we write the connection form ω as (ωi ,ωj,ωb ,ω∗,ωa), with ωi = ωa =0.

ko lo As basis for the subalgebra h = sl(lo) ⊕ sl(ko) ⊕ IR ⊕ L(IR , IR )wechoose i a a (ej,eb ,e∗,ei ). The structure equations of Cartan on P are now defined as 606 P.F. Dhooghe

a − a ∧ k − a ∧ b − a ∧ a (1) dωj = ωk ωj ωb ωj ωj ω∗ +Ωj 1 (2) dωi = −ωi ∧ ωk − ωi ∧ ωb + δi ωk ∧ ωc +Ωi j k j b j l j c k j 1 (3) dωa = −ωa ∧ ωk − ωa ∧ ωc + δa ωc ∧ ωk +Ωa (22) b k b c b k b k c b i − i ∧ k − i ∧ b i ∧ i (4) dωa = ωk ωa ωb ωa + ωa ω∗ +Ωa k + l o o a ∧ i (5) dω∗ = ωi ωa +Ω∗, kolo

i a i a with ωi = ωa =Ωi =Ωa =0. In analogy with the projective case described by Kobayashi and Nagano, the a i a i form Ωi is called the torsion form while ( Ωj, Ωb , Ωa, Ω∗ ) are called the curvature forms of the connection. The connection form satisfies the following conditions : a ? i ? i a ? a ? i a i ∈ ωi (A )=0,ωj (A )=Aj, ωb (A )=Ab , ω∗(A )=A∗ for A =(Aj,Ab ,Aa,A∗) ∈X a h. Furthermore if X such that ωi (X)=0,thenX is vertical.

Proposition 3.1 The torsion and the curvature forms are basic forms on the bundle P . Hence we define :

a abc j ∧ k i ibc l ∧ k Ωi = Kijk ωb ωc , Ωj = Kjlkωb ωc ,

a adc j ∧ k i ibc j ∧ k bc j ∧ k Ωb = Kbjk ωd ωc , Ωa = Kajk ωb ωc , Ω∗ = K∗,jk ωb ωc (23)

Proof ∈ a Let Fx,x M, be the fibre above x. The restriction of ωi to Fx is identically i a i zero and the forms ωj,ωb ,ωa,ω∗ are linearly independent on Fx as a consequence of [(21 (1)(3))]. Because the form ω sends the fundamental vectorfields A∗ which are tangent to Fx into the left invariant vectorfields A on the group H,theforms i a i ωj ,ωb ,ωa,ω∗ satisfy the equations of Maurer cartan on H. The combination of these equations and equations [(22)] implies the vanishing of the curvature forms when restricted to Fx. @A

a From now on we assume the torsion Ωi to be zero.

Proposition 3.2 Let P be a principal fibre bundle over M with structure group H i i a a and ( ωb,ωj ,ωb ,ωj ) a Cartan connection on P satisfying the structure equations [(22)]. The curvature forms possess the following properties :

a ∧ k a ∧ b a ∧ (1) 0 = ωk Ωj +Ωb ωj + ωj Ω∗ − a ∧ i (2) 0 = dΩ∗ ωi Ωa (24) Grassmannian structures on manifolds 607

Proof These equations are obtained by taking the exterior differential of equations [(22,(1) and (5))]. @A

B. The normal Cartan connection

a The first equation of the structure equations of Cartan with Ωi = 0 is called the i torsion zero equation and does not contain the form ωa, while the other equations de- a i a fine the curvature forms. A natural question then arises, namely : let (ωi ,ωj ,ωb ,ω∗) be given a priori on P which satisfy the torsion equation, does there then exists a i ωa such that ω is a Cartan connection on P and if so is there a canonical one.

a a i Theorem 3.1 Let the bundle P be given as defined and (ωi ,ωb ,ωj,ω∗) be 1-forms satisfying : a ∗ a ∗ a i ∗ i ∗ (1) ωi (A )=0,ωb (A )=Ab ,ωj (A )=Aj,ω∗(A )=A∗ ∀ i a i ∈ A =(Aj,Ab ,Aa,A∗) h

∗ a i a −1 a i a ∀ ∈ (2) (Ra) (ωi ,ωj ,ωb ,ω∗)=ad(a )(ωi ,ωj ,ωb ,ω∗), a H ∈X a (3) If X (P ) such that ωi (X)=0,thenX is vertical. a − a ∧ b − a ∧ j − a ∧ (4) dωi = ωb ωi ωj ωi ωi ω∗.

If lo =16 and ko =16 then there exists an unique Cartan connection ω on P

a i a i ω =(ωi ,ωj ,ωb ,ω∗,ωa), such that : ilk dki Ω∗ =0 and Klab = Kadb. (25)

Proof The existence of a Cartan connection satisfying the given conditions follows from a classical construction using the partition of unity. Because the manifold is supposed to be paracompact there exists a locally finite cover {Uα} of M such that P (Uα)is trivial, for each α.Let{(fα, Uα)} then be a subordinate partition of unity. If for each U a a i Pα the form ωα is a Cartan connection on P ( α) with prescribed (ωi ,ωb ,ωj ,ω∗), then → α(fα oπ)ωα is a Cartan connection in P (π being the bundle projection P M). Hence the problem is reduced to a local problem for a trivial P .Letσ : U⊂ → i i M P be a local section we define the 1-form ωa over σ as ωa(X) = 0 for all i ∗ i ∈ tangent vectors to σ and ωa(A )=Aa for A h. Now any vectorfield Y on P can be written uniquely as Y = Ra(X)+V ,whereX is tangent to σ and a ∈ H and V is tangent to the fibre. Hence the condition

ω(Y )(p.a)=ad(a−1)(ω(X))(p)+A, p = σ(x),x∈ M with ∗A the unique fundamental vectorfield corresponding to A, such that ∗A(p.a)= i V (p.a), determines ωa(Y ). 608 P.F. Dhooghe

We will prove the existence of a Cartan connection satisfying the required con- ditions [(25)] by means of a set of propositions. @A

Proposition 3.3 Let ω be a Cartan connection on P . Then there exists a Cartan connection satifying the condition Ω∗ =0. Two Cartan connections satisfying this i i − ik b ik ki same condition are related by ω¯a = ωa Aabωk,withAab = Aba.

Proof Using conditions [(21,(1) (3))] the unkown form can be written as

i i − ik b ω¯a = ωa Aabωk. Equation [(22, (5))] then yields

k + l o o a ∧ ik b − ¯ 0= ωi Aabωk +Ω∗ Ω∗. kolo

6 ik If Ω∗ =0chooseAab such that

ko + lo a ∧ ik b 0= ωi Aabωk +Ω∗ kolo or 2k l ik − ki − o o ik Aab Aba = K∗ ab. ko + lo As follows directly from this equation two Cartan connections satisfying the curva- i i − ik b ik − ki ture condition Ω∗ = 0 are related byω ¯a = ωa Aabωk,withAab Aba =0. @A

Proposition 3.4 Let ω beaCartanconnectiononP satisfying condition Ω∗ =0. Then the Bianchi identities [(24)] become

klm a alm k mkl a akl m lmk a amk l (1) Kjcb δd + Kdcb δj + Kjdc δb + Kbdc δj + Kjbd δc + Kabd δj =0

ikl lik kli (2) Kacb + Kbac + Kcba =0. (26)

Consequences : From equation [(26,(1))] we find by contraction of the indices ko & j and a & d mkl dml − lkm − dlm Kkbc + Kbdc Kkcb Kcdb = 0 (27) and by contraction of ko & j and a & c the expression mjl − aml − ljm aml kKjdc lKdac Kjdc + Kcad =0. (28) Lemma 3.1 The expression mkl − dlm Kkbc Kcdb is symmetric in the pair ((m, b), (l, c)). Grassmannian structures on manifolds 609

Proposition 3.5 Let ω be a Cartan connection on P satisfying Ω∗ =0. Then there exists a unique Cartan connection satisfying the curvature conditions [(25)].

Proof It is sufficient to consider the class of Cartan connections determined by the condi- tion Ω∗ = 0. Two such connections are related by

i i − ik b ω¯a = ωa Aabωk,

ik ki with Aab = Aba [(3.3)]. Equation [(22, (2))] then gives

i − ¯ i − il b ∧ a Ωj Ωj Aab ωl ωj =0 or h i ilk − ¯ ilk − il k b ∧ a Kjba Kjba Aab δj ωl ωk =0, which yields 1   Kilk − K¯ ilk − δkAil − δl Aik =0. jba jba 2 j ab j ba Summation on the indices l and j gives :

1   Kilk − K¯ ilk − Aik − lAik =0. (29) lba lba 2 ab ba

From [(22),(3)] we derive in a similar way the following equation :

1   δd Alk − δd Akl + Kdkl − K¯ dkl =0. 2 a bc c ba bca bca

Contraction on d and c yields

1   Aik − kAki + Kdki − K¯ dki =0. (30) 2 ab ab adb adb

From the lemma [(3.1)] we know that the expression

ilk − dki Klba Kadb is symmetric in the pair ((i, b), (k, a)). If

ilk − dki 6 Klba Kadb =0

ik we define Aab such that

1 Kilk − Kdki = Aik − (l + k )Aik . (31) lba adb ab 2 o o ba 610 P.F. Dhooghe

Or   ki 4 ki 1 ik Aab = 2 ∆ba + (lo + ko)∆ba , (32) 4 − (lo + ko) 2 with ik ilk − dki ∆ba = Klba Kadb. (33) Substitution of [(31)] in the sum of equations [(29)] and [(30)] gives ¯ ilk − ¯ dki Klba Kadb =0.

This proves the theorem for lo =1and6 ko =1.If6 lo or ko equals one we refer to the projective case treated by Kobayashi S., Nagano T. [3]. The uniqueness follows from the same considerations. @A

Definition 3.1 The unique Cartan connection ω on P satisfying the curvature con- ditions [(25)], will be called the normal Grassmannian connection on P .

Proposition 3.6 Let ω be a normal Cartan connection on the bundle P .The following curvature equations are identities :

mjl aml koKjdc = loKdac mjl ljm koKjdc = loKjcd alm aml koKcad = loKdac (34)

Proof These relations follow from the conditions [(25)] and the identities [(28)]. @A

i a Proposition 3.7 Let P and ω be as above. If Ωj =0and Ωb =0, then it follows i that Ωa =0.

Proof

If ko =1and6 lo =6 1 then the manifold M has dimension larger than 3. The proposition follows from differentiation of equations [(22, (2)(3))] :

1 i − i ∧ k i ∧ k i k ∧ c − i ∧ b dΩj Ωk ωl + ωk Ωj + δj Ωc ωk Ωb ωj = 0 (35) lo and 1 a − a ∧ c a ∧ c − a c ∧ k a ∧ k dΩb Ωc ωb + ωc Ωb δb ωk Ωc + ωk Ωb =0. (36) ko From equation [(35)] one finds

1 k ∧ c ∧ a − i ∧ b ∧ a Ωc ωk ωj Ωb ωj ωi =0. lo Grassmannian structures on manifolds 611

While from equation [(36)] one has

1 k ∧ c ∧ a − k ∧ a ∧ b Ωc ωk ωj Ωb ωk ωj =0. ko Combining the two equations gives k ∧ a ∧ b (ko + lo)Ωb ωk ωj =0, whichsubstitutedinequation[(35)]gives i ∧ b Ωb ωj =0 and in equation [(36)] a ∧ k ωk Ωb =0. Or in terms of the components we find the two equations :

ikl j ijk l ilj k Kbcdδm + Kdbc δm + Kcdbδm =0. and ikl a kli a lik a Kbcdδe + Kbdeδc + Kbecδd =0. ≥ ijk In case lo 3, let l be different from k and j. We find by taking m = l that Kdbc =0 .Incaseko ≥ 3, let c be different from e and d. One finds the same result by taking a = c.

The special case ko =2andlo = 2 is easily proven by consideration of the different cases k = j = l , k = l =6 j , e = c = d and e = c =6 d. @A

Proposition 3.8 Let P with ko ≥ 3, lo ≥ 3 and ω be as above. Then

i a Ωj =0 iff Ωb =0.

Proof

i From the Bianchi identities [(24,(1))] we find with Ωj =0

a ∧ b Ωb ωj =0. In terms of the components this equation is

alm k akl m amk l Kdcb δj + Kbdc δj + Kcbd δj =0.

akl Let m be different from k and l.Takingj equal to m yields Kbdc =0. a i Conversally, the condition Ωb = 0 implies Ωj = 0 by an analogous argument using the Bianchi equations [(24,(1))]. @A This proves the following theorem.

Theorem 3.2 Let P with ko ≥ 3, lo ≥ 3 and ω as above. The bundle P is locally i a flat iff Ωj =0or Ωb =0. Local flatness of P means vanishing of the structure functions [2]. 612 P.F. Dhooghe

4 The Ehresmann connection

A. Second order frames

Let M be a manifold of dimension mo and f a diffeomorphism of an neighborhood of 0 in IR mo onto an open neighborhood of M.Iff is a local diffeomorphism then r the r-jet j0 (f)isanr-frameatx = f(0). The set of r-frames of M will be denoted by r r F (M), while the set of r-frames at f(0) forms a group G (mo) with multiplication defined by the composition of jets :

r r r j0 (g1) .j0 (g2)=j0 (g1 og2).

r r The group G (mo)actsonF (M)ontheright:

r r r j0 (f) .j0 (g)=j0 (fog). (37)

r r The Lie algebra of G (mo) will be denoted by g (mo). We mainly will be interested in the bundle of 2-frames on M.Let(xα) be some local coordinates on M andx ¯α IR mo 2 the natural coordinates on . A 2-frame u then is given by u = j0 (f). From 1 f(¯x)=xαe + uαx¯βe + uα x¯βx¯γe , (38) α β α 2 βγ α α α α 2 we get a set of local coordinates (x ,uβ,uβγ)onF (M). α α 2 In a similar way we may use (sβ ,sβγ) as coordinates on G (mo). The multipli- 2 cation in G (mo)isgivenby

α α α α α σ α σ α σ ρ (¯sβ , s¯βγ).(sβ,sβγ)=(¯sσ sβ, s¯σ sβγ +¯sσρsβsγ), (39)

2 2 while the action of G (mo)onF (M)isgivenby

α α α α α α α σ α σ α σ ρ (x ,uβ ,uβγ).(sβ,sβγ)=(x ,uσ sβ,uσsβγ + uσρsβsγ). (40) Let ∂ ∂ (e = ,eα = ⊗ dx¯β) α ∂x¯α β ∂x¯α be a basis for the Lie algebra of affine transformations on IR mo . The canonical one form θ on F 2(M), which we write as

α α β θ = θ eα + θβ eα,

α α is given in local coordinates by (with vβ is the inverse matrix of uβ )[4]:

α α β θ = vβ dx , (41) α α γ − α γ ρ σ θβ = vγ duβ vγ uρβvσdx . (42) 2 2 2 Because the group G (mo)actsonF (M) on the right, with each A ∈ g (mo) ? ∈X 2 2 2 → 1 corresponds a fundamental vectorfield A (F (M). Let π1 : g (mo) g (mo), we have the following proposition [3] : Grassmannian structures on manifolds 613

Proposition 4.1 ? 2 ∈ 2 (1) θ(A )=π1(A) for A g (mo) ? −1 ∈ 2 (2) Raθ = ad(a )θ, a G (mo).

The canonical form satisfies the structure equation [4] :

α − α ∧ β dθ = θβ θ . (43)

B. The Grassmannian bundle Gr(ko,lo)(M) We will now define a subbundle of F 2(M) which is isomorphic with the bundle × ς P . In this section we use the identification IR ko lo = IR mo .

2 Proposition 4.2 The embedding H → G (mo), mo = kolo, defined by (   sα =ς αi βa i a k 7→ β jh b i βj,βb ,βc α ς − a l i a l i (44) sβγ = βb αj γlcαk + βc αkγlbαj ,

− − − i τ −1 i k with α =(a 1)lo + j, β =(b 1)lo + j, γ =(c 1)lo + k and αj = β j ,γkc = βc , 2 is a group morphism. Let H˜ designate image of the embedding in G (mo).

Proof The multiplication in H yields

ˆi ˆi ˆa j j b ˆi j ˆi j ˆi c ˆa b (βj, βc, βb ).(βk,βc ,βc)=(βjβk, βjβb + βcβb, βb βc ). (45)

Let h i α i a α − a l i a l i sβ = αjβb ,sβγ = βb αjγlcαk + βc αkγlbαj and h i α i ˆa α − ˆa l i ˆa l i sˆβ =ˆαjβb , sˆβγ = βb αˆjγˆlcαˆk + βc αkγˆlbαˆj . We find for the multiplication

α α α α i ˆa j b (¯sβ , s¯βγ).(sβ,sβγ)=(ˆαj βb αkβc, h i h i − i ˆa b l i b l j − ˆa l i ˆa l i j b m e αˆjβb βdαmγlcαk + βcαkγldαm βb αˆj γˆleαˆm + βe αmγˆlbαˆj αmβdαk βc ), which proves the group morphism. @A

Definition 4.1 A Grassmannian structure, Gr(ko,lo)(M), on a manifold M is a subbundle of F 2(M) with structure group H˜ .

Proposition [(4.2)] together with some classical results in bundle theory [9] proves the following theorem. 614 P.F. Dhooghe

Theorem 4.1 Let P be a H-bundle over M. Then there exists a Gr(ko,lo)(M), subbundle of F 2(M), which is isomorphic to P .

(ko,lo) Definition 4.2 A (ko,lo)-structure on a manifold, B (M), is a subbundle of 1 F (M) with structure group Go.

Theorem 4.2 Each Grassmannian structure, Gr(ko,lo)(M),onM is the prolon- gation of a (ko,lo)-structure. Moreover this structure has vanishing second prolon- gation.

Proof

(ko,lo) 1 Let B (M) be any subbundle of F (M) with structure group Go. The first prolongation of B(ko,lo)(M) is a subbundle of F 2(M) with structure group the semi mo direct product of Go and the group of automorphisms of V ' IR generated by the (1) Lie algebra g˜ [10]. Hence the first prolongation is a Gr(ko,lo)(M). 2 2 → 1 Let Gr(ko,lo)(M)begivenandπ1 : F (M) F (M) the bundle projection. 2 (ko,lo) Then π1(Gr(ko,lo)(M) is a bundle B (M) whose prolongation co¨ıncides with Gr(ko,lo)(M) by the isomorphism of the structure groups. The second prolongation of a B(ko,lo)(M) vanishes identically [(2.1)]. @A

We refer to S. Sternberg [10] for a detailled exposition of the relationship between connections on G structures and prolongations. In particular the set of adapted symmetric connections is parametrised by the first prolongation of the Lie algebra g˜(1). To make this clear we first need the following lemma on symmetric affine connections.

2 Lemma 4.1 Let Γ:M → F (M)/Gl(mo) be an affine symmetric connection. Then there exists a canonical homomorphism Γ:˜ F 1(M) → F 2(M) canonically associated with Γ.

Proof For a proof we refer to [5]. In local coordinates the map Γ is given by

˜ α α α α α − σ α ρ Γ:¯x = x ;¯uβ = uβ ;¯uβγ = uβΓσρuγ. (46) @A Remark that ˜∗ α α γ γ σ ρ Γ θβ = vγ (duβ +Γρσuβdx ). (47)

(ko,lo) Let B (M)bea(ko,lo) structure on M. An adapted affine symmetric connec- ∗ (ko,lo) → 2 ˜ α tion on B (M)isamapΓ:M F (M)/Gl(mo) such that Γ θβ restricted to B(ko,lo)(M) is a connection form with values in g0.LetΦ(B(ko,lo))(M)betheset of adapted affine symmetric connection and denote the set of associated homomor- phisms by Φ(˜ B(ko,lo))(M). Grassmannian structures on manifolds 615

In order to prove the next proposition we need some local expressions. Let ς × (¯xai =¯xα) be the coordinates on IR mo ' IR ko lo . The Lie algebra of the second order formal vector fields L on this space as given in [(19)] has the following basis

∂ ∂ ∂ ∂ e = ,ei δa + eaδi = δax¯ci + δix¯ak ,eai =¯xajx¯ci . (48) ai ∂x¯ai j b b j b ∂x¯cj j ∂x¯bk ∂x¯cj

In terms of local coordinates on M and taking the identification ς directly into account, a 2-frame is given by h i α α bj α bj ck f(¯x)= x + ubjx¯ + ubjckx¯ x¯ eα. (49)

Let σ be a local section of F 1(M), then σ is given by the functions

7→ α ∗ α σ :(x) Ebj(x)=σ ubj. (50)

The fundamental form along σ becomes

¯ai ∗ ai ai β θ = σ θ = Fβ (x)dx , (51) while the connection form with respect to a given Γ˜ ∈ Φ(B(ko,lo)(M)is

¯ai ∗ ai ai α ai α σ ρ θbj = σ θbj = Fα dEbj + Fα ΓρσEbjdx . (52)

¯ai The form θbj satisfies the structure equation [(43)]

¯ai −¯ai ∧ ¯bj dθ = θbj θ .

ˆai Let θbj be a second connection form with respect to a different morphism belonging to Φ(B(ko,lo)(M). This form satisfies the same equation [(43)]. Hence we find

¯ai − ˆai ∧ ˆbj 0=(θbj θbj ) θ . (53)

¯ai − ˆai → 0 ⊂ ⊗ ∗ ∈ The difference (θbj θbj ) defines a morphism V g V V at each x M, satisfying [(53)] and hence defines an element in g(1). This implies that at x ∈ M :

¯ai − ˆai a i a i θbj θbj = ubkδc δj + ucj δb δk, (54)

i ∈ with ua M(lo,ko).

Proposition 4.3 Any two adapted affine symmetric connections on B(ko,lo)(M) are locally related by : 0 γ − γ γ ck bj Γασ Γασ =2ubkEcj F(α Fσ). (55) with ubk an element of M(lo,ko). 616 P.F. Dhooghe

Proof

We know that any connection form on B(ko,lo)(M) takes values in g0. Hence

ai a i i a θbjα = θbαδj + θjαδb .

We find along the section σ :

∂ θ¯a Eα δi + θ¯i Eα δa = F ai( Eγ )Eα + F aiΓγ Eα Eσ . bα ck j jα ck b γ ∂xα bj ck γ ασ ck bj

From the theorem [(2.1)] and equation [(54)] it follows that for any two of such connection forms there exists an element ubk such that

a i a i ai 0 γ − γ α σ ubkδc δj + ucjδb δk = Fγ (Γασ Γασ)EckEbj.

Hence h i γ a i a i ck bj 0 γ − γ Eai ubkδc δj + ucj δb δk Fα Fσ =Γασ Γασ. @A

(1) Because the first prolongation g˜ can be identified with M(lo,ko) this describes the parametrisation of the set of adapted connections. This allows us to formulate the following theorem.

(ko,lo) Theorem 4.3 Let B (M) be a (ko,lo)-structure on M.Theset

{Γ(˜ B(k,l)(M)) | Γ˜ ∈ Φ(˜ B(k,l))(M)} (56) forms a Grassmannian structure on M.

Consequences :

(1) Each Gr(ko,lo)(M) is locally determined by a section

Γ˜ oσ : M → F 2(M) where Γ˜ ∈ Φ(˜ B(ko,lo))(M)andσ a section M → B(ko,lo)(M). 2 (2) The set of Gr(ko,lo)(M) bundles is given by F (M)/H. Each local section Γ˜ oσ determines locally an element of F 2(M)/H. (ko,lo) (3) Each Gr(ko,lo)(M) is equivalent with a B (M) together with its set of adapted connections. As alternative formulation of former theorem we have :

Theorem 4.4 Each Gr(ko,lo)(M) is locally uniquely defined by a section σ : M → F 1(M) and an identification IR mo =ς IR kolo .

C. The normal Grassmannian connection coefficients Grassmannian structures on manifolds 617

We will now investigate the coefficients of a normal Grassmannian connection in terms of an adapted frame and give an expression of the normal Grassman- nian curvature tensor. Let Gr(ko,lo)(M) be a Grassmannian structure defined as 2 α α a subbundle of F (M). Let θ ,θβ be the fundamental and the connection form on

mo ς ko×lo Gr(ko,lo)(M). Because of the identification IR = IR we write these forms as ai i a i − a (θ ,θj,θb )withkoθi loθa = 0 in order to fix their uniqueness in the decomposition. We then define on Gr(ko,lo)(M)

1 1 1 1 a ai i − τ i τ k i a a − c a − i − a ωi = θ ; ωj = θj + θk δj ; ωb = θb θcδb ; ω∗ = θi θa. (57) lo ko lo ko

As a consequence of theorem [(3.1)] there exists a unique normal connection form a i a i ω =(ωi ,ωj ,ωb ,ω∗,ωa)onGr(ko,lo)(M).

(ko,lo) Theorem 4.5 Let M be a manifold equipped with a (ko,lo) structure B (M) U⊂ a α and M an open subset carrying an adapted coframe Fiαdx .Letfurther (ko,lo) Gr(ko,lo)(M) be the Grassmannian structure on M determined by B (M) and

a i a i ω =(ωi ,ωj,ωb ,ω∗,ωa) the normal Cartan connection. Then there exists a unique local section ν : U→Gr(ko,lo)(M) determined by the conditions ∗ a α a α ∗ ν ωiαdx = Fiαdx ,νω∗ =0. (58)

Proof

Any section ν may be decomposed into a section σ of B(ko,lo)(M) and a section ∗ (ko,lo) → a α a α ϑ : B (M) Gr(ko,lo)(M). The requirement ν ωiαdx = Fiαdx implies ∗ a α a α ˜ 1 → σ ωiαdx = Fiαdx , which determines the section σ.LetΓ be a morphism F (M) F 2(M) defined by an adapted symmetric connection. Using proposition [(4.3)] and expression [(48)] the map ϑ can be written as

α − σ α α ck bj ρ uβγ = uβ[Γσρ +2ubkucjv(σvρ)]uγ, with ubk a function on U.Oralso

α − σ α α ck bj ρ ubj ai = Ebj [Γσρ +2ubkEcjF(σ Fρ) ]Eai,

α ai with Eai the local frame dual to the coframe Fα . α 1 ∗ α We remark that θ = − ω∗. The calculation of (ϑoσ) θ = 0 yields, with the α kolo α use of expression [(42)], the equation

ai γ γ ρ β ck bj ρ Fγ dEai +Γργdx +2ubkEcj F(ρ Fβ)dx =0 or 1 u F ckdxρ = − [F aidEγ +Γγ dxρ]. ck ρ 2 γ ai ργ 618 P.F. Dhooghe

The unicity follows from the same calculations. Any two morphisms of B(ko,lo)(M) into F 2(M) indeed are, as a consequence of proposition [(4.3)], defined by affine con- nections on B(ko,lo)(M) which are related by 0 γ − γ γ ck bj Γασ Γασ =2ubkEcj F(α Fσ). A simple substitution then yields the unicity. @A

The theorem allows us to introduce the normal Grassmannian connection coef- ficients. We set

∗ i i α ∗ a a α ∗ i i α σ ωj =Πjαdx ,σωb =Πbαdx ,σωa =Πaαdx . (59)

a α iα ∂ Dual to the coframe Fiαdx we define the frame Ea ∂xα by the conditions

a jα a j FiαEb = δb δi . (60) From equation [(22)(5)] we find ∗ a ∧ ∗ i σ ωi σ ωa =0 or a i − a i FiαΠaβ FiβΠaα =0. i kβ ik Define ΠaβEc =Πac. The former equation becomes

kl − lk Πcd Πdc =0. (61) Let i i i ∧ k Lj = dωj + ωk ωj (62) and a a a ∧ c Lb = dωb + ωc ωb. (63) The equations [(22)(2) and (3)] become

1 Ki = Li + (Πi F b − Πi F b ) jαβ jαβ 2 bα jβ bβ jα 1 Ka = La + (Πa F i − Πa F i ). (64) bαβ bαβ 2 iα bβ iβ bα Using the notations i kα jβ ikj LlαβEc Eb = Llcb (65) and a kα jβ akj LdαβEc Eb = Ldcb , (66) we find 1 Kikl = Likl + (Πikδbδl − Πil δbδk) jcd jcd 2 bc d j bd c j 1 Kakl = Lakl + (Πmlδaδk − Πmkδaδl ). (67) bcd bcd 2 bd c m bc d m Grassmannian structures on manifolds 619

From the condition ilk − dki Klba Kadb =0 we obtain k + l Lilk − Ldki − o o Πki +Πik =0. lba adb 2 ab ab This gives

2 h i Πki = (k + l )(Lilk − Ldki )+2(Lkli − Ldik ) . (68) ab 2 o o lba adb lba adb (ko + lo) − 4

Let M be equipped with an adapted symmetric affine connection on B(ko,lo)(M). jk dk We define the coefficients (γlc ,γbc )by

j di j ji l ∇ i EaEb = γbaEd + γlaEb, ik − dk together with koγic loγdc =0. A Grassmannian related covariant derivation is defined as h  j di i d j ∇˜ i Ea Eb = γba + ubδa δl   i ji j i d l + γla + uaδl δb Ed. (69)

Or j j i j j i ∇˜ i ∇ i EaEb = EaEb + ubEa + uaEb. (70) Using this expression we find

Proposition 4.4 Let X, Y ∈X(M), ∇ and ∇˜ be two adapted connections on the bundle B(ko,lo)(M).Letfurtherσ : U→B(k,l)(M) be a local section and σ¯(x) the corresponding identification of the tangent space Tx(M) at x ∈Uwith M(ko,lo). Then there exists a map µ : U→M(lo,ko) such that

−1 ∇˜ XY = ∇XY +¯σ [(µ.σ¯(X)) . σ¯(Y )+(µ.σ¯(Y )) . σ¯(X)]. (71)

Because µ ∈ M(lo,ko)and¯σ ∈ M(ko,lo) the composition (µ.σ¯(X)(x)) is an element of M(lo,lo) which acts onσ ¯(Y )(x) by composition, giving thus an element of M(ko,lo). Remark We can define the (2, 1)-tensorfield

µ˜ =¯σ−1.µ.σ.¯

The Grassmannian relationship of two symmetric affine adapted connections is then given by ∇ˆ XY = ∇XY +˜µ(X)(Y )+˜µ(Y )(X).

We define the splitting of the coefficients γ into the trace free parts and the trace ak ik k ak ik part as (¯γbc , γ¯jc, γ¯∗ c), with (¯γac =¯γic = 0). A Grassmannian related covariant derivation is then given by 620 P.F. Dhooghe

  j di i d 1 i d j ∇˜ i E = γ¯ + u δ − u δ δ Ea b ba b a a b l ko   1 ji j i − i j d + γ¯la + uaδl uaδl δb lo ! # i ko + lo i d j l + γ¯∗ a + ua δb δl Ed (72) kolo with

i 1 ji ci γ¯∗ a = (koγja + loγca). (73) kolo The normal Cartan connection is defined by the requirement

1 i − ji ci ua = (koγja + loγca) ko + lo and the coefficients of this connection are

1 ji − ji − j j i j Πla = γ¯la uaδi + uaδl lo di di i d − 1 i d Πba =¯γba + ubδa uaδb . (74) ko

We now are able to investigate the Grassmannian curvature tensor. Because the 2 bundle Gr(ko,lo)(M) is a subbundle of F (M) the restriction of the homomorphism 2 2 → 1 π1 : F (M) F (M)toGr(ko,lo)(M) is the homomorphism :

(ko,lo) η : Gr(ko,lo)(M) → B (M). (75) ∗ The fibres of η are isomorphic to the kernel M of the homomorphism H → Go. i a The following theorem proves that the curvature forms Ωj and Ωb are defined on the bundle B(k,l)(M) ⊂ F 1(M).

Proposition 4.5 Let Gr(ko,lo)(M) be a Grassmannian structure equipped with a i a normal Grassmannian connection. Then the curvature forms Ωj and Ωb satisfy the following conditions. Let A∗ be a fundamental vectorfield with A ∈ g1. Then (1) i a L ∗ L ∗ A (Ωj)= A (Ωb )=0. (76) (2) The tensor α i a jα b a iα Kβγσ = KjγσFiβEa + KaγσFiβEb (77) is a (1, 3)-tensorfield on M, which we call the Grassmannian curvature tensor. Grassmannian structures on manifolds 621

Proof The relations (1) are a direct consequence of the equations [(36)], while (2) is a con- sequence of the fact that B(ko,lo)(M) is a subbundle of F 1(M) together with proposi- i i α ⊗ β a α ⊗ β tion [(3.1)]. Writing the curvature forms as Ωj = Kjαβdx dx and Ωbαβdx dx , jα Eb , the Grassmannian curvature tensorfield is defined as h i α i b b i a jα Kβγσ = Kjγσδa + Kaγσδj FiβEb which is equivalent with [(77)]. @A We call a Grassmannian structure on M locally flat if the structure has vanishing structure constants, which means that the structure is locally isomorphic with a flat structure [2]. The flat structure here means the structure of a Grassmannian. As a consequence of proposition [(3.7)] and because the dimension of the manifold admitting a Grassmannian structure is larger than 3, we have

Theorem 4.6 A Grassmannian structure on M is locally flat iff the Grassmannian curvature equals zero.

References

[1] Guillemin V.W., Sternberg S., ‘An Algebraic Model of Transitive Differential Geometry’, Bull. Am. Math. Soc., 70, 1964, 16-47

[2] Guillemin V.W., ‘ The integrability problem for G-structures’,Trans.Am. Math. Soc., 116, 4, 1965, 544-560

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[4] Kobayashi S., ‘Canonical Forms on Framebundles of higher order contact’ , Proc. Symp. Pure Math., Vol. 3., Amer. Math. Soc., 1961, 186-193

[5] Kobayashi S., Nomizu K., ‘Foundations of Differential Geometry, Vol I, Inter- science Publ., 1963, J. Wiley & Sons, New York - London

[6] Manin Y.I., ‘Gauge Field Theory’ , Springer Verlag, Berlin New York, 1988

[7] Matsushima Y., ‘Sur les alg`e bres de Lie lineaires semi-involutives’, Colloque de topologie de Strassbourg (1954)

[8] Singer I.M., Sternberg S, ‘On the infinite groups of Lie and Cartan’, Journal d’Analyse Math., XV, 1965, 306-445

[9] Steenrod N., ‘The Topology of Fibrebundles’, Princeton Univ. Press, 1951, Princeton 622 P.F. Dhooghe

[10] Sternberg S., ‘Lectures on Differential Geometry’, Prentice Hall, 1965, N.Y.

[11] Tanaka N., ‘Projective connections and projective transfomations, Nagoya Math. J., 12, 1957, 1-24

P.F. Dhooghe DEPARTMENT OF MATHEMATICS, KULeuven, Celestijnenlaan, 200B, B 3001 LEUVEN, BELGIUM Email : paul dhooghe@[email protected]