Affine immersions in homogeneous spaces

Consider the isotropic representationAd :H → GL(m), the group isomorphism I_i : GL(Rⁿ)→GL(m)defined byI_i(τ) =i◦τ◦i⁻¹, for allτ ∈GL(m)and set:

Gⁱ =I_i⁻¹ Ad(H)

⊂GL(Rⁿ).

Consider the smooth left action ofAon FR T(A/H)

defined as in (3.2.3) and let Pⁱ ⊂ FR T(A/H)

be the A-orbit of i. ThenPⁱ is aGⁱ-structure on A/H (see Exercise 1.73). The groupGⁱ_¯1of allGⁱ-structure preserving endomorphisms ofT¯1(A/H) =mis justAd(H); its Lie algebragⁱ_¯1 is thus equal toad(h).

We will now determine the inner torsionI^P_¯1ⁱ :m → gl(m)/ad(h)of theGⁱ -structurePⁱ onA/H. By the result of Exercise 2.24, we may as well compute the inner torsion of theG-structure:

(3.3.4) P =γ_i⁻¹(Pⁱ) =

dLg :g∈A ⊂FRm T(A/H) ,

whereG=I_i(Gⁱ) = Ad(H). Notice thatP is just the image of the morphism of principal spacesφdefined in (3.2.15).

PROPOSITION 3.3.3. Let ∇ be an A-invariant connection on A/H corre-sponding to a linear mapλ, as in Proposition 3.2.3. The inner torsionI^P_¯ⁱ

1 :m→ gl(m)/ad(h) of theAd(H)-structure(3.3.4)onA/H is equal to the composition ofλ:m→gl(m)with the quotient mapgl(m)→gl(m)/ad(h).

PROOF. Lets : U → Abe a smooth local section of the quotient mapA → A/H with¯1∈U,s(¯1) = 1andds¯1(m) =m; notice thatds¯1is just the inclusion map of min a. Notice thatφ◦sis a smooth local section of P → A/H. Set

¯

ω = (φ◦s)^∗ω, whereω denotes the connection form of∇. By diagram (2.10.2), in order to conclude the proof, it suffices to show thatω¯¯1 is equal toλ. We have

¯

ω=s^∗(φ^∗ω)and therefore:

¯

ω¯1 = (φ^∗ω)₁◦ds¯1.

The conclusion follows directly from (3.2.17).

3.4. Affine immersions in homogeneous spaces

LetM be ann-dimensional differentiable manifold,Gbe a Lie subgroup of GL(Rⁿ) and assume thatM is endowed with a connection∇and aG-structure P ⊂FR(T M). For eachx∈M we denote byGxthe Lie subgroup ofGL(TxM) consisting ofG-structure preserving endomorphisms of T_xM, byg_x ⊂gl(T_xM) the Lie algebra ofGxand byI^P_x :TxM → gl(TxM)/gxthe inner torsion of the G-structureP (recall Section 1.8). The triple (M,∇, P)will be called an affine manifold withG-structure. Given pointsx, y ∈ M and aG-structure preserving map σ : TxM → TyM then the Lie group isomorphism I_σ : GL(TxM) → GL(T_yM)defined by:

I_σ : GL(T_xM)3T 7−→σ◦T◦σ⁻¹ ∈GL(T_yM)

carriesGx ontoGy. Its differential at the identityAdσ : gl(TxM) → gl(TyM) carriesg_xontog_y and therefore it induces a linear isomorphism

Ad_σ :gl(T_xM)/g_x −→gl(T_yM)/g_y.

DEFINITION3.4.1. LetV,V⁰be real vector spaces andσ :V →V⁰be a linear isomorphism. Given a multilinear map B⁰ ∈ Link(V⁰, V⁰) then thepull-back of B⁰ byσis the multilinear mapσ^∗B ∈Lin_k(V, V)defined by:

(σ^∗B)(v1, . . . , vk) =σ⁻¹

B σ(v1), . . . , σ(vk) ,

for allv1, . . . , v_k∈V. Given multilinear mapsB∈Lin_k(V, V),B⁰∈Lin_k(V⁰, V⁰) and a (not necessarily invertible) linear mapσ : V → V⁰ thenB is said to be σ-relatedwithB⁰if:

(3.4.1) B⁰ σ(v1), . . . , σ(vk)

=σ B(v1, . . . , vk) ,

for allv1, . . . , vk ∈ V. More generally, ifV1, . . . , Vk are subspaces of V and if B ∈ Lin(V1, . . . , V_k;V),B⁰ ∈ Lin_k(V⁰, V⁰)are multilinear maps thenB is said to beσ-relatedwithB⁰ if (3.4.1) holds for allv₁∈V₁, . . . ,v_k ∈V_k.

Clearly, if σ : V → V⁰ is a linear isomorphism and ifB⁰ ∈ Lin_k(V⁰, V⁰) then σ^∗B⁰ is the only multilinear mapBinLink(V, V)that isσ-related withB⁰.

DEFINITION3.4.2. LetM be ann-dimensional differentiable manifold,M be ann-dimensional differentiable manifold and let¯ π :E → M be a vector bundle overM with typical fiberR^k, wheren¯ = n+k. SetEb = T M ⊕E, so thatEb is a vector bundle overM with typical fiberR^n¯. Let∇b and∇be connections on Eb and onT M respectively. By anaffine immersionof(M, E,∇)b into the affine manifold(M ,∇) we mean a pair (f, L), wheref : M → M is a smooth map, L:Eb→f^∗T M is a connection preserving vector bundle isomorphism and:

(3.4.2) L_x|_TxM = df_x,

for allx ∈ M, wheref^∗T M is endowed with the connection f^∗∇. By a local affine immersionof(M, E,∇)b into (M ,∇)we mean an affine immersion(f, L) of(U, E|_U,∇)b into(M ,∇), whereUis an open subset ofM; we callUthedomain of the local affine immersion(f, L).

Observe that if(f, L)is a (local) affine immersion, condition (3.4.2) implies thatf is an immersion.

There exists in the literature a notion of affine immersion between affine man-ifolds (see [11, Definition 1.1, Chapter II]). Using our terminology, such notion of affine immersion is:

DEFINITION 3.4.3. Given affine manifolds (M,∇), (M ,∇) then a smooth mapf :M →Mis said to be anaffine immersionof(M,∇)into(M ,∇)if there exists a vector bundleπ :E→M, a connection∇b onEb=T M⊕Eand a vector bundle isomorphismL : Eb → f^∗T M such that(f, L)is an affine immersion of

3.4. AFFINE IMMERSIONS IN HOMOGENEOUS SPACES 179

(M, E,∇)b into(M ,∇)and such that∇is a component of∇b with respect to the decompositionEb=T M⊕E, i.e.:

∇_XY = pr₁(∇b_XY),

for allX, Y ∈Γ(T M), wherepr₁ :Eb →T M denotes the first projection.

LEMMA 3.4.4. Fix objectsM,M, π : E → M, E,b ∇b and∇as in Defini-tion 3.4.2. Let s : U → FR(E)b be a smooth local frame of E,b f : U → M be a map and L : E|b _U → f^∗T M be a bijective fiberwise linear map. Define F :U →FR(T M)by setting:

(3.4.3) F(x) =Lx◦s(x)∈FR(T_f(x)M),

for allx ∈U. Denote byω^M the connection form onFR(T M)corresponding to the connectionHor FR(T M)

associated to∇and byω^M the connection form onFR(E)b corresponding to the connectionHor FR(E)b

associated to∇. Denoteb also by θ^M the canonical form ofFR(T M) and byθ^M theι-canonical form of FR(E), whereb ι : T M → Eb denotes the inclusion map. Then(f, L)is a local affine immersion with domainU if and only if the mapF is smooth and:

F^∗θ^M =s^∗θ^M, (3.4.4)

F^∗ω^M =s^∗ω^M. (3.4.5)

PROOF. Denote by L∗ : FR(E)b → FR(f^∗T M) = f^∗FR(T M) the map induced by L and by f¯ : f^∗FR(T M) → FR(T M) the canonical map of the pull-backf^∗FR(T M). Clearly:

(3.4.6) F = ¯f◦L∗◦s.

We claim thatF is smooth if and only if bothf andLare smooth. Namely, if both f andLare smooth then equality (3.4.6) implies thatF is smooth. Conversely, if F is smooth thenf is also smooth, sincef = Π◦F, whereΠ : FR(T M) →M denotes the projection. Moreover,F is a local section ofFR(T M)alongf and:

L∗◦s=←− F ,

so thatL∗◦sis smooth by Corollary 1.3.19. Sincesis an atlas of local sections for the principal bundleFR(E)|b _U, it follows from the result of Exercise 1.45 that L∗ : FR(Eb)|_U → FR(f^∗T M) is a (smooth) isomorphism of principal bundles whose subjacent Lie group homomorphism is the identity map ofGL(Rⁿ). Hence Lis smooth by Lemma 1.5.18.

Now, assuming that F, f and L are smooth, we prove that Lis connection preserving if and only if (3.4.5) holds. Recall from (c) of Lemma 2.5.10 thatLis connection preserving if and only ifL∗ : FR(E)b → FR(f^∗T M) is connection preserving. By definition, the connection form of the pull-back FR(f^∗T M) =

f^∗FR(T M)is equal tof¯^∗ω^M; thus, by part (d) of Lemma 2.2.11,L∗is connection preserving if and only if:

(3.4.7) (L∗◦s)^∗( ¯f^∗ω^M) =s^∗ω^M. But (3.4.7) is obviously the same as (3.4.5), by (3.4.6).

Finally, let us prove thatL_x|_TxM = df_x for allx ∈ U if and only if (3.4.4) holds. Using (2.9.12), we see that (3.4.4) holds if and only if:

(3.4.8) F(x)⁻¹◦dΠ_F(x)◦dF_x=s(x)⁻¹◦ι_x,

for allx∈U. SinceΠ◦F =f, we see that (3.4.8) holds if and only if:

(3.4.9) F(x)⁻¹◦dfx =s(x)⁻¹|_TxM,

for allx∈U. Finally, sinceF(x) =L_x◦s(x), it is clear that (3.4.9) holds if and only ifLx|_TxM = dfx. This concludes the proof.

COROLLARY 3.4.5 (uniqueness of affine immersions with initial data). Let M, M, π : E → M, E,b ∇b and ∇be as in Definition 3.4.2; assume that M is connected. If (f¹, L¹),(f², L²) are both affine immersions of(M, E,∇)b into (M ,∇)and if there existsx₀∈M with:

f¹(x₀) =f²(x₀), L¹_x0 =L²_x0, then(f¹, L¹) = (f², L²).

PROOF. Denote byf¯ⁱ : (fⁱ)^∗T M →T M the canonical map of the pull-back (fⁱ)^∗T M,i= 1,2. Clearly(f¹, L¹) = (f², L²)if and only if the maps:

(3.4.10) f¯¹◦L¹:M −→T M , f¯²◦L² :M −→T M

are equal. The set of points ofM where the maps (3.4.10) coincide is obviously closed and, by our hypotheses, nonempty. Let us check that such set is also open.

Letx ∈M be a point at which the maps (3.4.10) coincide. Lets :U → FR(E)b be a smooth local frame ofEbwhereU is a connected open neighborhood ofxin M. Fori= 1,2, defineFⁱ :U →FR(T M)by settingFⁱ(y) =Lⁱ_y◦s(y), for all y ∈ U. ThenF¹(x) =F²(x)and Lemma 3.4.4 implies thatFⁱ is a smooth map satisfying:

(Fⁱ)^∗(θ^M, ω^M) = (s^∗θ^M, s^∗ω^M), fori= 1,2. Since for eachp∈FR(T M), the linear map:

(θ^M_p , ω_p^M) :TpFR(T M)−→R^¯n⊕gl(R^n¯)

is an isomorphism (recall (2.11.7)) then Lemma A.4.9 implies that F¹ = F². Hence the maps (3.4.10) coincide inU and we are done.

DEFINITION3.4.6. An affine manifold withG-structure(M,∇, P)is said to beinfinitesimally homogeneousif for allx, y∈ M and allG-structure preserving mapσ :T_xM →T_yM, the following conditions hold:

• Ad_σ◦I^P_x =I^P_y ◦σ;

• T_xisσ-related withT_y;

3.4. AFFINE IMMERSIONS IN HOMOGENEOUS SPACES 181

• Rxisσ-related withRy.

The condition of infinitesimal homogeneity means that curvature, torsion and inner torsion are constant with respect to frames that are in theG-structure. This statement is made more precise in the following:

LEMMA 3.4.7. Let (M,∇, P) be an n-dimensional affine manifold with G-structure, whereGis a Lie subgroup ofGL(Rⁿ). Then(M,∇, P)is infinitesimally homogeneous if and only if there exists multilinear maps R0 ∈ Lin3(Rⁿ,Rⁿ), T₀∈Lin₂(Rⁿ,Rⁿ)and a linear mapI₀ :Rⁿ→gl(Rⁿ)/gsuch that:

(3.4.11) p^∗Rx=R0, p^∗Tx=T0, Adp◦I₀ =I^P_x ◦p, for allx∈M and allp∈Px.

PROOF. Assume the existence ofR₀, T₀, I₀ such that (3.4.11) holds for all x ∈ M and allp ∈ Px. Let x, y ∈ M and a G-structure preserving map σ : T_xM → T_yM be fixed. Choose anyp ∈ P_x and setq = σ◦p, so that q ∈ P_y. Then:

p^∗Rx =R0=q^∗Ry =p^∗σ^∗Rx,

and thenRx =σ^∗Ry, i.e.,Rxisσ-related withRy. Similarly,Txisσ-related with T_y. MoreoverAd_p◦I₀ =I^P_x ◦p,Ad_q◦I₀=I^P_y ◦qand therefore:

I^P_y ◦σ◦p=I^P_y ◦q= Adq◦I₀ = Adσ◦Adp◦I₀= Adσ◦I^P_x ◦p, provingAd_σ◦I^P_x =I^P_y ◦σ. Conversely, assume that(M,∇, P)is infinitesimally homogeneous. Choose anyx∈M and anyp∈P_xand set:

R₀ =p^∗R_x, T₀=p^∗T_x, I₀ = (Ad_p)⁻¹◦I^P_x ◦p.

Given anyy ∈ M, q ∈ Py thenσ = q◦p⁻¹ : TxM → TyM is aG-structure preserving map and thereforeσ^∗Ry = Rx,σ^∗Ty =Tx andAdσ ◦I^P_x = I^P_y ◦σ.

Then:

q^∗R_y =p^∗σ^∗R_y =p^∗R_x =R₀, q^∗T_y =p^∗σ^∗T_y =p^∗T_x =T₀; moreover:

Ad_q◦I₀ = Ad_q◦(Ad_p)⁻¹◦I^P_x ◦p= Ad_σ◦I^P_x ◦p=I^P_y ◦σ◦p=I^P_y ◦q.

This concludes the proof.

Roughly speaking, an affine manifold withG-structure(M,∇, P)is infinites-imally homogeneous if one can describe the inner torsionI^P, the torsion tensorT and the curvature tensorRby formulas that involve only theG-structure. A bet-ter understanding of this statement can be obtained by considering the following examples.

EXAMPLE3.4.8. Let(M, g)be ann-dimensional semi-Riemannian manifold withn−(g) =rhavingconstant sectional curvaturec∈R. This means that:

g_x R_x(v, w)v, w

=c g_x(v, w)²−g_x(v, v)g_x(w, w) ,

for allx ∈ M and allv, w ∈ TxM, whereRdenotes the curvature tensor of the Levi-Civita connection ∇of (M, g). It is well-known (see Exercise 3.8) that if (M, g)has constant sectional curvaturecthen the curvature tensorRis given by:

(3.4.12) Rx(v, w)u=c gx(w, u)v−gx(v, u)w , for allx∈M and allv, w, u∈T_xM.

IfP = FR^o(T M)is theOr(Rⁿ)-structure onMconsisting of all orthonormal frames then the triple(M,∇, P)is infinitesimally homogeneous. Namely,I^P = 0, T = 0and formula (3.4.12) says that the curvature tensorRis constant on frames that belong to theG-structure (the curvature tensorRcan be described using only theG-structureP, that can be identified with the metric tensorg). In this situation, the multilinear mapsR₀,T₀,I₀of Lemma 3.4.7 are given byT₀ = 0,I₀ = 0and:

R₀ :Rⁿ×Rⁿ×Rⁿ3(v, w, u)7−→ hw, uiv− hv, uiw∈Rⁿ, whereh·,·idenotes the Minkowski bilinear form of indexrinRⁿ.

EXAMPLE3.4.9. LetAbe ann-dimensional Lie group and∇be a left invari-ant connection onA, i.e., the left translations ofAare affine maps. Denote byathe Lie algebra ofA. The connection∇is determined by a linear mapΓ :a→Lin(a) and it is given by:

(3.4.13) ∇_vX =g dXeg(v) + Γ(g⁻¹v)·X(g)e ,

for all g ∈ A, v ∈ TgA and all X ∈ Γ(T A), where X(g) =e g⁻¹X(g). The curvature tensor of∇at1∈Ais easily computed as:

R₁(X, Y) = [Γ(X),Γ(Y)]−Γ [X, Y] ,

for all X, Y ∈ a. Choose any linear isomorphism p0 : Rⁿ → a. Consider the global smooth sections:A→FR(T A)defined by:

(3.4.14) s(g) = dL_g(1)◦p₀ ∈FR(T_gA),

for all g ∈ A, where Lg : A → A denotes left translation by g. Then P = s(A) is a G-structure on A with G = {Id_Rⁿ}. Since the left translations of A are affine G-structure preserving diffeomorphisms, it follows that(A,∇, P)is a homogeneous (and infinitesimally homogeneous) affine manifold withG-structure.

The Christoffel tensor of∇with respect tosis given by:

TgA3v7−→dLg(1)◦Γ(g⁻¹v)◦dLg(1)⁻¹ ∈Lin(TgA)

for allg ∈ A. The inner torsionI^P coincides with the Christoffel tensor (Exam-ple 2.11.2).

EXAMPLE3.4.10. Let(M₁, g¹),(M₂, g²)be semi-Riemannian manifolds with dim(Mi) =ni,n−(gⁱ) =ri,i= 1,2. Assume that(Mi, gⁱ)has constant sectional curvaturec_i ∈ R,i = 1,2. Consider the productM = M₁×M₂endowed with the metricgobtained by taking the orthogonal sum ofg¹andg², i.e.:

g_(x1,x2) (v₁, v₂),(w₁, w₂)

=g_x¹₁(v₁, w₁) +g²_x2(v₂, w₂),

3.4. AFFINE IMMERSIONS IN HOMOGENEOUS SPACES 183

for all x1 ∈ M1, x2 ∈ M2, v1, w1 ∈ Tx1M1 and all v2, w2 ∈ Tx2M2. The curvature tensorRof the Levi-Civita connection ∇of(M, g) is given by (recall (3.4.12)):

(3.4.15) R_(x1,x2) (v₁, v₂),(w₁, w₂)

(u₁, u₂)

=c1 g¹_x1(w1, u1)v1−g¹_x1(v1, u1)w1

+c₂ g_x²₂(w₂, u₂)v₂−g²_x2(v₂, u₂)w₂ , for allx₁ ∈M₁,x₂∈M₂,u₁, v₁, w₁∈T_x1M₁and allu₂, v₂, w₂ ∈T_x2M₂. Set:

P = FR^o T M;Rⁿ¹⊕ {0}ⁿ²,pr^∗₁(T M₁) ,

where pr₁ : M → M₁ denotes the first projection and Rⁿ¹⁺ⁿ² = Rⁿ¹ ⊕Rⁿ² is endowed with the orthogonal sum of the Minkowski bilinear forms of indexes r₁ andr₂. More explicitly, for all(x₁, x₂) ∈ M, P_(x1,x2) is the set of all linear isometriesp :Rⁿ¹⁺ⁿ² → T_(x1,x2)M such thatp Rⁿ¹ ⊕ {0}ⁿ²

=Tx1M1⊕ {0}

and (automatically)p {0}ⁿ¹⊕Rⁿ²

={0} ⊕Tx2M2. ThenP is aG-structure on M with:

G= O Rⁿ¹⁺ⁿ²;Rⁿ¹ ⊕ {0}ⁿ²∼= Or1(Rⁿ¹)×Or2(Rⁿ²).

We claim that (M,∇, P) is infinitesimally homogeneous. Since ∇ is compati-ble with g and the covariant derivative of sections of pr^∗₁(T M₁) are sections of pr^∗₁(T M₁), it follows from Example 2.11.5 that the inner torsionI^P is zero. More-over, the torsion of ∇is zero and formula (3.4.15) implies thatR is constant on frames that belong to theG-structureP.

EXAMPLE3.4.11. Let(M, g)be a semi-Riemannian manifold and letJbe an almost complex structure onM such thatJ_xis anti-symmetric with respect tog_x, for allx∈M. Assume thatJ is parallel with respect to the Levi-Civita connection

∇. Then (M, g, J) is called a semi-K¨ahler manifold; when g is a Riemannian metric, we call(M, g, J) aK¨ahler manifold. We say that(M, g, J)hasconstant holomorphic curvature c∈Rif:

gx

Rx v, J(v) v, J v

=−cg_x(v, v)²,

for all x ∈ M and all v ∈ TxM. It is well-known (see Exercise 3.9) that if (M, g, J)has constant holomorphic curvaturecthen the curvature tensorRis given by:

(3.4.16) Rx(v, w)u=−₄^c

gx(v, u)w−gx(w, u)v−gx v, Jx(u) Jx(w) +g_x w, J_x(u)

J_x(v)−2g_x v, J_x(w) J_x(u)

, for allx∈M and allv, w, u∈T_xM. If(M, g, J)is a semi-K¨ahler manifold with constant holomorphic curvature and ifP = FR^u(T M)then(M,∇, P)is infinites-imally homogeneous. Namely, the inner torsionI^P is zero (Example 2.11.8), the torsion is zero and formula (3.4.16) shows thatRis constant in frames that belong toP.

EXAMPLE3.4.12. Let(M, g)be ann-dimensional semi-Riemannian manifold where g has index r and let ξ ∈ Γ(T M) be a smooth vector field on M with gx ξ(x), ξ(x)

= 1, for allx∈M. Let us endowRⁿwith the Minkowski bilinear formh·,·iof indexr; denote bye₁, . . . ,e_nthe canonical basis ofRⁿ. Assume that there exists a trilinear map R0 : Rⁿ×Rⁿ×Rⁿ → Rⁿand a linear map L0 : Rⁿ → Rⁿsuch that for everyx ∈ M and every linear isometryp :Rⁿ → TxM withp(e₁) =ξ(x), the following conditions holds:

(a) R₀isp-related withR_x; (b) p◦L0 = (∇ξ)_x◦p.

SetP = FR^o(T M;e1, ξ), so thatP is aG-structure onM withG = O(Rⁿ;e1) (Example 2.11.6). Then(M,∇, P)is infinitesimally homogeneous. Namely, this follows from Lemma 3.4.7, keeping in mind that, since∇is compatible with g, the inner torsionI^P can be identified with∇ξ (Example 2.11.6). It will also be interesting to consider the case whereMis oriented and (a) and (b) above hold only for orientation preserving linear isometriesp:Rⁿ→T_xM withp(e₁) =ξ(x). In this case, one considers the open subset ofP consisting of orientation preserving frames, which is a principal bundle with structural group:

T ∈O(Rⁿ;e1) : det(T) = 1 .

Interesting examples of Riemannian manifolds satisfying the conditions above are the homogeneous3-dimensional Riemannian manifolds with an isometry group of dimension4(see, for instance, [7]).

DEFINITION 3.4.13. Fix objects M, M, π : E → M, E,b ∇b and ∇ as in Definition 3.4.2. LetGbe a Lie subgroup ofGL(R^¯n)and assume thatEbandT M are endowed withG-structuresPbandP, respectively. A (local) affine immersion (f, L) of (M, E,∇)b into (M ,∇) is said to be G-structure preserving if L is a G-structure preserving isomorphism of vector bundles, wheref^∗T M is endowed with theG-structuref^∗P (recall Example 1.8.3).

THEOREM 3.4.14. Fix objectsM, M,π : E → M,E,b ∇,b ∇,G, Pb andP as in Definition 3.4.13. Denote byTb,R,b T,R, respectively theι-torsion of ∇, theb curvature of ∇, the torsion ofb ∇and the curvature of ∇, whereι : T M → Eb denotes the inclusion map. Assume that(M ,∇, P)is infinitesimally homogeneous and that for allx∈ M,y ∈M and everyG-structure preserving mapσ :Ebx → T_yM, the following conditions hold:

(a) Adσ◦I^P_x^b =I^P_y ◦σ|_TxM;

(b) Tbx :TxM×TxM →Ebxisσ-related withTy :TyM×TyM →TyM;

(c) Rbx : TxM ×TxM ×Ebx → Ebxisσ-related withRy :TyM ×TyM× TyM →TyM.

Then, for all x0 ∈ M, all y0 ∈ M and for every G-structure preserving map σ0 : Ebx0 → Ty0M there exists a G-structure preserving local affine immersion (f, L)of(M, E,∇)b into(M ,∇)whose domain is an open neighborhoodU ofx₀ inM and such thatf(x₀) =y₀,L_x0 =σ₀.

3.4. AFFINE IMMERSIONS IN HOMOGENEOUS SPACES 185

PROOF. Denote by ω^M the connection form on FR(T M) corresponding to the connectionHor FR(T M)

associated to ∇and by ω^M the connection form onFR(E)b corresponding to the connectionHor FR(E)b

associated to∇. Denoteb also by θ^M the canonical form ofFR(T M) and byθ^M theι-canonical form of FR(E), whereb ι : T M → Eb denotes the inclusion map. Lets : V → Pb be a smooth local section withx0 ∈ V. Denote byλ^P the1-form on P obtained by restricting theR^¯n⊕gl(R^¯n)-valued1-form(θ^M, ω^M)and byλ^V theR^n¯⊕gl(R^¯n )-valued1-form onV defined by:

λ^V =s^∗(θ^M, ω^M) = (s^∗θ^M, s^∗ω^M).

Since(M ,∇, P) is infinitesimally homogeneous, by Lemma 3.4.7, there exists a linear mapI₀:R^n¯ →gl(R^n¯)/gsuch that:

(3.4.17) Ad_p¯◦I₀ =I^P_y ◦p,¯

for ally ∈M and allp¯∈ P_y. Let us show that for allx ∈M and allp ∈P_x we have:

(3.4.18) (Ad_p)⁻¹◦I^P_x^b =I₀◦p⁻¹|_TxM.

Namely, choose anyy∈M,p¯∈P_y and setσ = ¯p◦p⁻¹, so thatσ:Eb_x →T_yM isG-structure preserving (notice thatp¯=σ◦pand use Remark 1.1.14). Then:

Ad_σ◦I^P_x^b =I^P_y ◦σ|_TxM =I^P_y ◦p¯◦p⁻¹|_TxM ^(3.4.17)= Ad_p¯◦I₀◦p⁻¹|_TxM, and:

Adσ◦I^P_x^b= Adp¯◦(Adp)⁻¹◦I^P_x^b, so that:

Ad_p¯◦(Ad_p)⁻¹◦I^P_x^b = Ad_p¯◦I₀◦p⁻¹|_TxM, proving (3.4.18).

We divide the rest of the proof into steps.

Step 1. The thesis of the theorem follows once it is shown the existence of a smooth mapF :U →P defined in an open neighborhoodU ofx₀ inV such thatF^∗λ^P =λ^V|_U andF(x₀) =σ₀◦s(x₀).

Assume that we are given a smooth mapF :U → P defined in an open neighborhoodUofx0inV such thatF^∗λ^P =λ^V|_UandF(x0) =σ0◦s(x0).

Setf = Π◦F : U → M, whereΠ denotes the projection of the principal bundleP. We define a fiberwise linear mapL:E|b_U →f^∗T M by setting:

L_x=F(x)◦s(x)⁻¹ :Eb_x −→T_f(x)M = (f^∗T M)_x,

for allx∈U; thus (3.4.3) holds. Clearlyf(x0) =y0andLx0 =σ0. SinceF is smooth and:

F^∗(θ^M, ω^M) =λ^V|_U = (s|_U)^∗θ^M,(s|_U)^∗ω^M ,

Lemma 3.4.4 implies that the pair(f, L)is a local affine immersion of(M, E,∇)b into(M ,∇)with domainU. Since, for allx ∈ U,s(x)is inPb_xandF(x)is in P_f(x), equality (3.4.3) implies that L is G-structure preserving (see Re-mark 1.1.14).

Step 2. For all p ∈ P, the linear mapλ^P_p mapsTpP isomorphically onto the space:

(3.4.19)

(u, X)∈R^n¯⊕gl(R^¯n) :I₀(u) =X+g . Follows directly from Remark 2.11.9 and from equality (3.4.17).

Step 3. The1-formλ^V takes values in the space(3.4.19).

Letx∈V andv∈T_xM be fixed. We have:

λ^V_x(v) = (s^∗θ^M)x(v),(s^∗ω^M)x(v)(2.9.12)

= s(x)⁻¹·v,(s^∗ω^M)x(v) . We have to check that:

I₀ s(x)⁻¹·v

= (s^∗ω^M)_x(v) +g.

By the definition ofI^P_x^b, we have:

(Ad_s(x))⁻¹ I^P_x^b(v)

= (s^∗ω^M)x(v) +g.

But formula (3.4.18) withp=s(x)gives:

(Ad_s(x))⁻¹ I^P_x^b(v)

=I₀ s(x)⁻¹·v . Step 4. There exists a smooth mapF :U →P as in step 1.

We apply Proposition A.4.7. Observe that, since σ0 isG-structure pre-serving and s(x0) ∈ Pb, we have σ0 ◦s(x0) ∈ P; thus, once the hypothe-ses of Proposition A.4.7 have been checked, its thesis will give us a smooth map F : U → P defined in an open neighborhood U of x₀ in V with F(x₀) = σ₀ ◦s(x₀) andF^∗λ^P = λ^V|_U. Letx ∈ V, y ∈ M, p¯ ∈ P_y be fixed. By step 3, the linear mapλ^V_x mapsTxM to (3.4.19) and by step 2 the linear map λ^P_p¯ mapsT_p¯P isomorphically onto (3.4.19); therefore, we get a linear map:

τ = (λ^P_p¯)⁻¹◦λ^V_x :TxM −→Tp¯P . In order to apply Proposition A.4.7, we need to check that:

(3.4.20) τ^∗dλ^P_p¯ = dλ^V_x. Obviously (3.4.20) is the same as:

(3.4.21) τ^∗dθ^M_p¯ = (s^∗dθ^M)_x, τ^∗dω_p^M_¯ = (s^∗dω^M)_x. Clearly:

τ^∗θ^M_p¯ = (s^∗θ^M)_x, τ^∗ω_p^M_¯ = (s^∗ω^M)_x,

3.4. AFFINE IMMERSIONS IN HOMOGENEOUS SPACES 187

so that (3.4.21) is equivalent to:

(3.4.22) τ^∗(dθ^M+ω^M∧θ^M)p¯= s^∗(dθ^M+ω^M∧θ^M)

x, τ^∗(dω^M +¹₂ω^M ∧ω^M)_p¯= s^∗(dω^M +¹₂ω^M ∧ω^M)

x. But, by (2.9.2) and (2.9.13), (3.4.22) is the same as:

(3.4.23) τ^∗Θ^M_p¯ = (s^∗Θ^M)_x, τ^∗Ω^M_p¯ = (s^∗Ω^M)_x,

whereΘ^M denotes the torsion form ofFR(T M),Ω^M denotes the curvature form of the connection ofFR(T M),Θ^M denotes theι-torsion form ofFR(E)b andΩ^M denotes the curvature form of the connection ofFR(E). Equalitiesb (3.4.23) hold if and only if:

(3.4.24) Θ^M_p¯ τ(v), τ(w)

= Θ^M_s(x) ds_x(v),ds_x(w) , Ω^M_p¯ τ(v), τ(w)

= Ω^M_s(x) dsx(v),dsx(w) ,

for all v, w ∈ TxM. Denote by Π : FR(b E)b → M the projection; using (2.9.20) and (2.9.14), keeping in mind that dΠb_s(x) ◦ds_x is the identity of TxM, we obtain that (3.4.24) is equivalent to:

(3.4.25) p¯⁻¹

Ty dΠp¯[τ(v)],dΠp¯[τ(w)]

=s(x)⁻¹ Tbx(v, w) ,

¯

p⁻¹◦Ry dΠp¯[τ(v)],dΠp¯[τ(w)]

◦p¯=s(x)⁻¹◦Rbx(v, w)◦s(x).

Let us computedΠp¯◦τ :TxM → TyM. Givenu ∈R^n¯,X ∈gl(R^n¯)with (u, X)in (3.4.19) then(λ^P_p¯)⁻¹(u, X) =ζ, whereζ ∈Tp¯P satisfies:

θ_p^M_¯ (ζ) = ¯p⁻¹ dΠp¯(ζ)

=u;

thus:

dΠp¯◦(λ^P_p¯)⁻¹

(u, X) = ¯p(u).

Givenv∈T_xMthen, using (2.9.12), we see that the first component ofλ^V_x(v) iss(x)⁻¹·v; therefore:

(dΠ_p¯◦τ)(v) = dΠ_p¯◦(λ^P_p¯)⁻¹◦λ^V_x

(v) = ¯p◦s(x)⁻¹ (v).

Settingσ = ¯p◦s(x)⁻¹ :Eb_x →T_yMthen (3.4.25) is equivalent to:

¯ p⁻¹

T_y σ(v), σ(w)

=s(x)⁻¹ Tb_x(v, w) ,

¯

p⁻¹◦Ry σ(v), σ(w)

◦p¯=s(x)⁻¹◦Rbx(v, w)◦s(x), which is the same as:

(3.4.26) T_y σ(v), σ(w)

=σ Tb_x(v, w) , Ry σ(v), σ(w)

=σ◦Rbx(v, w)◦σ⁻¹.

Finally, sinceσisG-structure preserving, our hypotheses say thatσ^∗T_y =Tb_x andσ^∗Ry =Rbx, i.e., (3.4.26) holds. This concludes the proof.

3.4.1. The global affine immersions theorem.

THEOREM3.4.15. Under the assumptions of Theorem 3.4.14, ifM is simply-connected and(M ,∇) is geodesically complete then, for allx0 inM, all y0 ∈ M and for allG-structure preserving mapσ₀ : Eb_x0 → T_y0M there exists a G-structure preserving affine immersion(f, L) of(M, E,∇)b into (M ,∇) such that f(x0) =y0,Lx0 =σ0. Moreover ifM is connected then, by Corollary 3.4.5, such affine immersion(f, L)is unique.

LEMMA3.4.16. Let(M,∇),b (M ,∇)ben-dimensional affine manifolds,Gbe a Lie subgroup ofGL(Rⁿ),Pb⊂FR(T M)be aG-structure onM,P ⊂FR(T M) be aG-structure on M ands : V → Pbbe a smooth local section ofPb. Denote byθ^M,ω^M,θ^M,ω^M respectively the canonical form ofFR(T M), the connection form of FR(T M), the canonical form ofFR(T M) and the connection form of FR(T M). Set:

λ^V = (s^∗θ^M, s^∗ω^M)

and denote byλ^P the restriction toP of(θ^M, ω^M). Letγ :I →V,µ:I →M be geodesics andµ˜:I →P be a parallel lifting ofµ. Assume thats◦γ is a parallel lifting ofγ and that:

(3.4.27) s γ(t0)−1

·γ⁰(t0) = ˜µ(t0)⁻¹·µ⁰(t0), for somet0∈I. Then:

(3.4.28) λ^P_µ(t)˜ µ˜⁰(t)

=λ^V_γ(t) γ⁰(t) , for allt∈I.

PROOF. Sinces◦γ andµ˜are both parallel, we have:

(s^∗ω^M)_γ(t) γ⁰(t)

=ω^M_(s◦γ)(t) (s◦γ)⁰(t)

= 0, ω_µ(t)^M_˜ µ˜⁰(t)

= 0, for allt∈I, so that (3.4.28) is equivalent to:

(s^∗θ^M)_γ(t) γ⁰(t)

=θ^M_µ(t)˜ µ˜⁰(t) , for allt∈I. By (2.9.12), we have:

(s^∗θ^M)_γ(t) γ⁰(t)

=s γ(t)−1

·γ⁰(t), for allt∈I; moreover:

θ_µ(t)^M_˜ µ˜⁰(t)

= ˜µ(t)⁻¹·µ⁰(t),

for allt∈I. Sinceγ andµare geodesics, the curvesγ⁰ :I →T M andµ⁰ :I → T M are parallel; sinces◦γ : I → FR(T M)andµ˜ : I → FR(T M)are also parallel, the maps:

I 3t7−→s γ(t)−1

·γ⁰(t)∈Rⁿ, I 3t7−→µ(t)˜ ⁻¹·µ⁰(t)∈Rⁿ are constant and therefore (3.4.27) implies that:

s γ(t)−1

·γ⁰(t) = ˜µ(t)⁻¹·µ⁰(t),

3.4. AFFINE IMMERSIONS IN HOMOGENEOUS SPACES 189

for allt∈I. The conclusion follows.

LEMMA3.4.17. Let(M,∇),b (M ,∇)ben-dimensional affine manifolds,Gbe a Lie subgroup ofGL(Rⁿ),Pb⊂FR(T M)be aG-structure onM,P ⊂FR(T M) be aG-structure onM; assume that(M ,∇)is geodesically complete. Denote by Tb,R,b T, R, respectively the torsion of ∇, the curvature ofb ∇, the torsion ofb ∇ and the curvature of ∇. Assume that for allx∈M,y ∈M and everyG-structure preserving mapσ:T_xM →T_yM, the following conditions hold:

(a) Ad_σ◦I^P_x^b =I^P_y ◦σ;

(b) Tb_x :T_xM×T_xM →T_xMisσ-related withT_y :T_yM×T_yM →T_yM; (c) Rb_x :T_xM×T_xM×T_xM →T_xMisσ-related withR_y :T_yM×T_yM×

T_yM →T_yM.

Letx1 ∈ M be fixed and letV0be an open subset ofTx1M that is star-shaped at the origin and such thatexp_x1 mapsV₀diffeomorphically onto an open subsetV ofM. Then, for allx0 ∈V, ally0 ∈M and for everyG-structure preserving map σ₀ :T_x0M →T_y0M there exists aG-structure preserving affine mapf :V →M such thatf(x0) =y0,dfx0 =σ0.

REMARK3.4.18. Observe that, ifM is nonempty, conditions (a), (b) and (c) in the statement of Lemma 3.4.17 imply that(M ,∇, P) is infinitesimally homo-geneous. A similar observation does not holds in the case of Theorem 3.4.14, because the relations that appear in conditions (a), (b) and (c) in the statement of Theorem 3.4.14 involve restrictions of the tensors.

PROOF. By Lemma 2.2.30, there exists a smooth local sections : V → Pb such that for allv ∈T_x1M, the curvet7→ s exp_x1(tv)

∈Pbis a parallel lifting of the geodesict 7→ exp_x1(tv). Defineθ^M, ω^M,θ^M,ω^M, λ^V andλ^P as in the statement of Lemma 3.4.16. Our strategy is to employ Proposition A.4.10 to obtain a smooth mapF :V → P such thatF(x₀) =σ₀◦s(x₀)andF^∗λ^P =λ^V. Once this map F is obtained, we set f = Π◦F, where Π : P → M denotes the projection; then, arguing as in step 1 of the proof of Theorem 3.4.14, it will follow thatf is a G-structure preserving affine map such thatf(x₀) = y₀, df_x0 = σ₀. Let us check the validity of the hypotheses of Proposition A.4.10. Hypothesis (a) is obtained as in the proof of steps 2 and 3 of Theorem 3.4.14 and hypothesis (b) is obtained as in the proof of step 4 of Theorem 3.4.14. Hypothesis (c) (i.e., the simply-connectedness of V) follows from the fact that V is homeomorphic to a star-shaped open subset ofT_x1M. To prove that hypothesis (d) holds, we consider the setCof all geodesics γ : [0,1] → V such thats◦γ is a parallel lifting ofγ.

The fact thatCis rich follows by considering the map:

H : [0,1]×V 3(t, x)7−→exp_x1 texp⁻¹_x1(x)

∈V.

Finally, givenγ ∈ Candp¯∈P, we have to show that there exists a smooth curve

˜

µ : [0,1]→ P such thatµ(0) = ¯˜ pand such that (3.4.28) holds, for allt ∈[0,1].

Since (M ,∇) is geodesically complete, there exists a geodesicµ : [0,1] → M

withµ(0) = Π(¯p)and:

µ⁰(0) =

¯

p◦s γ(0)−1

·γ⁰(0).

Letµ˜: [0,1]→P be a parallel lifting ofµwithµ(0) = ¯˜ p(Proposition 2.2.28). By Lemma 3.4.16, (3.4.28) holds, for allt∈[0,1]. This concludes the proof.

We can now prove a global version of Theorem 3.4.14 in codimension zero.

PROPOSITION3.4.19. Under the conditions of Lemma 3.4.17, ifM is simply-connected then for allx₀ ∈M, ally₀ ∈ M and for everyG-structure preserving map σ0 : Tx0M → Ty0M there exists aG-structure preserving affine mapf : M → M such thatf(x₀) = y₀ anddf_x0 =σ₀. IfM is connected then suchf is unique, by Corollary 3.4.5.

PROOF. We may assume without loss of generality thatM is connected. Our plan is to use the globalization theory explained in Section B.4. Let us define a pre-sheaf onM as follows: for every open subsetU ofM,P(U)is the set of all G-structure preserving affine mapsf :U →Mand given open subsetsU, V ⊂M withV ⊂ U, the map P_U,V : P(U) → P(V)is given byf 7→ f|_V. The fact that the pre-sheafPhas the localization property is trivial. The fact thatPhas the uniqueness property follows from Corollary 3.4.5. Moreover, givenx₁ ∈ M, if V0 is an open subset of Tx1M, star-shaped at the origin, such that exp_x1 maps V₀ diffeomorphically onto an open subset V of M then it follows easily from Lemma 3.4.17 thatV has the extension property with respect to P. Thus, Phas the extension property. We are therefore under the hypotheses of Corollary B.4.22.

Now, let f¯ : V → M be a G-structure preserving affine map defined on a con-nected open neighborhoodV ofx0withf¯(x0) =y0andd ¯fx0 =σ0(the existence of f¯can be obtained either from Lemma 3.4.17 or from Theorem 3.4.14). By Corollary B.4.22, there existsf ∈P(X)such that f|_V = ¯f. This concludes the

proof.

REMARK 3.4.20. Under the conditions of Proposition 3.4.19, if in addition (M,∇)is geodesically complete,M is simply-connected and bothM andM are connected then the mapf given by the thesis of the proposition is a smooth diffeo-morphism. Namely, one can interchange the roles ofM andM to obtain a smooth inverse for the mapf.

PROPOSITION3.4.21. Let(M,∇) be an affine manifold endowed with a G-structure P. If M is connected and simply-connected, (M,∇) is geodesically complete and(M,∇, P)is infinitesimally homogeneous then(M,∇, P) is a ho-mogeneous affine manifold withG-structure.

PROOF. Take (M ,∇, P) = (M,∇, P) in Proposition 3.4.19 and use

Re-mark 3.4.20.

PROOF OFTHEOREM3.4.15. We can assume without loss of generality that M is connected. We will first prove the theorem under the additional assump-tion that M is simply-connected so that, by Proposition 3.4.21, (M ,∇, P) is a

3.5. ISOMETRIC IMMERSIONS INTO SEMI-RIEMANNIAN MANIFOLDS 191

homogeneous affine manifold withG-structure. We will use the globalization the-ory explained in Section B.4. Let us define a pre-sheaf on M as follows: for every open subset U of M, P(U) is the set of all G-structure preserving local affine immersions(f, L) of(M, E,∇)b into (M ,∇)with domainU; given open subsets U, V ⊂ M with V ⊂ U, the map P_U,V : P(U) → P(V) is given by (f, L)7→(f|_V, L|

E|b_V). The fact that the pre-sheafPhas the localization property is trivial. The fact thatPhas the uniqueness property follows from Corollary 3.4.5.

Let us now show that every open subsetU ofM such thatP(U)is nonempty has the extension property with respect toP; since, by Theorem 3.4.14, the set of such open setsU coverM, it will follow that the pre-sheafPhas the extension property.

Let thenU be an open subset ofM such thatP(U)is nonempty and let( ˆf ,L)ˆ in P(U) be fixed. Given a nonempty connected open subsetV ofU and an affine immersion(f, L)inP(V), we show that(f, L)admits an extension toU. Choose anyx₀∈V; the linear map:

(3.4.29) L_x0◦Lˆ⁻¹_x0 :T_fˆ(x

0)M −→T_f(x0)M

isG-structure preserving. Thus, by the homogeneity of(M ,∇, P), there exists a affineG-structure preserving diffeomorphismg :M →M such thatg f(xˆ 0)

= f(x₀)anddg_x0 is equal to (3.4.29). Then:

( ¯f ,L) =¯ g◦f ,ˆ ( ˆf^∗←− dg)◦Lˆ

is in P(U) and f(x¯ 0) = f(x0), L¯x0 = Lx0. Since V is connected, by Corol-lary 3.4.5, the restriction of ( ¯f ,L)¯ to V is equal to (f, L). This concludes the proof thatPhas the extension property. We are therefore under the hypotheses of Corollary B.4.22 which allows us to extend aG-structure preserving local affine immersion given by Theorem 3.4.14 to the desired G-structure preserving affine immersion of(M, E,∇)b into(M ,∇). The general case in whichMis not simply-connected can be obtained by considering the universal covering ofM.

No documento The theory of connections and G-structures. Applications to affine and isometric immersions Paolo Piccione Daniel V. Tausk (páginas 185-199)