AN EXISTENCE THEOREM FOR G-STRUCTURE PRESERVING AFFINE IMMERSIONS

AFFINE IMMERSIONS

PAOLO PICCIONE AND DANIEL V. TAUSK

ABSTRACT. We prove an existence result for local and globalG-structure pre- serving affine immersions between affine manifolds. Several examples are dis- cussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

1. INTRODUCTION

In this paper we prove an existence result forG-structure preserving affine im- mersions between affine manifolds, with special attention to the class of isometric immersions in the context of Riemannian and semi-Riemannian geometry. The original idea was to find a unifying language for several isometric immersion theorems that appear in the classical literature [2] (immersions into Riemannian manifolds with constant sectional curvature, immersions into K¨ahler manifolds of constant holomorphic curvature), and also some recent results (see for instance [3, 4]) concerning the existence of isometric immersions in more general Riemann- ian manifolds. Given an isometric immersion, the celebrated equations of Gauss, Codazzi and Ricci relate the curvature of the environment with the curvature of the submanifold, the curvature of the normal bundle and the second fundamental form (and its covariant derivative). A folk theorem says that such equations are nec- essary conditions for the existence of isometric immersions, however the reader should observe that, unless the isometric immersion has already been given, the equations cannot in general even be written down. Nevertheless, when the target manifold is “sufficiently homogeneous” (or, more precisely,infinitesimally homo- geneousin the sense of Definition 6.1), the Gauss, Codazzi and Ricci equations do make sensea prioriand then they are indeed necessary conditions for the existence of the isometric immersion. Sufficient conditions for the existence of an isometric immersion involve additional assumptions depending on the context; the starting point of our theory was precisely the interpretation of such additional assumptions in terms ofG-structures andinner torsion, which is a kind of covariant derivative of aG-structure.

The central result of the paper is an affine immersion theorem into infinitesi- mally homogeneous affine manifolds endowed with aG-structure. Infinitesimally

Date: October 2006.

2000Mathematics Subject Classification. 53A15, 53B05, 53C10, 53C40.

Key words and phrases. G-structures, affine immersions.

1

homogeneousmeans that the curvature and the torsion of the connection, as well as the inner torsion of theG-structure, can be written uniquely in terms of theG- structure, i.e., are constant in frames that belong to theG-structure. For instance, consider the case thatM is a Riemannian manifold endowed with the Levi-Civita connection of its metric tensor, Gis the orthogonal group and theG-structure is given by the set of orthonormal frames. Since the Levi-Civita connection is com- patible with the Riemannian metric, the inner torsion of this G-structure is zero.

The condition that the curvature tensor should be constant in orthonormal frames is equivalent to the condition thatMhas constant sectional curvature, and we recover in this case the classical “fundamental theorem of isometric immersions in spaces of constant curvature”. Similarly, ifMis a Riemannian manifold endowed with an orthogonal almost complex structure, then one has aG-structure onM, whereGis the unitary group, by considering the set of orthonormal complex frames ofT M. In this case, the inner torsion of theG-structure relatively to the Levi-Civita con- nection of the Riemannian metric is the covariant derivative of the almost complex structure, which vanishes if and only if M is K¨ahler. Requiring that the curva- ture tensor be constant in orthonormal complex frames means thatM has constant holomorphic curvature; in this context, our immersion theorem reproduces the clas- sical result of isometric immersions into K¨ahler manifolds of constant holomorphic curvature. Another interesting example ofG-structure that will be considered in some detail is the case of Riemannian manifolds endowed with a distinguished unit vector fieldξ; in this case, we obtain an immersion theorem into Riemannian manifolds with the property that both the curvature tensor and the covariant de- rivative of the vector field at a general pointpcan be written in terms only of the Riemannian metric at p and of the vectorξ(p). This is the case in a number of important examples, like for instance all manifolds that are Riemannian products of a space form with a copy of the real line, as well as all homogeneous, simply- connected3-dimensional manifolds whose isometry group has dimension4. These examples were first considered in [3]. Two more examples will be studied in some detail. First, we will consider isometric immersions into Lie groups endowed with a left invariant semi-Riemannian metric tensor. These manifolds have an obvious 1-structure, given by the choice of a distinguished orthonormal left invariant frame;

clearly, the curvature tensor is constant in this frame. Moreover, the inner torsion of the structure is simply the Christoffel tensor associated to this frame, which is also constant. It should be observed that a different immersion theorem into a class of nilpotent and solvable Lie groups has been recently proved in [8]. In spite of many analogies both in the statement and in the proof of the result, the setup con- sidered by the author in [8] does not fit into the infinitesimally homogeneous case considered in the present paper. Another example discussed is the case of isomet- ric immersions into products of manifolds with constant sectional curvature; in this situation, theG-structure considered is the one consisting of orthonormal frames adapted to such product. More generally, products of infinitesimally homogeneous affine manifolds withG-structures are infinitesimally homogeneous.

The proof of (the local version of) the main theorem relies on an application of Frobenius in the language of differential forms. More precisely, assume that we

are given affine manifolds(M,∇),(M ,∇)and a vector bundleE⁰overM. Given a suitable set of data (a connection∇⁰ on E⁰, and “second fundamental forms”

α⁰ andA⁰) we assemble a connection ∇b on the Whitney sum Eb = T M ⊕E and we look for an immersionf : M → M and a connection preserving vector bundle isomorphism L : Eb → f^∗T M such thatL|_{T M} = df. We assume that G-structuresPb andP are given on Eb and on T M, respectively, and we require thatL be G-structure preserving. Given a smooth local frame of Eb (in Pb), the problem of determiningLis reduced to the problem of determining a smooth map F :U →P, withU open inM, such thatF pulls back the canonical form and the connection form ofP to, respectively, the canonical form and the connection form ofPb. We then employ a version of the Frobenius theorem that allows one to guar- antee the existence of a smooth mapFsatisfying a PDE of the formF^∗λ^M =λ^M, whereλ^M,λ^M are vector-valued1-forms taking values in the same vector space.

The integrability condition for such PDE corresponds to the Gauss, Codazzi and Ricci equations, as well as to sometorsion equations; moreover, the condition that λ^M,λ^M take value in the same vector space corresponds to an equation relating the inner torsions of Pb and ofP. Finally, the proof of the global version of the affine immersion theorem employs a general globalization principle stated in the language of pre-sheafs.

2. NOTATIONS AND TERMINOLOGY

Vector spaces. LetV be a real finite-dimensional vector space. We denote by GL(V)the general linear group ofV and bygl(V)its Lie algebra. IfW is another real finite-dimensional vector space andp :V →W is a linear isomorphism then I_p : GL(V)→ GL(W)denotes the Lie group isomorphism given by conjugation withpandAd_p= dI_p(Id) :gl(V)→gl(W)denotes the Lie algebra isomorphism given by conjugation withp. ByLin(V, W) we denote the space of linear maps fromV toW.

Vector bundles, frame bundles and connections. LetEbe a vector bundle over a differentiable manifoldM. We denote byΓ(E) the set of all smooth sections ofE. Given a connection∇on E then the curvatureof∇is the smooth tensor R∈Γ(T M^∗⊗T M^∗⊗E^∗⊗E)defined by:

R(X, Y)=∇_X∇_Y− ∇_Y∇_X− ∇_[X,Y],

for allX, Y ∈Γ(T M),∈Γ(E); if ι:T M → E is a vector bundle morphism then theι-torsionof∇is the smooth tensorT^ι ∈Γ(T M^∗ ⊗T M^∗⊗E)defined by:

T^ι(X, Y) =∇_X ι(Y)

− ∇_Y ι(X)

−ι [X, Y] ,

for allX, Y ∈ Γ(T M). WhenE = T M andιis the identityT = T^ι coincides with the usualtorsionof∇.

Letk be therank ofE, i.e., the dimension of the fibers of E. We denote by FR(E) = S

x∈MFR(E_x) the frame bundle of E, which is the set of all linear

isomorphismsp:R^k →Ex, withx∈M. The frame bundleFR(E)is aGL(R^k)- principal bundle overM. A local sections : U → FR(E)(where U is an open subset ofM) is called alocal frameforE. A smooth local frames:U →FR(E) defines a connection dI^s on E|_U which corresponds via the trivialization of E|_U defined bysto the standard derivative. More explicitly, we set:

dI^s_v=s(x) d˜_x(v) ,

for allx ∈ U,v ∈ TxM, ∈ Γ(E|_U), where˜: U → R^k is defined by˜(x) = s(x)⁻¹ (x)

, for allx∈U. If∇is a connection inEthen theChristoffel tensor of∇with respect to the smooth local framesis the tensorΓ = ∇ −dI^s; more explicitly,Γ :U →T M^∗⊗E^∗⊗Eis the smooth local section such that:

∇_v= dI^s_v+ Γx v, (x) ,

for allx ∈U,v ∈ T_xM and all∈ Γ(E|_U). Forx ∈U,v ∈ T_xM, we also set Γ_x(v) = Γ_x(v,·) ∈ gl(E_x), so thatΓ_x :T_xM → gl(E_x)is a linear map. IfHor is the horizontal distribution inFR(E) corresponding to∇andω is thegl(R^k)- valued connection form onFR(E)whose kernel isHorthen, settingω¯ =s^∗ω, we have:

(2.1) Γ_x(v) = Ad_s(x) ω¯_x(v) ,

for allx∈U and allv∈T_xM. Ifι:T M →E is a vector bundle morphism then theι-canonical formofFR(E)is theR^k-valued1-formθonFR(E)defined by:

θ_p(ζ) =p⁻¹

ι dΠ_p(ζ)

∈R^k,

for allp∈FR(E),ζ ∈T_pFR(E), whereΠ : FR(E)→Mdenotes the projection.

WhenE =T M andιis the identity thenθis simply the usualcanonical formof FR(T M). Theι-torsion formΘand thecurvature formΩare defined respectively by:

(2.2) Θ = dθ+ω∧θ, Ω = dω+ω∧ω,

where the wedge product inω∧ωis taken with respect to the associative product ofgl(R^k). The following equalities hold:

p Θ_p(ζ₁, ζ₂)

= T^ι_x dΠ_p(ζ₁),dΠ_p(ζ₂)

∈E_x, (2.3)

Adp Ωp(ζ1, ζ2)

=Rx dΠp(ζ1),dΠp(ζ2)

∈gl(Ex), (2.4)

for allp∈FR(E),ζ1, ζ2 ∈TpFR(E), wherex= Π(p).

Covariant derivative along curves. Letp :I → FR(E)be a smooth curve and setγ = Π◦p, whereΠ : FR(E) →M denotes the projection. For allt∈ I, we denote by(∇₁p)(t) ∈ Lin(R^k, E_γ(t)) thecovariant derivativeofp at the instant t, which is just the vertical component of p⁰(t) ∈ T_p(t)FR(E) (observe that the vertical space ofFR(E)atp(t)is identified withLin(R^k, E_γ(t))). We have:

(2.5) ω_p(t) p⁰(t)

=p(t)⁻¹◦(∇₁p)(t),

for allt∈I. Ifε:I →E is a smooth section ofEalongγ (i.e.,ε(t)∈E_γ(t), for allt∈I), we denote by(∇₁ε)(t)∈ E_γ(t) the covariant derivative ofεalongγ at

the instantt. Given a smooth curveu:I →R^kthenε:I 3t7→p(t)·u(t)∈E_γ(t) is a smooth section ofEalongγ and the following “Leibniz rule” holds:

(∇₁ε)(t) = (∇₁p)(t)·u(t) +p(t)·u⁰(t),

for allt∈ I. Ifs: U → FR(E) is a smooth local section withγ(I) ⊂U and if

¯

ω=s^∗ωthen:

(2.6) (∇₁p)(t) =s γ(t)

◦

˜

p⁰(t) + ¯ω_γ(t) γ⁰(t)

◦p(t)˜ , for allt∈I, wherep˜:I →GL(R^k)is defined byp(t) =˜ s γ(t)−1

◦p(t).

Vector subbundles. If F is a vector subbundle of E then the absolute second fundamental formofF inEwith respect to the connection∇is the tensorα^F ∈ Γ T M^∗⊗F^∗⊗(E/F)

defined byα^F(X, ) =q ∇_X), for allX ∈Γ(T M), ∈Γ(F), whereq:E →E/F denotes the quotient map.

3. AFFINE IMMERSIONS AND THEIR INVARIANTS

Let(M,∇), (M ,∇) be affine manifolds and f : M → M be a smooth im- mersion. We identify the differentialdf : T M → T M with a (injective) vector bundle morphism df : T M → f^∗T M. A vector subbundle E of f^∗T M with f^∗T M = df(T M)⊕E will be called a normal bundle for f. Let a normal bundle E for f be fixed; we denote by π^E : f^∗T M → E the projection onto E corresponding to the decompositionf^∗T M = df(T M) ⊕E and by π^{T M} : f^∗T M →T M the composition of the projection ontodf(T M)with the isomor- phism df⁻¹ : df(T M) → T M. Given smooth vector fields X, Y ∈ Γ(T M) inM, we setα(X, Y) = π^E ∇_Xdf(Y)

∈Γ(E), so thatα is identified with a smooth section ofT M^∗⊗T M^∗⊗E. We callαthesecond fundamental formof the immersionf with respect to the normal bundle E. Notice that in the case of Riemannian (or semi-Riemannian) geometry,fhas a canonical normal bundle (the orthogonal complement ofdf(T M) with respect to the metric), so there is also a canonical notion of second fundamental form. Also, if (M, g)and(M ,¯g) are semi-Riemannian,∇and∇are the corresponding Levi-Civita connections,f is an isometric immersion andE is the orthogonal complement ofdf(T M)inf^∗T M with respect tog¯then:

(3.1) ∇_Xdf(Y) = df(∇_XY) +α(X, Y),

for allX, Y ∈ Γ(T M). In the general affine case, we say thatf is anaffine im- mersion with respect toE if (3.1) holds, for allX, Y ∈ Γ(T M). Following [11]

we say simply thatf is anaffine immersionif there exists a normal bundleE for f such thatf is an affine immersion with respect toE. We define thenormal con- nection∇^⊥of the immersionf corresponding to the normal bundleE by setting

∇^⊥_X=π^E(∇_X), for allX ∈Γ(T M)and all∈Γ(E). In the semi-Riemannian case, αand∇^⊥ are the only invariants associated to an isometric immersion. In the general affine case, we have an additional invariant associated to the immersion.

We setA(X, ) =π^{T M}(∇_X), for allX∈Γ(T M)and all∈Γ(E); clearlyAis identified with a smooth section ofT M^∗⊗E^∗⊗T M. We callAtheWeingarten

formof the immersion f relatively toE and for allx ∈ M and alle ∈ Ex, the linear endomorphismA(e) = A_x(·, e) : T_xM → T_xM is called theWeingarten operatorin the direction ofe. In the semi-Riemannian caseαandAare related by the equality:

¯

g_x α_x(v, w), e

=−¯g_x A(e)·v, w

, x∈M, v, w∈T_xM, e∈E_x, so thatAis determined byα. In the affine case, there is no relation betweenαand A; moreover,αis not in general symmetric unless the connection∇is symmetric.

We are interested in studying the existence of affine immersions with prescribed invariants∇^⊥,αandA. More precisely, let(M,∇),(M ,∇)be affine manifolds, E⁰ be a vector bundle overM,∇⁰ be a connection inE⁰ andα⁰,A⁰ be smooth sections ofT M^∗⊗T M^∗⊗E⁰ andT M^∗ ⊗(E⁰)^∗⊗T M respectively. We look for an affine immersion f : M → M, a normal bundle E for f and a connec- tion preserving vector bundle isomorphism S : (E⁰,∇⁰) → (E,∇^⊥) such that S α⁰(·,·)

=αandA(·, S·) =A⁰. The pair(f, S)will be called asolution for the affine immersion problem with data∇⁰,α⁰andA⁰. More generally, iff :U →M is an affine immersion defined in an open subsetU ofM andS : E⁰|_U → E|_U is a connection preserving vector bundle isomorphism such thatS α⁰(·,·)

= α andA(·, S·) =A⁰then the pair(f, S)will be called alocal solution for the affine immersion problem with data∇⁰,α⁰,A⁰ and with domainU.

An important special situation is the one of isometric immersions. Assume that(M, g), (M ,g)¯ are semi-Riemannian manifolds, E⁰ is a vector bundle over M endowed with a semi-Riemannian structureg⁰ (i.e.,g⁰ is a smooth section of (E⁰)^∗⊗(E⁰)^∗ andg⁰_x is a nondegenerate symmetric bilinear form onE_x⁰, for all x ∈ M), g⁰ is∇⁰-parallel and α⁰ is a smooth symmetric section of the vector bundle T M^∗ ⊗T M^∗ ⊗E⁰; by a solution for the isometric immersion problem with data ∇⁰, α⁰, g⁰ we mean a pair(f, S)wheref : M → M is an isometric immersion, S : (E⁰,∇⁰, g⁰) → (E,∇^⊥, g^⊥) is a connection preserving vector bundle isometry and S α⁰(·,·)

= α, where E denotes the orthogonal comple- ment ofdf(T M)inf^∗T M with respect tog¯andg^⊥denotes the restriction ofg¯to E. As in the affine case, one defines the concept oflocalsolution for the isometric immersion problem by replacingM with an open subsetU ofM.

Notice that if(f, S)is a (local) solution for the isometric immersion problem, (M, g)and(M ,¯g) are endowed with their respective Levi-Civita connections∇,

∇and a smooth sectionA⁰ofT M^∗⊗(E⁰)^∗⊗T M is defined by the equality:

(3.2) g_x⁰ α⁰_x(v, w), e

=−g_x A⁰_x(e)·v, w

, x∈M, v, w∈TxM, e∈E_x⁰, then (f, S) is also a (local) solution for the affine immersion problem with data

∇⁰,α⁰ andA⁰.

4. THE COMPONENTS OF A CONNECTION

The following general construction gives a convenient language for discussing the theory of affine immersions. Let E be a vector bundle over a differentiable manifoldM endowed with a connection ∇and letE = E¹⊕ E² be a direct sum

decomposition ofE; denote byπⁱ :E → Eⁱ,i= 1,2, the projections. We set:

∇ⁱ_Xi=πⁱ(∇_Xi), i= 1,2

α²(X, ₁) =π²(∇_X1), α¹(X, ₂) =π¹(∇_X2),

for allX ∈Γ(T M),_i ∈Γ(Eⁱ),i= 1,2, so that∇ⁱis a connection inEⁱ,i= 1,2, andα²(resp.,α¹) is identified with a smooth section ofT M^∗⊗(E¹)^∗⊗E²(resp., of T M^∗⊗(E²)^∗⊗ E¹). We call∇¹,∇²,α²andα¹thecomponentsof the connection

∇with respect to the decompositionE =E¹⊕ E². Clearly, one recovers∇from its components using the formula:

∇_X=∇¹_X π¹()

+α¹ X, π²()

+∇²_X π²()

+α² X, π¹() , whereX∈Γ(T M)and∈Γ(E).

Notice that if f : (M,∇) → (M ,∇) is an affine immersion with respect to a normal bundle E then, identifying for a moment T M with df(T M), the components of the connectionf^∗∇on f^∗T M with respect to the decomposition f^∗T M = df(T M)⊕Eare the connection∇ofM, the normal connection∇^⊥, the second fundamental formαand the Weingarten formAoffwith respect toE.

Remark4.1. Ifg¹,g²are semi-Riemannian structures onE¹andE², respectively and ifgis the semi-Riemannian structure onEgiven by the orthogonal direct sum ofg¹andg²then a connection∇with components∇¹,∇²,α²,α¹ is compatible with g(i.e., ∇g = 0) if and only if∇ⁱ is compatible with gⁱ, i = 1,2, and the following relation betweenα²andα¹holds:

(4.1) g_x α²_x(v, e₁), e₂

+g_x e₁, α¹_x(v, e₂)

= 0,

for allx ∈M,v ∈TxM,e1 ∈ E_x¹ ande2 ∈ E_x². Notice that relation (4.1) implies thatα¹ is uniquely determined fromα², so that in a context where we are dealing with connections compatible with a semi-Riemannian structure, we will talk only about the components∇¹,∇²andα²of∇, where one should understand implicitly thatα¹is determined by condition (4.1).

Denote byR, R¹, R² the curvature tensors of∇,∇¹ and∇², respectively. A straightforward computation gives the following:

π¹ R_x(v, w)e₁

=R¹_x(v, w)e₁+α¹_x v, α²_x(w, e₁)

−α¹_x w, α²_x(v, e₁) (4.2) ,

π² Rx(v, w)e2

=R²_x(v, w)e2+α²_x v, α¹_x(w, e2)

−α²_x w, α¹_x(v, e2) , (4.3)

for allx ∈M,e1 ∈ E_x¹,e2 ∈ E_x² and allv, w∈TxM. Moreover, given a connec- tion∇^M onT M with torsionTand denoting by∇^⊗the induced connections on T M^∗⊗(E²)^∗⊗ E¹and onT M^∗⊗(E¹)^∗⊗ E²then:

π² R_x(v, w)e₁

= (∇^⊗α²)_x(v, w, e₁)−(∇^⊗α²)_x(w, v, e₁) +α²_x Tx(v, w), e1

, (4.4)

π¹ R_x(v, w)e₂

= (∇^⊗α¹)_x(v, w, e₂)−(∇^⊗α¹)_x(w, v, e₂) +α¹_x Tx(v, w), e2

, (4.5)

for allx ∈ M, e1 ∈ E_x¹,e2 ∈ E_x² and allv, w ∈ TxM. Ifι = (ι¹, ι²) : T M → E = E¹⊕ E²is a vector bundle morphism then theι-torsionT^ιof∇satisfies the following identities:

π¹ T^ι_x(v, w)

= T^ι_x¹(v, w) +α¹_x v, ι²(w)

−α¹_x w, ι²(v) , (4.6)

π² T^ι_x(v, w)

= T^ι_x²(v, w) +α²_x v, ι¹(w)

−α²_x w, ι¹(v) , (4.7)

for allx∈ M,v, w ∈T_xM, whereT^ι1,T^ι2 denote respectively theι¹-torsion of

∇¹and theι²-torsion of∇².

Let us look at equations (4.2), (4.3), (4.4), (4.5), (4.6) and (4.7) in the context of affine immersions. More precisely, let f : (M,∇) → (M ,∇) be an affine immersion, E be a normal bundle for f, and ∇^⊥, α and A denote respectively the normal connection, the second fundamental form and the Weingarten form.

Since ∇, ∇^⊥, α andA are (up to the identification of T M with df(T M)) the components off^∗∇with respect to the decompositionf^∗T M = df(T M)⊕E, equation (4.2) gives:

(4.8) π^{T M}

R_f(x) df_x(v),df_x(w)

df_x(u)

=R_x(v, w)u+A_x v, α_x(w, u)

−Ax w, αx(v, u) , for allx∈Mand allv, w, u∈TxM, whereRandRdenote the curvature tensors of∇and∇respectively. We call (4.8) theGauss equationof the affine immersion f with respect toE. Similarly, equation (4.3) gives:

(4.9) π^E

R_f(x) df_x(v),df_x(w) e

=R^⊥_x(v, w)e+α_x v, A_x(w, e)

−α_x w, A_x(v, e) , for all x ∈ M, v, w ∈ T_xM and all e ∈ E_x, where R^⊥ denotes the curvature tensor of the normal connection∇^⊥. We call (4.9) theRicci equationof the affine immersionf with respect toE. Equations (4.4) and (4.5) (with∇^M =∇) give:

π^E

R_f(x) df_x(v),df_x(w) u

= (∇^⊗α)_x(v, w, u)−(∇^⊗α)_x(w, v, u) +α_x T_x(v, w), u (4.10) ,

π^{T M}

R_f(x) dfx(v),dfx(w) e

= (∇^⊗A)x(v, w, e)−(∇^⊗A)x(w, v, e) +A_x T_x(v, w), e (4.11) ,

for allx∈M,v, w, u∈T_xMand alle∈E_x, whereTdenotes the torsion tensor of∇. We call (4.10) and (4.11) the Codazzi equationsof the affine immersionf with respect toE. Finally, if ι : T M → f^∗T M = df(T M)⊕E is the map identified withdf :T M → T M thenι¹ = df :T M → df(T M),ι² = 0and equations (4.6) and (4.7) read¹:

π^{T M}

T_f(x) df_x(v),df_x(w)

= T_x(v, w), (4.12)

π^E

T_f(x) dfx(v),dfx(w)

=αx(v, w)−αx(w, v), (4.13)

1By taking such mapι, theι-torsion off^∗∇is just the pull-back byfof the torsion of∇.

for allx∈M,v, w∈TxM, whereTdenotes the torsion of∇. We call (4.12) and (4.13) thetorsion equationsof the affine immersionfwith respect toE.

Now assume that we are given affine manifolds(M,∇),(M ,∇), a vector bun- dleE⁰ overM endowed with a connection∇⁰ and smooth sectionsα⁰,A⁰of the vector bundlesT M^∗⊗T M^∗⊗E⁰andT M^∗⊗(E⁰)^∗⊗T Mrespectively. Assume that there exists a local solution(f, S)for the affine immersion problem with data

∇⁰,α⁰,A⁰defined on an open subsetU ofM, whereS :E⁰|_U → E|_U andEis a normal bundle forf. Clearly, (4.8), (4.9), (4.10), (4.11), (4.12) and (4.13) imply:

π^{T M}

R_f(x) dfx(v),dfx(w)

dfx(u)

=Rx(v, w)u+A⁰_x v, α⁰_x(w, u)

−A⁰_x w, α⁰_x(v, u) (4.14) ,

π^E

R_f(x) df_x(v),df_x(w) S_x(e)

=S_x

R_x⁰(v, w) +α⁰_x v, A⁰_x(w, e)

−α⁰_x w, A⁰_x(v, e) (4.15) ,

π^E

R_f(x) df_x(v),df_x(w)

df_x(u)

=S_x

(∇^⊗α⁰)_x(v, w, u)

−(∇^⊗α⁰)x(w, v, u) +α⁰_x Tx(v, w), u (4.16) ,

π^{T M}

R_f(x) dfx(v),dfx(w) Sx(e)

= (∇^⊗A⁰)x(v, w, e)

−(∇^⊗A⁰)_x(w, v, e) +A⁰_x T_x(v, w), e (4.17) ,

π^{T M}

T_f(x) dfx(v),dfx(w)

= Tx(v, w), (4.18)

π^E

T_f(x) df_x(v),df_x(w)

=S_x α⁰_x(v, w)−α⁰_x(w, v) (4.19) ,

for allx ∈ U, v, w, u ∈ T_xM and alle ∈ E_x⁰, whereR⁰ denotes the curvature tensor of∇⁰.

Notice that in the case of isometric immersions, the torsion equation (4.18) is trivial and (4.19) says thatα⁰ is symmetric; moreover, using (3.2), it can be seen that the Codazzi equations (4.16) and (4.17) are equivalent to each other.

One may think that Gauss, Ricci, Codazzi and the torsion equations are “neces- sary conditions” for the existence of a solution(f, S)of the affine immersion prob- lem, although such statement is obviously meaningless because one cannot write down equations (4.14), (4.15), (4.16), (4.17), (4.18) and (4.19) unlessf andSare already given. Notice that in the special case that(M ,¯g) is a semi-Riemannian manifold with constant sectional curvaturec ∈ Rand(f, S)is a solution for the isometric immersion problem then the lefthand side of equations (4.14), (4.15), (4.16) and (4.17) can be written only in terms ofcandg, i.e., without usingf and S; more explicitly, the lefthand side of (4.14) isc gx(w, u)v−gx(v, u)w

, while the lefthand sides of (4.15), (4.16) and (4.17) are zero (the possibility of writing down Gauss, Ricci and Codazzi equations without usingf andS depends on the fact that the curvature tensorRof the target manifold isconstant in orthonormal frames). Thus, in this case, the Gauss, Ricci and Codazzi equations are indeed necessary conditions for the existence of a solution of the isometric immersion problem(f, S). By the celebrated fundamental theorem of isometric immersions

into space forms (see, for instance, [2, 5, 12, 14]), the Gauss, Ricci and Codazzi equations are also sufficient conditions for the existence of local solutions for the isometric immersion problem (provided that one assumes thatα⁰is symmetric and thatg⁰is∇⁰-parallel).

Using the notion ofinfinitesimally homogeneous affine manifold with G-struc- tureintroduced in Section 6 we will describe a very general situation in which the lefthand side of (4.14), (4.15), (4.16), (4.17), (4.18) and (4.19) can be described without explicit use off andS.

5. G-STRUCTURES AND INNER TORSION

LetEbe a vector bundle of rankkover a differentiable manifoldM. IfGis a Lie subgroup ofGL(R^k)then by aG-structureonEwe mean aG-principal subbundle P ofFR(E). By aG-structure onMwe mean aG-structure on the tangent bundle ofM. Let∇be a connection inE. We denote byHorthe corresponding horizontal distribution on FR(E) and by ω thegl(R^k)-valued connection form on FR(E) whose kernel isHor. We say that∇iscompatiblewith aG-structureP ifHorp⊂ T_pP, for allp∈P, i.e., if parallel transport carries frames inP to frames inP. In the general case, there is a tensor that measures the lack of compatibility of∇with P called theinner torsionofP with respect to∇, which is defined as follows.

For each x ∈ M, denote by G_x the subgroup of GL(E_x) consisting of G- structure preserving maps, i.e., maps σ : Ex → Ex such thatσ ◦p ∈ Px for some (and hence for all) p ∈ P_x. ClearlyG_x = I_p(G), for allp ∈ P_x, so that Gx is a Lie subgroup of GL(Ex). We denote by g_x ⊂ gl(Ex) the Lie algebra of G_x, so that Ad_p(g) = g_x, for all p ∈ P_x, where g ⊂ gl(R^k) denotes the Lie algebra of G. For eachx ∈ M and each p ∈ FR(Ex), we can identify the tangent spaceTpFR(E)with the direct sumTxM⊕gl(R^k)via the isomorphism (dΠ_p, ω_p), where Π : FR(E) → M denotes the projection. For p ∈ P, the subspaceV_p = (dΠp, ωp)(TpP)ofTxM⊕gl(R^k)corresponding toTpP satisfies the conditions pr₁(V_p) = T_xM and V_p ∩ {0} ⊕ gl(R^k)

= {0} ⊕g, where pr₁ : TxM ⊕gl(R^k) → TxM denotes the first projection. Thus, there exists a unique linear mapL_p:T_xM →gl(R^k)/gsuch that:

(5.1) V_p =

(v, X)∈TxM⊕gl(R^k) :L_p(v) =X+g .

Ifs: U → P is a smooth local section withx ∈ U andω¯ = s^∗ω thenL_p is the composition ofω¯x :TxM → gl(R^k)with the quotient mapgl(R^k) →gl(R^k)/g.

It follows from the usual properties of connection forms that, givenp, q∈P_x, the mapsL_pandL_qare related byL_q= Ad_g◦ L_p, whereg∈Gis such thatp=q◦g, andAdg : gl(R^k)/g → gl(R^k)/gis obtained from Adg : gl(R^k) → gl(R^k) by passing to the quotient. It follows that the linear mapI^P_x : T_xM → gl(E_x)/g_x defined by:

(5.2) I^P_x = Adp◦ L_p, p∈Px,

does not depend on the choice ofp ∈ Px; hereAdp : gl(R^k)/g → gl(Ex)/gx is obtained fromAd_p :gl(R^k) →gl(E_x)by passing to the quotient. We callI^P_x the

inner torsionof theG-structureP at the pointxwith respect to the connection∇.

Obviously,I^P = 0if and only if∇is compatible withP. It follows from (2.1) that ifs:U → P is a smooth local section withx∈ U andΓdenotes the Christoffel tensor of∇with respect tosthen the inner torsionI^P_x is precisely the composition of the mapΓ_x:T_xM →gl(E_x)with the quotient mapgl(E_x)→gl(E_x)/g_x. This observation gives a simple method for computing inner torsions.

Let us compute inner torsions in some specific examples.

5.1.Example. IfEis trivial ands:M →FR(E)is a smooth global frame then P = s(M)is aG-structure onE withG = {Id_Rk}. For eachx ∈ M, we have Gx={Id_Ex}andg_x={0}; the inner torsionI^P_x :TxM →gl(Ex)is equal to the Christoffel tensorΓ_x:T_xM →gl(E_x)of∇with respect tos.

5.2.Example. Let g be a semi-Riemannian structure on E of index r, i.e., g is a smooth section ofE^∗⊗E^∗ such thatg_x is a nondegenerate symmetric bilinear form onEx of indexr, for allx ∈ M; denote byh·,·i_r the standard Minkowski inner product of indexrinR^kdefined by:

(5.3) hv, wi_r=

k−r

X

i=1

v_iw_i−

k

X

i=k−r+1

v_iw_i, v, w∈R^k.

We denote byFR^o(E)the set of allp∈FR(E)that are linear isometries, so that P = FR^o(E)is aG-structure onE, whereG= O(k−r, r)denotes the Lie group of linear isometries of(R^k,h·,·i_r). Letx ∈M be fixed. Clearly,G_xis the group of linear isometries of(Ex, gx)andg_x is the Lie algebra of linear endomorphisms ofE_xthat areg_x-anti-symmetric. We identifygl(E_x)/g_xwith the spacesym(E_x) of all linear endomorphisms ofExthat aregx-symmetric via the map:

(5.4) gl(Ex)/gx3T+g_x7−→ ¹₂(T +T^∗)∈sym(Ex),

whereT^∗ :E_x→E_xdenotes the transpose ofTwith respect tog_x. Thus, the inner torsionI^P_x is identified with a linear map fromTxM tosym(Ex). Lets:U →P be a smooth local section withx ∈ U and lete, e⁰ ∈ E_x be fixed; consider the local sections, ⁰ :U → Esuch that(x) = e,⁰(x) =e⁰ and such that theR^k- valued mapsy 7→s(y)⁻¹ (y)

,y7→ s(y)^{−1 0}(y)

are constant. ThendI^s= 0, dI^s0= 0and:

∇_v= Γx(v)·e, ∇_v⁰ = Γx(v)·e⁰,

for allv ∈TxM. Sinces(y) : (R^k,h·,·i_r) →(Ey, gy)is a linear isometry for all y∈U, the real-valued mapg(, ⁰)is constant. Thus:

0 =v g(, ⁰)

= (∇_vg)(e, e⁰) +gx(∇_v, e⁰) +gx(e,∇_v⁰)

= (∇_vg)(e, e⁰) +gx Γx(v)·e, e⁰

+gx e,Γx(v)·e⁰ , for allv∈TxM. Then:

g_x

Γ_x(v) + Γ_x(v)^∗

·e, e⁰

=−(∇_vg)(e, e⁰) and (using (5.4)):

g_x I^P_x(v),·

= ¹₂g_x

Γ_x(v) + Γ_x(v)^∗ ,·

=−¹₂∇_vg,

for allx∈M,v∈TxM. Usinggxto identify∇_vg:Ex×Ex →Rwith a linear endomorphism ofE_xwe obtain:

I^P_x(v) =−¹₂∇_vg.

Thus, the inner torsion of P is essentially the covariant derivative of the semi- Riemannian structureg. In particular,I^P = 0if and only if∇g= 0.

5.3.Example. LetF be a vector subbundle ofEof rankl. Forx∈M, set:

FR(Ex;Fx) =

p∈FR(Ex) :p(R^l⊕ {0}) =Fx

andFR(E;F) = S

x∈MFR(Ex;Fx). ThenP = FR(E;F) is aG-structure on E with G the Lie subgroup of GL(R^k) consisting of linear isomorphisms that preserve R^l⊕ {0}. Letx ∈ M be fixed. ClearlyGx is the Lie group of linear isomorphisms ofE_xthat preserveF_xandg_xis the Lie algebra of linear endomor- phisms ofExthat preserveFx. We identify the quotientgl(Ex)/gxwith the space Lin(F_x, E_x/F_x)via the map:

(5.5) gl(Ex)/gx3T+g_x7−→q◦T|_Fx ∈Lin(Fx, Ex/Fx),

whereq :E_x → E_x/F_x denotes the quotient map. Thus, the inner torsionI^P_x is identified with a linear map fromT_xM toLin(F_x, E_x/F_x). Lets :U → P be a smooth local section withx∈U ande∈Fxbe fixed. Defineas in Example 5.2, so that∇_v= Γ_x(v)·e, for allv∈T_xM. We have(U)⊂F and therefore:

Γ_x(v)·e+F_x=∇_v+F_x=α^F_x(v, e)∈E_x/F_x,

whereα^F denotes the absolute second fundamental form of the vector subbundle F. Hence, using (5.5):

I^P_x(v) =α^F_x(v,·)∈Lin(F_x, E_x/F_x),

for allx∈Mand allv∈T_xM. In particular,I^P = 0if and only ifα^F = 0, i.e., if and only if the covariant derivative of any smooth section ofF is a smooth section ofF.

5.4.Example. Letg be a semi-Riemannian structure onE of indexr andF be a vector subbundle ofE such that the restriction of gtoF is a semi-Riemannian structure onF of indexs≤ r; denote byF^⊥theg-orthogonal complement ofF inE, so thatE =F⊕F^⊥. Defineh·,·i_ras in (5.3) and fix any subspaceF₀ofR^k such that the restriction ofh·,·i_rtoF0is a nondegenerate symmetric bilinear form of indexs. ThenP = FR^o(E;F) =S

x∈MFR^o(Ex;Fx), where FR^o(E_x;F_x) =

p∈FR^o(E_x) :p(F₀) =F_x ,

is aG-structure onEwhereGis the Lie group of linear isometries of(R^k,h·,·i_r) that preserveF₀. Denote byq : E → F^⊥the projection with respect to the de- compositionE = F ⊕F^⊥. Let x ∈ M be fixed. ClearlyG_x is the Lie group

of linear isometries of(Ex, gx)that preserveFx andg_x is the Lie algebra ofgx- anti-symmetric linear endomorphisms ofE_xthat preserveF_x. We have an isomor- phism:

gl(Ex)/gx −→sym(Ex)⊕Lin(Fx, F_x^⊥)

T +g_x 7−→ ¹₂(T +T^∗),¹₂q_x◦(T −T^∗)|_Fx ,

so that we identifyI^P_x with a linear map fromT_xM tosym(E_x)⊕Lin(F_x, F_x^⊥).

Consider the component α ∈ Γ(T M^∗ ⊗F^∗ ⊗F^⊥) of the connection ∇ with respect to the decompositionE =F ⊕F^⊥. Arguing as in Examples 5.2 and 5.3, one easily computes:

I^P_x(v) = −¹₂∇_vg, α_x(v,·) +¹₂q◦ ∇_vg|_Fx ,

for allx ∈M,v ∈TxM, where∇_vgis identified with a linear endomorphism of E_x usingg_x. In particular,I^P = 0if and only if∇g = 0andα = 0, i.e., if and only if∇g= 0and the covariant derivative of any smooth section ofFis a smooth section ofF.

5.5.Example. Let ∈ Γ(E) be a smooth section of E with (x) 6= 0, for all x∈M. Fix a nonzero vectore₀∈R^k; then:

P = [

x∈M

p∈FR(Ex) :p(e0) =(x)

is aG-structure onE whereGis the subgroup ofGL(R^k)consisting of isomor- phisms that fixe₀. Let x ∈ M be fixed. Then G_x is the subgroup ofGL(E_x) consisting of isomorphisms that fix (x) and g_x is the Lie algebra of linear en- domorphisms T : E_x → E_x such that T (x)

= 0. We identify the quotient gl(E_x)/g_xwithE_xvia the map:

gl(E_x)/g_x 3T +g_x 7−→T (x)

∈E_x,

so thatI^P_x is identified with a linear map fromTxM toEx. Lets :U → P be a smooth local section withx ∈U. We haves(y)⁻¹ (y)

=e₀, for ally ∈ U, so thatdI^s_v= 0and∇_v= Γx(v)·(x), for allv∈TxM. Then:

I^P_x(v) =∇_v,

for allv ∈TxM, i.e., the inner torsionI^P is identified with the covariant derivative of. In particular,I^P = 0if and only if the sectionis parallel.

Assume now that g is a semi-Riemannian structure onE of index r, h·,·i_r is defined as in (5.3) and thatg_x (x), (x)

=he₀, e₀i_r, for allx∈M. Then:

P⁰= [

x∈M

p∈FR^o(Ex) :p(e0) =(x)

is aG-structure onE where Gis the Lie subgroup of O(k−r, r) consisting of linear isometries that fix e₀. Let x ∈ M be fixed. Then G_x is the Lie group

of linear isometries of(Ex, gx)that fix(x) andg_xis the Lie algebra of gx-anti- symmetric linear endomorphisms T ofE_x such thatT (x)

= 0. We have the following linear isomorphism:

gl(E_x)/g_x 3T+g_x7−→ ¹₂(T +T^∗),¹₂(T−T^∗)·(x)

∈sym(E_x)⊕(x)^⊥ where (x)^⊥ denotes the g_x-orthogonal complement of (x) in E_x. Arguing as before, we obtain:

(5.6) I^P_x⁰(v) =

−¹₂∇_vg,∇_v+¹₂(∇_vg) (x) ,

for allx∈M and allv ∈T_xM. In particular,I^P0 = 0if and only if∇g= 0and

∇= 0.

5.6.Example. Assumek= 2land letJbe an almost complex structure onE, i.e., Jis a smooth section ofE^∗⊗EandJ_xis a complex structure onE_xfor allx∈M. Consider the complex structureJ₀ :R^k∼=R^l⊕R^l3(v, w)7→(−w, v)onR^kand setFR^c(E) =S

x∈MFR^c(Ex), where for eachx∈M,FR^c(Ex)denotes the set of all complex linear isomorphismsp: (R^k, J₀)→ (E_x, J_x). ThenP = FR^c(E) is aG-structure onEwhereG= GL(R^k, J₀)is the Lie group of complex linear isomorphisms of (R^k, J0). Letx ∈ M be fixed. Then Gx = GL(Ex, Jx) and g_x is the Lie algebra of complex linear endomorphisms of(E_x, J_x). We have an isomorphism:

gl(Ex)/gx3T +g_x 7−→[T, Jx]∈Lin(Ex, Jx),

where [T, Jx] = T ◦ Jx−Jx ◦T and Lin(Ex, Jx) denotes the space of linear maps T : Ex → Ex such that T ◦Jx +Jx ◦T = 0. Let s : U → P be a smooth local section with x ∈ U and let e ∈ E_x be fixed. We define a local section:U → Eas in Example 5.2, so that∇_v= Γx(v)·e, for allv ∈TxM. SinceU 3 y 7→ s(y)⁻¹ J_y·(y)

is constant, it follows thatdI^s_v J()

= 0and

∇_v J()

= Γ_x(v)· J_x(e)

; then:

Γ_x(v)· J_x(e)

=∇_v J()

= (∇_vJ)(e) +J_x Γ_x(v)·e , for allv∈T_xM. We therefore obtain:

Γ_x(v)◦J_x =∇_vJ+J_x◦Γ_x(v) and hence:

I^P_x(v) =∇_vJ,

for allx∈M and allv∈TxM. In particular,I^P = 0if and only ifJ is parallel.

5.7.Example. Assumek= 2l. LetJbe an almost complex structure onE,gbe a semi-Riemannian structure onE of indexr = 2s,J₀ be the complex structure on R^kconsidered in Example 5.6 andh·,·ibe the nondegenerate symmetric bilinear form of indexronR^kdefined by:

hv, wi=

l−s

X

i=1

viwi−

l

X

i=l−s+1

viwi+

k−s

X

i=l+1

viwi−

k

X

i=k−s+1

viwi, v, w∈R^k;

observe that J0 is anti-symmetric with respect to h·,·i. Assume that Jx is gx- anti-symmetric for allx ∈ M. SetFR^u(E) = S

x∈MFR^u(E_x), where for each x ∈ M, FR^u(E_x) is the set of complex linear isometries from(R^k, J₀,h·,·i)to (Ex, Jx, gx). ThenP = FR^u(E)is aG-structure onEwhereG= U(s, l−s)is the Lie group of complex linear isometries of(R^k, J0,h·,·i). Letx∈M be fixed.

ClearlyG_xis the Lie group of complex linear isometries of(E_x, J_x, g_x)andg_xis the Lie algebra of complex lineargx-anti-symmetric endomorphisms of(Ex, Jx).

We have a linear isomorphism:

gl(Ex)/gx −→sym(Ex)⊕Lina(Ex, Jx) T +g_x 7−→ ¹₂(T +T^∗),¹₂[T −T^∗, Jx]

,

whereLina(Ex, Jx)denotes the space ofgx-anti-symmetric linear endomorphisms T ofE_xsuch thatT ◦J_x+J_x◦T = 0. Arguing as in Examples 5.2 and 5.6 we obtain:

I^P_x(v) = −¹₂∇_vg,∇_vJ−[∇_vg, Jx] ,

for allx ∈M and allv ∈TxM. In particular,I^P = 0if and only if bothgandJ are parallel.

We conclude the section with a technical lemma that will be used later on.

Lemma 5.1. Let(M₁,∇¹),(M₂,∇²) ben-dimensional affine manifolds,Gbe a Lie subgroup ofGL(Rⁿ)andP¹ ⊂FR(T M1),P² ⊂FR(T M2)beG-structures onM₁ andM₂, respectively. Assume that for allx ∈M₁,y ∈M₂ and for every G-structure preserving mapσ :T_xM₁ → T_yM₂ we haveI^P_y² ◦σ = Ad_σ ◦I^P_x¹. Let γ : I → M₁, µ : I → M₂ be smooth curves and p : I → FR(T M₁), q : I → FR(T M2) be horizontal liftings of γ and µ, respectively. For each t ∈ I, setσ(t) = q(t)◦p(t)⁻¹ : T_γ(t)M₁ → T_µ(t)M₂. Ifσ(t) γ⁰(t)

= µ⁰(t), for all t ∈ I and ifσ(t₀)isG-structure preserving for some t₀ ∈ I thenσ(t)is G-structure preserving for allt∈I.

PROOF. By partitioningI, we may assume without loss of generality that there are smooth local sectionss₁ : U → P¹,s₂ : V → P² withγ(I) ⊂ U, µ(I) ⊂ V and such thatU is the domain of a local chart ofM1. Letωⁱdenote the connection form ofFR(T Mi)and setω¯ⁱ =s^∗_iω,i= 1,2. Sincep,qare horizontal, (2.6) gives us:

(5.7) p˜⁰(t) + ¯ω_γ(t)¹ γ⁰(t)

◦p(t) = 0,˜ q˜⁰(t) + ¯ω_µ(t)² µ⁰(t)

◦q(t) = 0,˜ for allt∈I, wherep(t) =˜ s1 γ(t)−1

◦p(t)andq(t) =˜ s2 µ(t)−1

◦q(t). Now setL(t) =s2 µ(t)−1

◦σ(t)◦s1 γ(t)

= ˜q(t)◦p(t)˜ ⁻¹, so thatσ(t)isG-structure preserving if and only ifL(t)∈G. Now (5.7) implies:

(5.8) L⁰(t) =L(t)◦ω¯¹_γ(t) γ⁰(t)

−ω¯_µ(t)² µ⁰(t)

◦L(t),

for allt∈ I. SinceU is the domain of a local chart ofM1, there exists a smooth time-dependent vector fieldX:I×U →T M₁inUsuch thatγis an integral curve ofX(for instance, letX(t, x)∈T_xM₁be the vector that has the same coordinates