On maximal totally real embeddings

TL;DR

This paper studies complex structures with totally real zero sections in tangent bundles, providing explicit integrability equations and fiberwise Taylor expansions under real-analytic assumptions, advancing understanding of their geometric properties.

Contribution

It introduces explicit integrability equations and fiberwise Taylor expansions for complex structures with totally real zero sections, under real-analytic conditions.

Findings

Explicit integrability equations derived

Fiberwise Taylor expansions computed

Enhanced understanding of complex structures in tangent bundles

Abstract

9We consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real-analytic along the fibers of the tangent bundle. This assumption is quite natural in view of a well known existence result by Bruhat and Whitney. We provide explicit integrability equations for such complex structures in terms of the fiberwise Taylor expansion. In a particular geometric case considered in the literature, we explicit much further the fiberwise Taylor expansion of the complex structure as well as the integrability equations.

Equations

$$d_0{\tau }_{\eta }:T_{E_p,0}⟶T_{E_p,\eta },$$

$$T_{\eta }:E_p⟶T_{E_p,\eta }.$$

$$T_{E,0_p}=d_p{0}_M\left(T_{M,p}\right)\oplus \mathrm{Ker}{d}_{0_p}{\pi }_E.$$

$$T_{E\mid M}\simeq T_M\oplus E.$$

$$A\in C^{\infty }\left(T_M,{\pi }^{\ast }\text{mathbbm}C{T}_M^{\ast }{\otimes }_{{}_{\text{mathbbm}C}}\text{mathbbm}C{T}_{T_M}\right),$$

$$\alpha ,\beta \in C^{\infty }\left(T_M,{\pi }^{\ast }{T}_M^{\ast }{\otimes }_{{}_{\text{mathbbm}R}}{T}_{T_M}\right),$$

$$T_{T_M,J_A,\eta }^{0,1}=A_{\eta }\left(\text{mathbbm}C{T}_{M,\pi \left(\eta \right)}\right)\subset \text{mathbbm}C{T}_{T_M,\eta },$$

$$A_{\eta }\left(\text{mathbbm}C{T}_{M,\pi \left(\eta \right)}\right)\cap \overline{A_{\eta }\left(\text{mathbbm}C{T}_{M,\pi \left(\eta \right)}\right)}=0.$$

$$\overline{A_{\eta }\left({\bar{\xi }}_1\right)}=A_{\eta }\left({\xi }_2\right),$$

$$B\in C^{\infty }\left(T_M,{\pi }^{\ast }\mathrm{GL}\left(T_M\right)\right).$$

$$T\in C^{\infty }\left(T_M,{\pi }^{\ast }{T}_M^{\ast }{\otimes }_{{}_{\text{mathbbm}R}}{T}_{T_M}\right),$$

$$\alpha \in C^{\infty }\left(U,{\pi }^{\ast }{T}_M^{\ast }{\otimes }_{{}_{\text{mathbbm}R}}{T}_{T_M}\right),$$

$$B\in C^{\infty }\left(U,{\pi }^{\ast }\mathrm{GL}\left(T_M\right)\right),$$

$$T_{T_M,J_A,\eta }^{0,1}=A_{\eta }\left(\text{mathbbm}C{T}_{M,\pi \left(\eta \right)}\right)\subset \text{mathbbm}C{T}_{T_M,\eta },$$

$${\xi }_{J_A}^{0,1}$$

$${\xi }_{J_A}^{1,0}$$

$$J_{A,\eta }=-{\alpha }_{\eta }{B}_{\eta }^{-1}{T}_{\eta }^{-1}\left(\text{mathbbm}{I}_{T_{T_M}}-{\alpha }_{\eta }{d}_{\eta }\pi \right)+T_{\eta }{B}_{\eta }{d}_{\eta }\pi .$$

$$J_{A,\eta }\xi$$

$$J_{A,\eta }{\alpha }_{\eta }v=T_{\eta }{B}_{\eta }v,$$

$$J_{A,\eta \mid \mathrm{Ker}{d}_{\eta }\pi }=-{\alpha }_{\eta }{B}_{\eta }^{-1}{T}_{\eta }^{-1}.$$

$$DB\in C^{\infty }\left(E,{\pi }_E^{\ast }\left(E^{\ast }\otimes T_M^{\ast }\otimes E\right)\right),$$

$$D_{\eta }B\left(v\right):={\frac{d}{dt}}_{{\mid }_{t=0}}{B}_{\eta +tv}\in T_{M,p}^{\ast }\otimes E_p,$$

$$A\neg \beta :={\mathrm{Alt}}_2\left(\beta \circ A\right),$$

$$H_{\eta }^{\nabla }\neg {\left({\nabla }^{\mathrm{End}\left(T_M\right),\pi }S\right)}_{\eta }-S_{\eta }\neg D_{\eta }S+S_{\eta }{\tau }^{\nabla }+R^{\nabla }\cdot \eta =0,$$

$$\left(A\cdot \theta \right)\left(u_1,\dots ,u_p,v_1,\dots ,v_q\right)$$

$$\left(A\neg \theta \right)\left(u_1,\dots ,u_p,v_1,\dots ,v_q\right)$$

$$(\mathrm{Circ}\theta )\left(v_1,v_2,v_3,\bullet \right)$$

$$d_1^{\nabla }:C^{\infty }\left(M,T_M^{\ast ,\otimes k}{\otimes }_{{}_{\text{mathbbm}R}}\text{mathbbm}C{T}_M\right)⟶C^{\infty }\left(M,{\Lambda }^2{T}_M^{\ast }{\otimes }_{{}_{\text{mathbbm}R}}{T}_M^{\ast ,\otimes \left(k-1\right)}{\otimes }_{{}_{\text{mathbbm}R}}\text{mathbbm}C{T}_M\right),$$

$$d_1^{\nabla }A\left({\xi }_1,{\xi }_2,\mu \right)$$

$$A$$

$$B$$

$$A{\wedge }_{1}B$$

Full text

††Key words : Totally real embeddings, Integrability equations. Linear and non-linear connections over vector bundles.

AMS Classification : 53C25, 53C55, 32J15.

1 Introduction and statement of the main result

Let $\left(E,\pi_{E},M\right)$ be a smooth vector bundle over a manifold $M$. Let $E_{p}$ be the fiber of $E$ over a point $p\in M$ and let $\eta\in E_{p}$. We consider the transition map $\tau_{\eta}\left(v\right):=\eta+v$ acting over $E_{p}$ and we consider its differential

[TABLE]

at the point [math]. Composing $d_{0}\tau_{\eta}$ with the canonical isomorphism $E_{p}\simeq T_{E_{p},0}$ we obtain an isomorphism map

[TABLE]

We denote by $0_{M}$ the zero section of $E$. Differentiating the identity $\operatorname{id}_{M}=\pi_{E}\circ 0_{M}$ we obtain $\mathbbm{I}_{T_{M,p}}=d_{0_{p}}\pi_{E}\circ d_{p}0_{M}$. This implies the decomposition

[TABLE]

We notice also the obvious equalities $\operatorname{Ker}d_{\eta}\pi_{E}=d_{0}\tau_{\eta}\left(T_{E_{p},0}\right)=T_{\eta}\left(E_{p}\right)\simeq E_{p}$, for any $\eta\in E_{p}$. Now applying this to $\eta=0_{p}$, using the previous decomposition and the canonical isomorphism $d_{p}\,0_{M}\left(T_{M,p}\right)\simeq T_{M,p}$, we infer the existence of the canonical isomorphism $T_{E,0_{p}}\simeq T_{M,p}\oplus E_{p}$, that we rewrite as

[TABLE]

Definition 1

A real sub-manifold $M$ of an almost complex manifold $\left(X,J\right)$ is called totally real if $T_{M,p}\cap J\left(T_{M,p}\right)=0_{p}$ for all $p\in M$. A totally real sub-manifold $M$ of an almost complex manifold $\left(X,J\right)$ is called maximally totally real if $\dim_{{}_{\mathbbm{R}}}M=\dim_{{}_{\mathbbm{C}}}X$.

1.1 $M$-totally real almost complex structures over $T_{M}$

We consider $M$ included inside $T_{M}$ via the zero section. We know by the isomorphism (1.2) with $E=T_{M}$, that this embedding induces the canonical isomorphism $T_{T_{M}\mid M}\simeq T_{M}\oplus T_{M}$. The vector bundle $T_{T_{M}\mid M}$ is a complex one with the canonical complex structure $J^{\operatorname{can}}:\left(u,v\right)\longmapsto\left(-v,u\right)$ acting on the fibers.

Any almost complex structure which is a continuous extension of $J^{\operatorname{can}}$ in a neighborhood of $M$ inside $T_{M}$ makes $M$ a maximally totally real sub-manifold of $T_{M}$.

Over an arbitrary small neighborhood of $M$ inside $T_{M}$ the complex distribution $T_{T_{M}}^{0,1}$ is horizontal with respect to the natural projection $\pi:T_{M}\longrightarrow M$.

We remind that the data of a smoth complex horizontal distribution over $T_{M}$ coincides with the one of section

[TABLE]

such that $d\pi\cdot A=\mathbbm{I}_{\pi^{\ast}\mathbbm{C}T_{M}}$.

For any complex vector field $\xi\in C^{\infty}\left(M,\mathbbm{C}T_{M}\right)$ we will denote by abuse of notations $A\left(\xi\right)\equiv A\cdot\left(\xi\circ\pi\right)$. The section $A$ evaluated at the point $\eta\in T_{M}$ will be denoted by $A_{\eta}$.

We notice that we can write $A=\alpha+i\beta$, with

[TABLE]

such that $d\pi\cdot\alpha=\mathbbm{I}_{\pi^{\ast}T_{M}}$ and $\beta_{\eta}=T_{\eta}B_{\eta}$, with $B\in C^{\infty}\left(T_{M},\pi^{\ast}\operatorname{End}\left(T_{M}\right)\right)$. The section $A$ determines an almost complex structure $J_{A}$ over $T_{M}$ such that

[TABLE]

if and only if

[TABLE]

This condition is equivalent to the property:

[TABLE]

implies $\xi_{1}=\xi_{2}=0$. Taking $d_{\eta}\pi$ in the equality (1.4) we infer $\xi_{1}=\xi_{2}$. Thus equality (1.4) is equivalent to $\left(\overline{A}-A\right)\left(\xi_{1}\right)=0$ and the previous property is equivalent to $\operatorname{Ker}\left(\overline{A}-A\right)=0$, i.e.

[TABLE]

We notice that with respect to the canonical complex structure of $T_{T_{M}\mid M}$ we have  the equality $\left(u,v\right)^{0,1}=\left(\xi,i\xi\right)$, with $\xi:=\left(u-iv\right)/2$. Then $J_{A}$ is an extension of this complex structure over an open neighborhood $U\subseteq T_{M}$ of $M$ if and only if for any point $p\in M$ we have $\alpha_{0_{p}}=d_{p}0_{M}$ and $B_{0_{p}}=\mathbbm{I}_{T_{M,p}}$. We denote by

[TABLE]

the canonical section which at the point $\eta\in T_{M}$ takes the value $T_{\eta}$.

Definition 2

Let $M$ be a smooth manifold. An $M$-totally real almost complex structure over an open neighborhood $U\subseteq T_{M}$ of the zero section is a couple $\left(\alpha,B\right)$ with

[TABLE]

and

[TABLE]

such that $d\pi\cdot\alpha=\mathbbm{I}_{\pi^{\ast}T_{M}}$ over $U$ and such that $\alpha_{0_{p}}=d_{p}\,0_{M}$, $B_{0_{p}}=\mathbbm{I}_{T_{M,p}}$, for all $p\in M$. The almost complex structure $J_{A}$, with $A=\alpha+iTB$ associated to $\left(\alpha,B\right)$ is the one which satisfies

[TABLE]

for all $\eta\in U\subseteq T_{M}$.

Every almost complex continuous extension of the canonical complex structure $J^{\operatorname{can}}$ of $T_{T_{M}\mid M}$ over a neighborhood of $M$ inside $T_{M}$ writes as the almost complex structure associated to a continuous $M$-totally real almost complex structure defined over a sufficiently small neighborhood of $M$.

We provide below an explicit formula for the almost complex structure $J_{A}$. For this purpose we notice first that for any vector $\xi\in T_{T_{M},\eta}$,

[TABLE]

Indeed $\xi_{J_{A}}^{0,1}\in T_{T_{M},J_{A},\eta}^{0,1}$, $\xi_{J_{A}}^{1,0}\in T_{T_{M},J_{A},\eta}^{1,0}$ and $\xi=\xi_{J_{A}}^{1,0}+\xi_{J_{A}}^{0,1}$. We deduce the expression

[TABLE]

This shows that for any $\alpha$-horizontal vector $\xi\in T_{T_{M},\eta}$, i.e. $\xi=\alpha_{\eta}\,d_{\eta}\pi\,\xi$, we have

[TABLE]

In equivalent terms

[TABLE]

for any $\eta\in U\subset T_{M}$ and any $v\in T_{M,\pi\left(\eta\right)}$. Moreover (1.5) implies

[TABLE]

A well known theorem by Bruhat and Whitney [Br-Wh] states that for any compact real-analytic manifold $M$ there exist a complex manifold $\left(X,J\right)$ and a real-analytic embedding of $M$ in $X$ such that as a sub-manifold of $X$, $M$ is maximally totally real. In addition one can arrange that $X$ is an open neighborhood $U\subseteq T_{M}$ of the zero section and $J_{\mid M}=J^{\operatorname{can}}$.

Moreover Bruhat and Whitney show [Br-Wh] that if $X$ is a real-analytic manifold equipped with two different real-analytic complex structures $J_{1}$ and $J_{2}$ which contains a real analytic sub-manifold $M$ which is maximally totally real with respect to both $J_{1}$ and $J_{2}$, then there exist neighborhoods $U_{1}$ and $U_{2}$ of $M$ inside $X$ and a real-analytic diffeomorphism $\kappa:U_{1}\longrightarrow U_{2}$ which is the identity on $M$ and is a holomorphic mapping of $\left(U_{1},J_{1}\right)$ onto $\left(U_{2},J_{2}\right)$.

In other terms the structure $J$ constructed by Bruhat and Whitney in [Br-Wh] is unique up to complex isomorphisms.

In a long series of celebrated papers inspired by the work of Grauert [Gra], Guillemin-Stenzel [Gu-St], Lempert [Lem], Lempert-Szöke [Le-Sz1, Le-Sz2]

Szöke [Szo1, Szo2], Burns [Bu1, Bu2], Burns-Halverscheid-Hind [BHH], Aslam-Burns-Irvine [ABI] as well as Bielawski [Bie] put pluri-potential and metric constraints on $J$. Some of their results will be reminded in great detail in the next section.

Their results are needed in a crucial way in analytic micro-local analysis, in pluri-potential theory (see the impressive work by Zelditch [Zel]) as well as in Hamiltonian dynamics and in geometric quantization (see the work by Morao-Nunes [Mo-Nu] and Hall-Kirwin [Ha-Ki]).

We state below our results on the integrability conditions for $J$.

1.2 The integrability equations for $M$-totally real almost complex

structures

Let $\left(E,\pi_{E},M\right)$ be a vector bundle over a manifold $M$. For an arbitrary section $B\in C^{\infty}(E,\pi_{E}^{\ast}\left(T^{\ast}_{M}\otimes E\right))$, we define the derivative along the fiber

[TABLE]

by the formula

[TABLE]

for any $\eta,v\in E_{p}$. We denote by $\operatorname{Alt}_{2}$ the alternating operator (without normalizing coefficient!) which acts on the first two entries of a tensor. For any morphism $A:T_{M}\longrightarrow E$ and any bilinear form $\beta:E\times T_{M}\longrightarrow E$ we define the contraction operation

[TABLE]

where the composition operator $\circ$ act on the first entry of $\beta$.

Theorem 1

Let $M$ be a smooth manifold and let $J_{A}$ with $A=\alpha+iTB$ be a $M$-totally real almost complex structure over an open neighborhood $U\subseteq T_{M}$ of the zero section. Let also $\nabla$ be a covariant derivative operator acting on the smooth sections of $T_{M}$. Then $J_{A}$ is integrable over $U$ if and only if the complex section $S:=T^{-1}(H^{\nabla}-\overline{A}\,)$ satisfies the equation

[TABLE]

for any point $\eta\in U$, where $\nabla^{\operatorname{End}\left(T_{M}\right),\pi}$ is the covariant derivative operator acting on the smooth sections of $\pi^{\ast}\operatorname{End}\left(T_{M}\right)$ induced by $\nabla$ and where $\tau^{\nabla}$ and $R^{\nabla}$ are respectively the torsion and curvature forms of $\nabla$.

We notice that $S_{\mid M}=i\mathbbm{I}_{T_{M}}$ by the conditions $\alpha_{0_{p}}=H^{\nabla}_{0_{p}}=d_{p}\,0_{M}$ and $B_{0_{p}}=\mathbbm{I}_{T_{M,p}}$.

Notations for the statement of the main theorem.

For any $A\in T^{\ast,\otimes p}_{M}\otimes\operatorname{End}_{{}_{\mathbbm{C}}}\left(\mathbbm{C}T_{M}\right)$ and for any $\theta\in T^{\ast,\otimes q}_{M}\otimes\mathbbm{C}T_{M}$, the product operations of tensors $A\cdot\theta,A\neg\,\theta\in T^{\ast,\otimes\left(p+q\right)}_{M}\otimes\mathbbm{C}T_{M}$ are defined by

[TABLE]

We will denote for notation simplicity $R^{\nabla}.\,\theta:=R^{\nabla}\cdot\theta-R^{\nabla}\neg\,\theta$. We will denote by $\operatorname{Circ}$ the circular operator

[TABLE]

acting on the first three entries of any $q$-tensor $\theta$, with $q\geqslant 3$. We define also the permutation operation $\theta_{2}\left(v_{1},v_{2},\bullet\right):=\theta\left(v_{2},v_{1},\bullet\right)$.

For any covariant derivative $\nabla$ acting on the smooth sections of $\mathbbm{C}T_{M}$ we define the operator

[TABLE]

with $k\geqslant 1$ as follows

[TABLE]

with $\xi_{1},\xi_{2}\in T_{M}$ and with $\mu\in T^{\oplus\left(k-1\right)}_{M}$. Moreover for any

[TABLE]

we define the exterior product

[TABLE]

as

[TABLE]

with $\xi_{1},\xi_{2}\in T_{M}$, $\eta\in T^{\oplus l}_{M}$ and $\mu\in T^{\oplus\left(k-1\right)}_{M}$. We denote by $\operatorname{Sym}_{r_{1},\ldots,r_{s}}$ the symmetrizing operator (without normalizing coefficient!) acting on the entries $r_{1},\ldots,r_{s}$ of a multi-linear form. We use in this paper the common convention that a sum and a product running over an empty set is equal respectively to 0 and 1.

With theese notations we can state our main theorem.

Theorem 2

$\left(\text{{Integrability in the fiberwise real analytic case}}\right)$.

Let $M$ be smooth manifold equipped with a torsion free covariant derivative operator $\nabla$ acting on the smooth sections of the tangent bundle $T_{M}$, let $U\subseteq T_{M}$ be an open neighborhood of the zero section with connected fibers and let $J_{A}$ be a $M$-totally real almost complex structure over $U$, which is real-analytic along the fibers of $U$. Consider the fiberwise Taylor expansion at the origin

[TABLE]

for any $\eta\in T_{M}$ in a neighborhood of the zero section and any $\xi\in T_{M,\pi\left(\eta\right)}$, with

[TABLE]

and with $\eta^{k}:=\eta^{\times k}\in T^{\oplus k}_{M,\pi\left(\eta\right)}$ and let $\nabla^{S_{1}}$ be the complex covariant derivative operator acting on the smooth sections of $\mathbbm{C}T_{M}$ defined by

[TABLE]

Then $J_{A}$ is integrable over $U$ if and only if $S_{1}\in C^{\infty}\left(M,S^{2}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M}\right)$, $($i.e. $\nabla^{S_{1}}$ is torsion free$)$ and for all $k\geqslant 2$,

[TABLE]

where $\sigma_{1}:=0$, $\beta_{1}\left(\sigma_{0}\right):=R^{\nabla^{S_{1}}}$, $\beta_{2}\left(\sigma_{1}\right):=-\frac{i}{3}(\nabla^{S_{1}}R^{\nabla^{S_{1}}})_{2}$ and for all $k\geqslant 3$,

[TABLE]

In more explicit terms

[TABLE]

[TABLE]

The assumption that the complex structure tensor is real-analytic along the fibers of the tangent bundle is quite natural. Indeed in the case $M$ is real analytic then the $M$-totally real complex structure constructed by Bruhat and Whitney [Br-Wh] is also real analytic with respect to the real analytic structure of the tangent bundle induced by $M$.

In this paper we request from the readers some knowledge of the geometric theory of linear connections. Basics of such theory can be found in the appendix.

2 Some old and new facts

2.1 The almost complex structure associated to a connection over the tangent bundle

It is well known (see [Dom]) that we can construct an $M$-totally real almost complex structure over $T_{M}$ by using the horizontal distribution $\mathcal{H}\subset T_{M}$ associated to a linear connection $\nabla$ acting on the sections of $T_{M}$. Indeed in this case we set $\alpha_{\eta}:=H_{\eta}$ and $B_{\eta}:=\mathbbm{I}_{T_{M},\pi\left(\eta\right)}$, where $\eta\longmapsto H_{\eta}$ is the horizontal map associated to $\mathcal{H}$. We will denote $J_{\mathcal{H}}:=J_{A}$. If we define for any $\eta\in T_{M,p}$ the vertical projection $\operatorname{Vert}_{\eta}:T_{T_{M},\eta}\longrightarrow T_{T_{M,p},\eta}$ as

[TABLE]

where $\pi:T_{M}\longrightarrow M$ is the canonical projection, then

[TABLE]

If we decompose any vector $\xi\in\mathbbm{C}T_{T_{M},\eta}$ in its horizontal and vertical parts $\xi=\xi^{h}+\xi^{v}$ with $\xi^{v}:=\operatorname{Vert}_{\eta}\left(\xi\right)$ then we have the expressions

[TABLE]

We infer

[TABLE]

with $\mu:=-i\,T^{-1}_{\eta}\xi^{v}$. We notice also the identity

[TABLE]

for any any $\eta\in T_{X,p}$. The distribution $T_{T_{M},J_{\mathcal{H}}}^{0,1}$ is horizontal, but the associated map does not satisfies the condition (8.5) of linear connections thanks to the identity (8.4). Therefore this distribution does not identify a linear connection. However its integrability implies that the vector bundle $T_{M}$ is flat. Indeed we have the following well known lemma due to Dombrowsky [Dom].

Lemma 1

The torsion form $\tau^{J_{\mathcal{H}}}$ of the almost complex structure $J_{\mathcal{H}}$ satisfies at the point $\eta\in T_{M}$ in the directions $V_{1},V_{2}\in T_{T_{M},J_{\mathcal{H}},\eta}^{0,1}$ the identity

[TABLE]

where $R^{\nabla}:=\nabla^{2}$ is the complex linear extension of the curvature tensor of $\nabla$, where $\tau^{\nabla}$ is the torsion of the complex connection $\nabla$ and where $v_{j}:=d_{\eta}\pi V_{j}$, $j=1,2$. In particular $J_{\mathcal{H}}$ is a complex structure if and only if the linear connection $\nabla$ is flat and torsion free.

Proof.

Let $\xi_{j}$ be vector field local extensions of $v_{j}$ such that $[\xi_{1},\xi_{2}]\pi\left(\eta\right)=0$. Then

[TABLE]

are local vector field extensions of $V_{j}$. We expand the bracket

[TABLE]

The last equality follows from to the computation at the end of the proof of lemma 19 and thanks to the identity (8.8) in the appendix. (We notice that $[T\xi_{1},T\xi_{2}]\equiv 0$, since the vector fields $T\xi_{1}$ are tangent constant along the fibers). Thanks to the assumption $[\xi_{1},\xi_{2}]\pi\left(\eta\right)=0$, we infer the equality

[TABLE]

The required formula follows from the identity

[TABLE]

The fact that that the distribution $T_{T_{M},J_{\mathcal{H}}}^{1,0}$ is horizontal implies that $\tau^{J_{\mathcal{H}}}\left(V_{1},V_{2}\right)\left(\eta\right)$ vanishes for all $V_{j}$ if and only if the quantity

[TABLE]

vanishes for all $v_{j}$. In particular for real vectors $v_{j}$ this implies that $R^{\nabla}$ and $\tau^{\nabla}$ vanish at the point $\pi\left(\eta\right)$. ∎

We observe that a connection over $T_{M}$ is flat and torsion free if and only if there exist local parallel frames with vanishing Lie brackets.

3 The symplectic approach

Let $M$ be a smooth manifold and let $\theta\in C^{\infty}(T^{\ast}_{M},T^{\ast}_{T^{\ast}_{M}})$ be the canonical $1$-form on the total space of the cotangent bundle defined as $\theta_{\lambda}:=\lambda\cdot d_{\lambda}\pi_{T^{\ast}_{M}}$, for any $\lambda\in T_{M}^{\ast}$. The canonical symplectic form over the total space $T_{M}^{\ast}$ is defined as $\Omega:=-d\theta$. Let now $g$ be a Riemann metric over $M$ viewed as a vector bundle map $g:T_{M}\longrightarrow T^{\ast}_{M}$. We define also the forms $\theta^{g}:=g^{\ast}\theta$ and $\Omega^{g}:=g^{\ast}\Omega=-d\theta^{g}$ over the total space of the tangent bundle. In explicit terms $\theta_{\eta}^{g}=g\left(\eta\right)\cdot d_{\eta}\pi_{T_{M}}$, for all $\eta\in T_{M}$, i.e.

[TABLE]

for all $\xi\in T_{T_{M},\eta}$. Let $\nabla^{g}$ be the Levi-Civita connection, defined as

[TABLE]

for any $\xi,\eta\in C^{\infty}\left(M,T_{M}\right)$. Let also $\gamma^{g}\in C^{\infty}(T_{M},T_{T_{M}}^{\ast}\otimes T_{T_{M}})$ be the Levi-Civita $1$-form, which is determined along any section $\eta\in C^{\infty}\left(M,T_{M}\right)$, by the identity $\gamma_{\eta}^{g}\cdot d\eta=T_{\eta}\nabla^{g}\eta$.

For any curve $\eta:t\longmapsto\eta_{t}\in T_{M}$, we define the covariant derivative

[TABLE]

We consider now two curves $\eta_{j}:t\longmapsto\eta_{j,t}\in T_{M}$, $j=1,2$, such that $\pi_{T_{M}}\left(\eta_{1,t}\right)=\pi_{T_{M}}\left(\eta_{2,t}\right)=x_{t}$. Then

[TABLE]

With the previous notations hold the following well known lemma (see also Klingenberg’s book [Kli] for a proof using local coordinates).

Lemma 2

The formula

[TABLE]

hold for any $\eta\in T_{M}$, $p=\pi_{T_{M}}\left(\eta\right)$ and for any $\xi_{1},\xi_{2}\in T_{T_{M},\eta}$.

Proof.

With respect to a local coordinate trivialization of the tangent bundle we can extend in a linear way the vectors $\xi_{1},\xi_{2}$ in to vector fields $\Xi_{1}$, $\Xi_{2}$ in a neighborhood of $T_{M,p}$ inside $T_{M}$. In this way $[\Xi_{1},\Xi_{2}]=0$ and thus $\Omega^{g}\left(\Xi_{1},\Xi_{2}\right)=\Xi_{2}.\theta^{g}\left(\Xi_{1}\right)-\Xi_{1}.\theta^{g}\left(\Xi_{2}\right)$. We denote by $\eta_{j,t}$, $j=1,2$ the corresponding flow lines starting from $\eta$. Then

[TABLE]

We distinguish two cases.

$\bullet$ In the case when $d_{\eta}\pi_{T_{M}}\xi_{j}=0$ for some $j$, say $j=1$, then $d_{\eta_{2,t}}\pi_{T_{M}}\Xi_{1}\left(\eta_{2,t}\right)=0$ and

[TABLE]

by the linear nature of the local extension. Then

[TABLE]

The case $j=2$ is quite similar.

$\bullet$ In the case when $d_{\eta}\pi_{T_{M}}\xi_{j}$, do not vanish for $j=1,2$, then the vector fields $\zeta_{j}:=d\pi_{T_{M}}\Xi_{j}$ are well defined and $[\zeta_{1},\zeta_{2}]=0$. Then

[TABLE]

which implies the required conclusion. ∎

We need to remind in detail also the following very well known lemma (see also [Kli]).

Lemma 3

Let $2\,\zeta^{g}:=\Omega^{g,-1}d\,|\cdot|^{2}_{g}$ and let $\Phi_{t}^{g}$ be the corresponding $1$-parameter sub-group of transformations of $T_{M}$. Then for any $\eta\in T_{M}$ the curve $c_{t}:=\pi_{T_{M}}\circ\Phi_{t}^{g}\left(\eta\right)$ is the geodesic with initial speed $\dot{c}_{0}=\eta$ and $\dot{c}_{t}=\Phi_{t}^{g}\left(\eta\right)$.

Proof.

For any $\eta\in T_{M}$ and for any $\xi\in T_{T_{M},\eta}$, let $t\longmapsto\eta_{t}\in T_{M}$ be the curve such that $\dot{\eta}_{0}=\xi$. Then

[TABLE]

and thus

[TABLE]

by the definition of the vector field $\zeta_{\eta}^{g}$. Using lemma 2 we infer

[TABLE]

In the case $d_{\eta}\pi_{T_{M}}\xi=0$, the identity (3.1) yields

[TABLE]

and thus $d_{\eta}\pi_{T_{M}}\zeta^{g}_{\eta}=\eta$. In the case $\gamma_{\eta}^{g}\,\xi=0$, the identity (3.1) yields

[TABLE]

and thus $\gamma_{\eta}^{g}\,\zeta^{g}_{\eta}=0$. We deduce the formula

[TABLE]

Thus the flow line $\eta_{t}:=\Phi_{t}^{g}\left(\eta\right)$ satisfies the identity

[TABLE]

We deduce

[TABLE]

and $\ddot{c}_{t}=H_{\dot{c}_{t}}^{g}\cdot\dot{c}_{t}$, which is the geodesic equation. ∎

We provide now a proof of the following well known result due to Lempert-Szöke [Le-Sz1]. See also Guillemin-Stenzel [Gu-St], Burns [Bu1, Bu2] and Burns-Halverscheid-Hind [BHH].

Corollary 1

Let $\left(M,g\right)$ be a smooth Riemannian manifold. A complex structure $J$ over the total space of the tangent bundle $T_{M}$ satisfies the conditions

[TABLE]

[TABLE]

if and only if for any $\eta\in T_{M}$, the complex curve $\psi_{\eta}:t+is\longmapsto s\Phi_{t}^{g}\left(\eta\right)$, defined in a neighborhood of $0\in\mathbbm{C}$, is $J$-holomorphic.

Proof.

We define the Reeb vector field $\Xi:=\Omega^{g,-1}\theta^{g}$. This vector field is independent of the metric $g$. Indeed by lemma 2 hold the identity

[TABLE]

for any $\xi\in T_{T_{M},\eta}$. Thus if $d_{\eta}\pi_{T_{M}}\xi=0$ we deduce the equality

[TABLE]

and thus $d_{\eta}\pi_{T_{M}}\Xi_{\eta}=0$. Then the identity (3.6) reduces as

[TABLE]

for any $\xi\in T_{T_{M},\eta}$. We infer the formula

[TABLE]

for all $\eta\in T_{M}$. We notice now that the identity (3.5) is equivalent to the identity

[TABLE]

and is also equivalent to the identity $\theta^{g}=-d_{{}_{J}}^{c}|\cdot|_{g}^{2}.$ Thus

[TABLE]

thanks to the fact that $J^{g}$ is integrable. We infer that the symplectic form $\Omega^{g}$ is $J$-invariant. Thus

[TABLE]

i.e.

[TABLE]

This combined with (3.7) and with (3.2) implies that (3.5) is equivalent to the identity

[TABLE]

We show now that the later combined with (3.4) is equivalent to the $J$-holomorphy of the maps $\psi_{\eta}$. For this purpose we observe that the differential of such maps is given by

[TABLE]

But

[TABLE]

thanks to (3.2). Then using the property (8.5) of the linear connection $\nabla^{g}$ we infer

[TABLE]

The complex curve $\psi_{\eta}$ is $J$-holomorphic if and only if

[TABLE]

thus, if and only if

[TABLE]

For $s_{0}\neq 0$ this is equivalent to (3.9). For $s_{0}=0$ this is equivalent to (3.4). We deduce the required conclusion. ∎

The condition (3.4) implies that $J$ is an $M$-totally real complex structure. We show now the following corollary of the main theorem 2.

Corollary 2

Let $\left(M,g\right)$ be a smooth Riemannian manifold and let $J\equiv J_{A}$ be an $M$-totally real almost complex structure over an open neighborhood $U$ of $M$ inside $T_{M}$, with connected fibers, which is real analytic along the fibers of $U\subset T_{M}$. Then $J$ is integrable over $U$ and for any $\eta\in U$, the complex curve $\displaystyle{\psi_{\eta}:t+is\longmapsto s\Phi_{t}^{g}\left(\eta\right)}$, defined in a neighborhood of $0\in\mathbbm{C}$, is $J$-holomorphic if and only if the fiberwise Taylor expansion at the origin

[TABLE]

for any $\eta\in T_{M}$ in a neighborhood of the zero section and any $\xi\in T_{M,\pi\left(\eta\right)}$, with

[TABLE]

and with $\eta^{k}:=\eta^{\times k}\in T^{\oplus k}_{M,\pi\left(\eta\right)}$, satisfies: $S_{1}=0$,

[TABLE]

for all $k\geqslant 2$, with $\Theta_{2}\left(g\right):=2R^{g}$ and with

[TABLE]

for all $k\geqslant 3$ and the metric $g$ satisfies the equations $\operatorname{Circ}\operatorname{Sym}_{3,\ldots,k+1}\Theta_{k}\left(g\right)=0$ for all $k\geqslant 4$.

Proof.

If we write $A=\alpha+iTB$ and $\alpha=H^{g}-T\Gamma$, then

[TABLE]

We set $S_{k}=\Gamma_{k}+iB_{k}$. From the proof of corollary 1 we know that in the case $J$ is integrable over $U$, the curve $\psi_{\eta}$ is $J$-holomorphic if and only if hold (3.9). The later rewrites as

[TABLE]

Using (1.7) we infer that the previous identity is equivalent to

[TABLE]

Taking $d_{\eta}\pi$ on both sides of (3.11) we deduce $\eta=B^{-1}_{\eta}\cdot\eta$. Therefore (3.11) is equivalent to the system

[TABLE]

Then the system (3.12) rewrites as

[TABLE]

and thus as $S_{k}\left(\eta^{k+1}\right)=0$ for all $k\geqslant 1$. We remind now that, according to theorem 2, the integrability of the structure $J$ implies the condition $S_{1}\in C^{\infty}(M,S^{2}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M})$. We infer $S_{1}=0$. We notice that, with the notations of the statement of theorem 2, the equation $\operatorname{Circ}\beta_{k}=0$ hold for all $k\geqslant 1$. This combined with the identity

[TABLE]

implies

[TABLE]

for all $k\geqslant 1$. So if  we apply the $\operatorname{Circ}$ operator to both sides of the definition of $S_{2}$ in the statement of theorem 2 we infer $\operatorname{Circ}S_{2}=\operatorname{Circ}\sigma_{2}=3\sigma_{2}$. If we evaluate this equality to $\eta^{3}$ we infer $S_{2}\left(\eta^{3}\right)=\sigma_{2}\left(\eta^{3}\right)$, which implies $\sigma_{2}=0$. We show now by induction that $\sigma_{k}=0$ for all $k\geqslant 2$. Indeed by the inductive assumption

[TABLE]

Applying the $\operatorname{Circ}$ operator to both sides of this identity and using the equation (3.13), we infer $\operatorname{Circ}S_{k+1}=\operatorname{Circ}\sigma_{k+1}=3\sigma_{k+1}$, which evaluated at $\eta^{k+2}$ gives $S_{k+1}\left(\eta^{k+2}\right)=\sigma_{k+1}\left(\eta^{k+2}\right)$. We deduce $\sigma_{k+1}=0$. Using the identity

[TABLE]

we infer from the statement of theorem 2 and with the notations there

[TABLE]

for $k\geqslant 2$, with $\theta_{1}:=2R^{g}$ and

[TABLE]

for all $k\geqslant 2$. Moreover we observe that the equation $\operatorname{Circ}\beta_{k}=0$, $k\geqslant 3$ rewrites as

[TABLE]

If we set $\Theta_{k}\left(g\right):=\theta_{k-1}$, for all $k\geqslant 2$ we obtain the required expansion.

On the other hand if the expansion in the statement of the lemma under consideration hold then $J$ is integrable thanks to theorem 2 and $\operatorname{Circ}S_{k}=0$ for all $k\geqslant 2$, ($S_{1}=0$). Indeed for $k=2,3$ this equality follows from the identities $\operatorname{Circ}\Theta_{k}\left(g\right)=0$ and

[TABLE]

For $k\geqslant 4$, we use the identities (3.15), (3.14) and the equations satisfied by the metric $g$. We deduce $S_{k}\left(\eta^{k+1}\right)=0$, for all $k\geqslant 1$, which is equivalent to (3.11) and so to the fact that the curves $\psi_{\eta}$ are $J$-holomorphic. ∎

We notice in particular that the equation $\operatorname{Circ}\operatorname{Sym}_{3,4,5}\Theta_{4}\left(g\right)=0$, writes as

[TABLE]

with $\tilde{R}^{g}:=\operatorname{Sym}_{2,3}R^{g}$. We will show below that the previous equation is an identity.

We remind first the following elementary and well known fact.

Lemma 4

For any covariant derivative operator $\nabla$ acting on the smooth sections of $\mathbbm{C}T_{M}$ and for any tensor $\theta\in C^{\infty}\left(X,T^{\ast,\otimes q}_{M}\otimes\mathbbm{C}T_{M}\right)$ holds the identity

[TABLE]

Proposition 1

Let $\nabla$ be a torsion free complex covariant derivative operator acting on the smooth sections of the bundle $\mathbbm{C}T_{M}$ with curvature operator $R^{\nabla}\left(\cdot,\cdot\right)\cdot\equiv R^{\nabla}\left(\cdot,\cdot,\cdot\right)$. Let $\tilde{R}^{\nabla}:=\operatorname{Sym}_{2,3}R^{\nabla}$. Then holds the identity

[TABLE]

Proof.

We expand first the term

[TABLE]

thanks to formula (3.17). Using the differential Bianchi identity we infer

[TABLE]

In order to simplify the notations in the computations that will follow we will use from now on the identification

[TABLE]

for any tensor $\theta$. We expand now the term

[TABLE]

We let

[TABLE]

and we observe the identities

[TABLE]

Summing up we obtain

[TABLE]

where we denote by $\nabla^{2}R^{\nabla}\left(\cdot\cdot\cdot\cdot\cdot\right)_{{}_{j}}$ the terms that summed up together equal zero thanks to the differential Bianchi identity for $j=1,3$ and thanks to the algebraic Bianchi identity for $j=2,4$. Using formula (3.17) we infer

[TABLE]

where as before we denote by $\nabla^{2}R^{\nabla}\left(\cdot\cdot\cdot\cdot\cdot\right)_{{}_{j}}$ the terms that summed up together equal zero thanks to the differential Bianchi identity. We deduce the expression

[TABLE]

We set now for notation simplicity $\rho:=R^{\nabla}.R^{\nabla}$ and let

[TABLE]

We observe that, by definition, the tensor

[TABLE]

satisfies the circular identity with respect to its last three entries. We expand now the term

[TABLE]

We observe the identities

[TABLE]

[TABLE]

[TABLE]

Summing up we obtain

[TABLE]

where we denote by $\rho\left(\cdot\cdot\cdot\cdot\cdot\right)_{{}_{j}}$ the terms that we sum up together using the symmetries of $\rho$. We obtain

[TABLE]

We conclude the expression

[TABLE]

We expand now the term

[TABLE]

From now on we will denote for notation simplicity $\left(123\right)\equiv R^{\nabla}\left(123\right)$ and

[TABLE]

We observe the identities

[TABLE]

[TABLE]

[TABLE]

Summing up using the symmetries of $[\cdots]$ and $(\cdots)$ we obtain

[TABLE]

We combine now the terms $[\cdot[\cdots]\cdot]_{{}_{j}}$ for each $j=1,2,3$ and we explicit and simplify them by using the algebraich Bianchi identity. We obtain

[TABLE]

Expanding further we obtain the complete expansion

[TABLE]

Expanding the terms $\rho$ present in the expression (3.20) we obtain the complete expansion of the term

[TABLE]

given by

[TABLE]

where as before we denote by $\left(\cdot\cdot\cdot\cdot\cdot\right)_{{}_{j}}$ the terms that we sum up together using the symmetries of the curvature tensor $R^{\nabla}$. All the terms summed up together cancel up. This is obvoious for all the sub indexes $j$ with the exeption of $j=3,6,9,13,16,19$ for which me must provide the detail of the computation. Indeed for $j=3$ we have

[TABLE]

For $j=6$ we have

[TABLE]

For $j=9$ we have

[TABLE]

For $j=13$ we have

[TABLE]

For $j=16$ we have

[TABLE]

For $j=19$ we have

[TABLE]

We infer the required identity (3.18). ∎

4 General connections over vector bundles

4.1 Basic definitions

Definition 3

Let $\left(E,\pi_{E},M\right)$ be a smooth vector bundle over a manifold $M$. A connection form over $E$ is a section $\gamma\in C^{\infty}\left(E,T^{\ast}_{E}\otimes T_{E}\right)$ such that $d\pi_{E}\cdot\gamma=0$ and $\gamma_{\mid\operatorname{Ker}d\pi_{E}}=\mathbbm{I}_{\operatorname{Ker}d\pi_{E}}$.

We will denote by $\gamma_{\eta}$ the connection form $\gamma$ evaluated at the point $\eta\in E$.

Lemma 5

For any connection $\gamma\in C^{\infty}\left(E,T^{\ast}_{E}\otimes T_{E}\right)$ the map

[TABLE]

is an isomorphism for all $\eta\in E$.

Proof.

The assumption $\gamma_{\mid\operatorname{Ker}d\pi_{E}}=\mathbbm{I}_{\operatorname{Ker}d\pi_{E}}$ implies $\gamma\cdot\left(\mathbbm{I}_{T_{E}}-\gamma\right)=0$. Thus $\operatorname{Im}\left(\mathbbm{I}_{T_{E}}-\gamma\right)\subseteq\operatorname{Ker}\gamma$. Then $\operatorname{Im}\left(\mathbbm{I}_{T_{E}}-\gamma\right)=\operatorname{Ker}\gamma$. Indeed if $\gamma\left(u\right)=0$ then $u=\left(\mathbbm{I}_{T_{E}}-\gamma\right)u$. On the other hand we notice that the condition $d\pi_{E}\cdot\gamma=0$ implies $d\pi_{E}\cdot\left(\mathbbm{I}_{T_{E}}-\gamma\right)=d\pi_{E}$ and thus

[TABLE]

This equality shows that the map (4.1) is surjective. The injectivity follows from the fact that if $u,v\in\operatorname{Ker}\gamma_{\eta}$ and $d_{\eta}\pi_{E}\left(u-v\right)=0$ then $u-v=\gamma\left(u-v\right)=0$ by the assumption $\gamma_{\mid\operatorname{Ker}d\pi_{E}}=\mathbbm{I}_{\operatorname{Ker}d\pi_{E}}$. ∎

We denote by $H_{\eta}^{\gamma}:=(d_{\eta}\pi_{E\mid\operatorname{Ker}\gamma_{\eta}})^{-1}$ the horizontal map. We deduce the existence of a section

[TABLE]

such that $d\pi_{E}\cdot H^{\gamma}=\mathbbm{I}_{\pi^{\ast}_{E}T_{M}}$. (We notice that $d\pi_{E}\in C^{\infty}\left(E,T^{\ast}_{E}\otimes\pi^{\ast}_{E}T_{M}\right)$). Composing both sides of (4.2) with $H_{\eta}^{\gamma}$ we infer

[TABLE]

and the smooth vector bundle decomposition $T_{E}=\operatorname{Ker}d\pi_{E}\oplus\operatorname{Ker}\gamma$.

The data of a connection form $\gamma$ is equivalent with the data of a horizontal form $H^{\gamma}$. The connection form is called linear if the horizontal form $H^{\gamma}$ satisfies

[TABLE]

where $sm_{{}_{E}}:E\oplus E\longrightarrow E$ is the sum bundle map where $\eta_{1},\eta_{2},\eta\in E$ with $\pi_{E}\left(\eta_{1}\right)=\pi_{E}\left(\eta_{2}\right)$, and $\lambda$ is a scalar.

Definition 4

The curvature form $\theta^{\gamma}\in C^{\infty}\left(E,\Lambda^{2}T^{\ast}_{E}\otimes T_{E}\right)$ of a connection form $\gamma$ is defined as

[TABLE]

for all $\xi_{1},\xi_{2}\in C^{\infty}\left(E,T_{E}\right)$.

The definition is tensorial. Indeed if $f\in C^{\infty}\left(E,\mathbbm{R}\right)$ then

[TABLE]

The conclusion follows from the fact that $\gamma\cdot\left(\mathbbm{I}_{T_{E}}-\gamma\right)=0$. We notice that

[TABLE]

and such element is uniquely determined by the curvature field $\Theta^{\gamma}$ defined as

[TABLE]

for all $\xi_{1},\xi_{2}\in T_{M,\pi_{E}\left(\eta\right)}$. In the case $\gamma$ is linear then

[TABLE]

is called the curvature operator. The terminology is consistent with the fact that if we denote by $\nabla^{\gamma}$ the covariant derivative associated to $\gamma$ then the identity $R^{\nabla^{\gamma}}=\Theta^{\gamma}$ holds, thanks to lemma 19 in the appendix.

4.2 Parallel transport

Given any horizontal form $\alpha\in C^{\infty}\left(E,\pi^{\ast}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}T_{E}\right)$ over a vector bundle $E$, the parallel transport with respect to $\alpha$ is defined as follows. We consider a smooth curve $c:\left(-\varepsilon,\varepsilon\right)\longrightarrow M$ and the section $\sigma\in C^{1}\left(\left(-\varepsilon,\varepsilon\right),c^{\ast}E\right)$ which satisfies the equation

[TABLE]

over $\left(-\varepsilon,\varepsilon\right)$ with $\sigma\left(0\right)=\eta\in E_{c\left(0\right)}$. We define the parallel transport map $\tau^{\alpha}_{c,t}:E_{c\left(0\right)}\longrightarrow E_{c\left(t\right)}$, $t\in\left(-\varepsilon,\varepsilon\right)$ along $c$ with respect to $\alpha$ as $\tau^{\alpha}_{c,t}\left(\eta\right)=\sigma\left(t\right)$.

We consider now a $C^{1}$-vector field $\xi$ over $M$ and let $\varphi_{\xi,t}$ be the associated $1$-parameter sub-group of transformations of $M$. Let $\Phi^{\alpha}_{\xi,t}:E\longrightarrow E$ be the parallel transport map along the flow lines of $\varphi_{\xi,t}$. In equivalent terms the map $\Phi^{\alpha}_{\xi,t}$ is determined by the ODE

[TABLE]

with initial condition $\Phi^{\alpha}_{\xi,0}\equiv\mathbbm{I}_{E}$. We observe that by definition of parallel transport, the map $\Phi^{\alpha}_{\xi,t}$ satisfies $\pi_{E}\circ\Phi^{\alpha}_{\xi,t}=\varphi_{\xi,t}\circ\pi_{E}$. This follows also from the equalities

[TABLE]

Moreover the vector field $\Xi^{\alpha}:=\alpha\cdot\left(\xi\circ\pi_{E}\right)$ over $E$ satisfies $\dot{\Phi}^{\alpha}_{\xi,t}=\Xi^{\alpha}\circ\Phi^{\alpha}_{\xi,t}$. Indeed

[TABLE]

We deduce that $t\longmapsto\Phi^{\alpha}_{\xi,t}$ is also a $1$-parameter sub-group of transformations of $E$.

4.3 The geometric meaning of the curvature field

The following result provides a clear geometric meaning of the curvature field.

Lemma 6

Let $\left(E,\pi_{E},M\right)$ be a smooth vector bundle over a manifold $M$ and consider a horizontal form $\alpha\in C^{\infty}\left(E,\pi^{\ast}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}T_{E}\right)$ over bundle $E$. Then the curvature field $\Theta^{\alpha}$ associated to $\alpha$ satisfies

[TABLE]

for any $\xi_{1},\xi_{2}\in C^{\infty}\left(M,T_{M}\right)$ such that $[\xi_{1},\xi_{2}]\equiv 0$ and for any $\eta\in E$.

Proof.

We observe first that if we have a family of transformations $\left(\Psi_{s}\right)_{s}$ over a manifold with $\Psi_{0}=\operatorname{id}$ and a curve $c$** then

[TABLE]

Applying the last equality to $\Psi_{s}=\varphi_{\xi_{2},-s}$ and $c_{s}:=\varphi_{\xi_{1},-t}\circ\varphi_{\xi_{2},s}\circ\varphi_{\xi_{1},t}$, we infer

[TABLE]

and thus

[TABLE]

In a similar way

[TABLE]

with $\Xi_{j}^{\alpha}:=\alpha\cdot\left(\xi_{j}\circ\pi_{E}\right)$, $j=1,2$. Let $\eta\in E_{p}$ and observe that

[TABLE]

for all parameters $t,s$, since $\varphi_{\xi_{1},-s}\circ\varphi_{\xi_{2},-t}\circ\varphi_{\xi_{1},s}\circ\varphi_{\xi_{2},t}\left(p\right)=p$ thanks to the assumption $[\xi_{1},\xi_{2}]\equiv 0$. We conclude the required geometric identity ∎

4.4 Comparison of the curvature fields of two connections

We consider now two connection forms $\gamma_{j}$, $j=1,2$ over $E$ and let $\alpha_{j}:=H^{\gamma_{j}}$ be the corresponding horizontal forms. The fact that $d\pi_{E}\left(\alpha_{1}-\alpha_{2}\right)=0$ implies that there exist a section

[TABLE]

which satisfies

[TABLE]

We want to compare the curvature fields $\Theta_{j}:=\Theta^{\gamma_{j}}$. We will denote by abuse of notations $\alpha_{j}\xi\equiv\alpha_{j}\cdot\left(\xi\circ\pi_{E}\right)$ and $B\xi\equiv B\cdot\left(\xi\circ\pi_{E}\right)$ for any $\xi\in C^{\infty}\left(M,T_{M}\right)$.

Lemma 7

In the above set up, the identity

[TABLE]

holds for any $\xi_{1},\xi_{2}\in C^{\infty}\left(M,T_{M}\right)$.

Proof.

We notice first the equalities

[TABLE]

In the last line we use the well known identity $d\pi_{E}[\alpha_{1}\xi_{1},\alpha_{1}\xi_{2}]=[\xi_{1},\xi_{2}]\circ\pi_{E}$, which follows from the fact that $d\pi_{E}\alpha_{1}\xi_{j}=\xi_{j}\circ\pi_{E}$, $j=1,2$. Let now $\Phi_{TB\xi_{2},t}$ be the $1$-parameter sub-group of transformations of $E$ associated to the vertical vector field $TB\xi_{2}$. It satisfies $\pi_{E}\circ\Phi_{TB\xi_{2},t}=\pi_{E}$. Using the standard expression of the Lie bracket

[TABLE]

we deduce that this vector field is vertical. In the same way $\left[TB\xi_{1},\alpha_{2}\xi_{2}\right]$ is vertical. It is obvious that the vector field $\left[TB\xi_{1},TB\xi_{2}\right]$ is also vertical. We infer the identity

[TABLE]

The required formula (4.3) follows from the identity

[TABLE]

that we show now. We remind first that for any vector space $V$, the canonical translation operator $T:C^{\infty}\left(V,V\right)\longrightarrow C^{\infty}\left(V,T_{V}\right)$ defined as $\left(T\xi\right)\left(v\right):=T_{v}\xi_{v}$ is a Lie algebra isomorphism, where the Lie algebra structure over $C^{\infty}\left(V,V\right)$ is defined by $[\xi,\eta]_{v}:=D_{v}\eta\cdot\xi_{v}-D_{v}\xi\cdot\eta_{v}$. Indeed if we define the action of $C^{\infty}\left(V,V\right)$ over $C^{\infty}\left(V,\mathbbm{R}\right)$ as

[TABLE]

then

[TABLE]

The fact that the bilinear form $D_{v}^{2}f$ is symmetric implies

[TABLE]

On the other hand by definition

[TABLE]

We conclude the required identity $[T\xi,T\eta]=T[\xi,\eta]$. We apply this remark to our set-up. For any point $p\in M$, we denote by $B\xi\left(p\right)\in C^{\infty}\left(E_{p},E_{p}\right)$ the map $\eta\in E_{p}\longmapsto B_{\eta}\xi\left(p\right)\in E_{p}$ and we denote by $TB\xi\left(p\right)\in C^{\infty}\left(E_{p},T_{E_{p}}\right)$ the section $\eta\in E_{p}\longmapsto T_{\eta}B_{\eta}\xi\left(p\right)\in T_{E_{p},\eta}$. Then for any $\eta\in E_{p}$

[TABLE]

which shows (4.4). ∎

We notice now that for any covariant derivative $\nabla$ over $E$, the identity (8.8) rewrites as

[TABLE]

for any vector field $\xi\in C^{\infty}\left(M,T_{M}\right)$ and any section $s\in C^{\infty}\left(M,E\right)$. We need to show the following more general formula.

Lemma 8

Let $\left(E,\pi_{E},M\right)$ be a smooth vector bundle over a manifold $M$ and let $\nabla$ be a covariant derivative operator acting on the smooth sections of $E$. Then the equality hold

[TABLE]

for any vector field $\xi\in C^{\infty}\left(M,T_{M}\right)$ and for any section $\sigma\in C^{\infty}\left(E,\pi_{E}^{\ast}E\right)$.

We observe that (4.6) implies (4.5), since $\nabla_{H^{\nabla}\xi}^{\pi_{E}}\sigma=\pi_{E}^{\ast}\left(\nabla_{\xi}s\right)$, thanks to the functorial property (8.6).

Proof.

In order to show the identity (4.6) we notice first that the assumption $\sigma\in C^{\infty}\left(E,\pi_{E}^{\ast}E\right)$ means that $\sigma$ is a map $\sigma:E\longrightarrow E$ such that $\pi_{E}\circ\sigma=\pi_{E}$. Then the $1$-parameter subgroup of transformations of $E$ associated to the vector field $T\sigma$ satisfies $\Phi_{T\sigma,t}\left(\eta\right)=\eta+t\sigma\left(\eta\right)$. Moreover with the notations in the proof of identity (8.8)

[TABLE]

The fact that $\Phi_{\xi,-t}$ is linear on the fibers of $E$ implies

[TABLE]

We infer

[TABLE]

for any $\eta\in E_{p}$. We observe that $\sigma\circ\Phi_{\xi,t}\left(\eta\right)\in E_{\varphi_{\xi,t}\left(p\right)}$. Indeed using the property $\pi_{E}\circ\sigma=\pi_{E}$ we deduce

[TABLE]

We remind now that if $t\longmapsto\eta_{t}\in E$ is a smooth curve such that $c_{t}:=\pi_{E}\left(\eta_{t}\right)$ then

[TABLE]

thanks to formula (8.7). We apply the previous identity to the curve $\eta_{t}:=\sigma\circ\Phi_{\xi,t}\left(\eta\right)\in E_{\varphi_{\xi,t}\left(p\right)}$. We obtain

[TABLE]

Moreover

[TABLE]

We conclude the equality

[TABLE]

which represents the required formula (4.6). ∎

We can show now the following result.

Lemma 9

Let $\left(E,\pi_{E},M\right)$ be a smooth vector bundle over a manifold $M$ and let $\nabla$ and $\nabla^{T_{M}}$ be covariant derivative operators acting respectively on the smooth sections of the bundles $E$ and $T_{M}$.

Then for any section $B\in C^{\infty}\left(E,\pi_{E}^{\ast}\left(T^{\ast}_{M}\otimes E\right)\right)$ the curvature field $\Theta^{\alpha}$ of the horizontal form $\alpha:=H^{\nabla}+TB$ satisfies

[TABLE]

where $\nabla^{T^{\ast}_{M}\otimes E,\pi_{E}}$ is the covariant derivative acting on the smooth sections of the bundle $\pi_{E}^{\ast}\left(T^{\ast}_{M}\otimes E\right)$, induced by $\nabla$ and $\nabla^{T_{M}}$ and where $\tau^{\nabla^{T_{M}}}$ is the torsion form of $\nabla^{T_{M}}$.

Proof.

In the case $\alpha_{2}=H^{\nabla}$ in the identity (4.3) we can apply the formula (4.6) to the sections $B\xi_{j}\in C^{\infty}\left(E,\pi_{E}^{\ast}E\right)$. We obtain

[TABLE]

Using functorial properties of the pull-back we have (with no abuse of the notations)

[TABLE]

We conclude by (4.3) that if $\alpha_{1}=\alpha=H^{\nabla}+TB$ then the curvature field $\Theta^{\alpha}$ of $\alpha$ satisfies the identity

[TABLE]

We infer the required formula (4.7). ∎

5 First reduction of the integrability equations

Proof of theorem  1.

Proof.

Let $\gamma^{A}$ be the connection form associated to the horizontal form $A$. Then the integrability of $J_{A}$ is equivalent to the condition

[TABLE]

for all smooth complex vector fields $\xi_{1}$, $\xi_{2}$ over $M$. (We remind here the use of the abusive notation $A\xi\equiv A\left(\xi\circ\pi\right)$). We denote respectively by $\Theta^{A}$ and $\Theta^{\alpha}$ the curvature fields of the horizontal distributions $A$ and $\alpha$. The integrability condition (5.1) is equivalent to the condition $\Theta^{A}\equiv 0$. Then applying the identity (4.3) with $\alpha_{1}=A$, $\alpha_{2}=\alpha$ and separating real and imaginary parts we deduce that the integrability of $J_{A}$ is equivalent to the system

[TABLE]

Let $\Gamma\in C^{\infty}\left(U,\pi^{\ast}\operatorname{End}\left(T_{M}\right)\right)$ such that $\alpha=H^{\nabla}-T\Gamma$. Using the formula (4.7) in the case $E=T_{M}$ and $\nabla=\nabla^{T_{M}}$ we can write the previous equation of the system (5.2) as

[TABLE]

We rewrite the second equation of the system (5.2) as

[TABLE]

Using formula (4.6) we infer

[TABLE]

which rewrites as

[TABLE]

We conclude that the system (5.2) is equivalent to the system

[TABLE]

It follows that, using the identification $S=\Gamma+iB$, the system (5.3) is equivalent to the complex equation (1.8). ∎

Remark 1

We notice that in the case $\left(\alpha,B\right)=\left(H^{\nabla},\mathbbm{I}_{\pi^{\ast}T_{M}}\right)$, i.e. in the case $J_{A}=J_{H^{\nabla}}$, the system (5.3) reduces to

[TABLE]

In this way we re-obtain the statement of lemma 1.

Lemma 10

Under the assumptions of the theorem 2 the $M$-totally real almost complex structure $J_{A}$ is integrable over $U$ if and only if

[TABLE]

$\left(\right.$i.e. $\nabla^{S_{1}}$ is torsion free$\left.\right)$,

[TABLE]

[TABLE]

for all $k\geqslant 2$ and for all $\xi_{1},\xi_{2},\eta\in T_{M,\pi\left(\eta\right)}$.

Proof.

Let $S:=T^{-1}(H^{\nabla}-\overline{A}\,)$. In the case the connection $\nabla$ is torsion free the equation (1.8) reduces to

[TABLE]

The identification $\mathcal{S}_{k,\eta}\cdot\xi\equiv S_{k}\left(\xi,\eta^{k}\right)$ shows that $\mathcal{S}_{k,\eta}\in T^{\ast}_{M,\pi\left(\eta\right)}\otimes\mathbbm{C}T_{M,\pi\left(\eta\right)}$, i.e.

[TABLE]

and

[TABLE]

We remind the formula

[TABLE]

for any vector field $\xi_{1}$, $\xi_{2}$ over $M$. On the other hand, by definition

[TABLE]

Let now $\bf{\eta}$ be the vector field over $\operatorname{Im}\left(\varphi_{\xi_{1},\bullet}\circ\pi\left(\eta\right)\right)$ defined by

[TABLE]

Then

[TABLE]

since $\nabla_{\xi_{1}}{\bf\eta}=0$. We conclude the identity

[TABLE]

$\xi_{1},\xi_{2}\in T_{M,\pi\left(\eta\right)}$. We infer the formula

[TABLE]

We notice now the equalities

[TABLE]

and

[TABLE]

We infer the equality

[TABLE]

Let $W\subset U$ be any set containing the zero section of $T_{M}$ such that $W\cap T_{M,p}$ is a neighborhood of $0_{p}$ for any $p\in M$ and such that the fiberwise expansion (5.7) converges over $W\cap T_{M,p}$. The fact that by assumption $U\cap T_{M,p}$ is connected implies by the fiberwise real analyticity of $S$ that $S$ is a solution of (5.6) over $U$ if and only if it satisfies (5.6) over $W$.

Using (5.8) we can write the equation (5.6) under the form

[TABLE]

over $W$. We decompose the sum

[TABLE]

thanks to the equality (5.9). If we denote by $\deg_{\eta}$the degree with respect to the fibre variable $\eta\in E_{\pi\left(\eta\right)}$ we have

[TABLE]

Thus by homogeneity the equation (5.10) is equivalent to the countable system

[TABLE]

The first equation in the system means $S_{1}\in C^{\infty}\left(M,S^{2}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M}\right)$, i.e. the complex connection $\nabla^{S_{1}}$ is torsion free. The second equation in the system (5.11) rewrites as (5.4). We show now that the equation for $k\geqslant 2$ in the system (5.11) rewrites as (5.5). Indeed using the formula

[TABLE]

where $\Gamma\in C^{\infty}(M,T^{\ast,\otimes 2}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M})$, $\theta\in C^{\infty}\left(M,T^{\ast,\otimes p}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M}\right)$ and $\xi,v_{k}\in T_{M}$, we infer

[TABLE]

since $S_{1}$ is symmetric and $S_{k}$ is symmetric in the last $k$ variables. We conclude (5.5). ∎

Remark 2

In the case $S_{k}=0$, for all $k\geqslant 2$, the previous system reduces to the equation

[TABLE]

The equation (5.12) means that the complex connection $\nabla^{S_{1}}$ acting on sections of $\mathbbm{C}T_{M}$ is flat. In the case $B=\mathbbm{I}_{\pi^{\ast}T_{M}}$, the second equation in the system (5.11) implies

[TABLE]

with $\Gamma_{1}:=S_{1}$. This means that the real connection $\nabla^{\Gamma_{1}}$ is flat.

6 Second reduction of the integrability equations

In this section we will prove the following result.

Proposition 2

Under the assumptions of the theorem 2 the $M$-totally real almost complex structure $J_{A}$ is integrable over $U$ if and only if

[TABLE]

$(\nabla^{S_{1}}R^{\nabla^{S_{1}}})_{2}\left(\xi_{1},\xi_{2},\xi_{3},\xi_{4}\right):=\nabla_{\xi_{2}}^{S_{1}}R^{\nabla^{S_{1}}}\left(\xi_{1},\xi_{3}\right)\xi_{4}$, for all $\xi_{1},\xi_{2},\xi_{3},\xi_{4}\in T_{M,\pi\left(\xi_{1}\right)}$ and for all $k\geqslant 3$,

[TABLE]

for all $\xi_{1},\xi_{2},\eta\in T_{M,\pi\left(\eta\right)}$.

We remind first that for any complex connection $\nabla$ acting over the sections of $\mathbbm{C}T_{M}$ its torsion $\tau^{\nabla}$ satisfies the identity

[TABLE]

where $d^{\nabla}$ is the covariant exterior differentiation and $\mathbbm{I}_{T_{M}}\in C^{\infty}\left(M,T^{\ast}_{M}\otimes T_{M}\right)$. Then

[TABLE]

and

[TABLE]

We conclude that if a connection is torsion free then then its curvature operator satisfies the algebraic Bianchi identity.

We denote by $\operatorname{Alt}_{p}$ the alternating operator (without normalizing coefficients!) acting on the first $p\geqslant 2$ entries of a tensor, counted from the left to the right. We notice the following very elementary fact.

Lemma 11

Let $V$ be a vector space over a field $\mathbbm{K}$ of characteristic zero. Then for any integer $p\geqslant 2$, the sequence

[TABLE]

is exact.

Proof.

The equality

[TABLE]

is obvious. We show now the equality

[TABLE]

We show first the inclusion $\subseteq$ in (6.1). We notice the equality

[TABLE]

Let now $\beta:=\operatorname{Alt}_{2}\alpha$, with $\alpha\in V^{\ast}\otimes S^{p}V^{\ast}$. Then summing up the two equalities

[TABLE]

we obtain

[TABLE]

which rewrites as

[TABLE]

i.e. $\operatorname{Circ}\beta=0$, which shows the inclusion $\subseteq$ in (6.1). In order to show the reverse inclusion in (6.1) we consider $\beta\in\Lambda^{2}V^{\ast}\otimes S^{p-1}V^{\ast}$ with $\operatorname{Circ}\beta=0$ and we will prove that $\beta=C_{p}\operatorname{Alt}_{2}\alpha$, with

[TABLE]

and with $C_{p}:=p/\left(p+1\right)!$. Indeed

[TABLE]

and

[TABLE]

Using the circular identity $\operatorname{Circ}\beta=0$, we obtain

[TABLE]

This combined with the fact that $\beta\in\Lambda^{2}V^{\ast}\otimes S^{p-1}V^{\ast}$ implies

[TABLE]

which shows the required identity. ∎

A direct consequence of the proof of lemma 11 is the following fact.

Corollary 3

Let $R\in C^{\infty}\left(M,\Lambda^{2}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M}\right)$ satisfying the algebraic Bianchi identity. Then a tensor $S\in C^{\infty}\left(M,T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}S^{2}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M}\right)$ satisfies $3R=\operatorname{Alt}_{2}S$ if and only if $S=\operatorname{Sym}_{2,3}R+\sigma$, with $\sigma\in C^{\infty}(M,S^{3}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M})$.

We infer by corollary 3 that the equation (5.4) is satisfied by $S_{2}=S^{0}_{2}+\sigma_{2}$, with

[TABLE]

and with $\sigma_{2}\in C^{\infty}\left(M,S^{3}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M}\right)$. We consider now the equation (5.5) for $k=2$, which writes as

[TABLE]

The fact that the tensor

[TABLE]

is symmetric in the last two variables implies that the equation (6.4) is equivalent to the equation

[TABLE]

that we can rewrite under the form

[TABLE]

with

[TABLE]

Then using the expression (6.3) we can rewrite equation (6.5) in the explicit form

[TABLE]

We notice that the fact that the complex connection $\nabla^{S_{1}}$ is torsion free implies that the tensor $\rho$ given by $\rho\left(\xi_{1},\xi_{2},\xi_{3},\xi_{4}\right):=\nabla^{S_{1}}_{\xi_{1}}R^{\nabla^{S_{1}}}\left(\xi_{2},\xi_{3}\right)\xi_{4}$ satisfies the circular identity with respect to the first and last three entries. Moreover $\rho$ is obviously skew-symmetric with respect to the variables $\xi_{2},\xi_{3}$.

Lemma 12

Let $\rho$ be a $4$-linear form which satisfies the circular identity with respect to the first and last three entries and which is skew-symmetric with respect to the second and third variables. Then a $4$-linear form $S$ which is symmetric with respect to the last three entries satisfies the equation

[TABLE]

if and only if

[TABLE]

with $\rho_{2}\left(\xi_{1},\xi_{2},\xi_{3},\xi_{4}\right):=\rho\left(\xi_{2},\xi_{1},\xi_{3},\xi_{4}\right)$, with $\rho_{3}\left(\xi_{1},\xi_{2},\xi_{3},\xi_{4}\right):=\rho\left(\xi_{2},\xi_{3},\xi_{1},\xi_{4}\right)$, for all $\xi_{1},\xi_{2},\xi_{3},\xi_{4}\in T_{M,\pi\left(\xi_{1}\right)}$ and with $\sigma$ a $4$-linear form which is symmetric with respect to all its entries.

Proof.

We observe first that the assumptions on $\rho$ imply $\operatorname{Circ}\operatorname{Alt}_{2}\operatorname{Sym}_{3,4}\rho=0$. Indeed

[TABLE]

and

[TABLE]

where we denote by $\rho\left(\cdot,\cdot,\cdot,\cdot\right)_{{}_{j}}$ the terms we group together. Using the assumption $\rho$ is skew-symmetric with respect to the second and third variables we infer

[TABLE]

Using the circular assumptions on $\rho$ we infer

[TABLE]

Then by the proof of lemma 11 in the case $p=3$, we infer that a $4$-linear form $S$ which is symmetric with respect to the last three entries satisfies the equation (6.7) if and only if

[TABLE]

with $\sigma$ any $4$-linear form which is symmetric with respect to all its entries, satisfies (6.7). We write now

[TABLE]

The fact that $\rho$ is skew-symmetric with respect to the second and third variables implies that $\operatorname{Sym}_{2,3,4}\rho=0$. We infer

[TABLE]

which shows the required expressions for $S$. ∎

By the equation (6.6) we can apply lemma 12 to the tensor $\rho:=\nabla^{S_{1}}R^{\nabla^{S_{1}}}$. We infer the equation

[TABLE]

We deduce that the equation (6.5) is equivalent to the equation (6.8). This concludes the proof of the proposition 2 thanks to lemma 10.

7 Third reduction of the integrability equations and proof of the main

theorem

In this section we will prove the following result.

Lemma 13

Under the assumptions of the theorem 2 the $M$-totally real almost complex structure $J_{A}$ is integrable over $U$ if and only if

[TABLE]

and for all $k\geqslant 3$,

[TABLE]

Proof.

We show that the statement of proposition 2 is equivalent to the statement of lemma 13. We show indeed by induction on $k\geqslant 3$ the following statement.

Statement 1

The tensors $S_{h}$, $h=3,\ldots,k+1$, satisfy the equations

[TABLE]

for all $h=3,\ldots,k$, for all $\xi_{1},\xi_{2},\eta\in T_{M,\pi\left(\eta\right)}$ and

[TABLE]

with $\sigma_{3}\in C^{\infty}\left(M,S^{4}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M}\right)$, if and only if the tensors $S_{h}$ satisfy for all $h=3,\ldots,k+1$, the identities

[TABLE]

with $\sigma_{h}\in C^{\infty}\left(M,S^{h+1}T^{\ast}_{M}\otimes_{{}_{\mathbbm{R}}}\mathbbm{C}T_{M}\right)$ and where for all $r=3,\ldots,k$,

[TABLE]

with $\beta_{2}:=-\frac{i}{3}(\nabla^{S_{1}}R^{\nabla^{S_{1}}})_{2}$ satisfies the equation $\operatorname{Circ}\beta_{r}=0$.

The statement 1 follows directly from the following fact.

Fact 1

Let $S_{h}$, for some $h=3,\ldots,k$, be the tensor given by (7.2). Then the tensor $S_{h+1}$ satisfies the equation (7.1) if and only if $S_{h+1}$ satisfies the identity (7.2), with $h$ replaced by $h+1$ and $\beta_{h}$ satisfies the equation $\operatorname{Circ}\beta_{h}=0$.

In order to show the fact 1 we observe first that (7.1) rewrites as

[TABLE]

Using the expression (7.2) for $S_{h}$ and the definition of $\beta_{h}$, we can rewrite the previous identity as

[TABLE]

By the proof of lemma 11 we deduce $\operatorname{Circ}\beta_{h}=0$ and

[TABLE]

Therefore the identity (7.3) is equivalent to; $\operatorname{Circ}\beta_{h}=0$ and $S_{h+1}$ satisfies (7.2), with $h$ replaced by $h+1$. This concludes the proof fact 1. We infer the required conclusion of lemma 13. ∎

Proof of the main theorem

Proof.

We show that the recursive definition of $\beta_{k}$ in the statement of lemma 13 yields the formula

[TABLE]

for all $k\geqslant 3$. We show (7.4) by induction on $k$. We notice first that the recursive definition of $\beta_{k}$ rewrites as

[TABLE]

and we write

[TABLE]

Using the inductive assumption (7.4) we infer the expressions

[TABLE]

This combined with the identity $\operatorname{Sym}_{3,\ldots,k+3}\operatorname{Sym}_{4,\ldots,k+3}=k!\operatorname{Sym}_{3,\ldots,k+3}$, yields

[TABLE]

Putting the terms together we obtain (7.4) for $\beta_{k+1}$. Then the obvious identity $d_{1}^{\nabla}\nabla=\operatorname{Alt}_{2}\nabla^{2}$ combined with the formula (3.17) allows to conclude the required expression of $\beta_{k}\equiv\beta_{k}\left(\sigma_{k-1}\right)$ in the statement of the main theorem. This concludes the proof of the main theorem. ∎

8 Appendix

In this appendix we provide some well known basic facts about the geometric theory of linear connections needed for the reading of the paper. (See also [Gau]).

8.1 The horizontal distribution associated to a linear connection

We start with the following fact.

Lemma 14

Let $\nabla$ be a linear connection acting on sections of a vector bundle $E$ over a manifold $M$. Then the linear map

[TABLE]

is independent of the sections $\sigma$ such that $\sigma\left(p\right)=\eta$.

Proof.

Let $e=\left(e_{k}\right)^{r}_{k=1}$ be a local frame of $E$ over an open set $U\subset M$. We consider the local expression $\sigma=e\cdot f$ with $f\in C^{1}\left(U,\mathbbm{R}^{r}\right)$. Let $A\in C^{\infty}\left(U,T^{\ast}_{M}\otimes\operatorname{Matrix}_{r\times r}\left(\mathbbm{R}\right)\right)$ be the connection form of $\nabla$ with respect to the local frame $e$, i.e $\nabla e=e\cdot A$. Then $\nabla\sigma=e\otimes(df+A\cdot f)$. If we denote by $\theta_{e}:U\times\mathbbm{R}^{r}\longrightarrow E_{\mid U}$ then the differential of this map at the point $\left(p,f\left(p\right)\right)$ provides an isomorphism

[TABLE]

With respect to it, the equality hold

[TABLE]

We observe now the linear identity $d\tau_{\sigma\left(p\right)}\cdot d_{p,0}\theta_{e\mid 0\oplus\mathbbm{R}^{r}}=d_{p,f\left(p\right)}\theta_{e\mid 0\oplus\mathbbm{R}^{r}}$. We infer

[TABLE]

and

[TABLE]

Thus

[TABLE]

i.e. if $\eta=e\cdot h$, then

[TABLE]

which shows the required conclusion. ∎

Let $\pi_{E}:E\longrightarrow M$ be the projection map and notice the equality $\operatorname{Ker}d_{\eta}\pi_{E}=T_{E_{p},\eta}$, for any $\eta\in E_{p}$. The identity $\pi_{E}\circ\sigma=\operatorname{id}_{M}$ implies

[TABLE]

We deduce the identity $d_{\eta}\pi_{E}\circ H_{\eta}\left(\xi\right)=\xi$. We define the horizontal distribution $\mathcal{H}\subset T_{E}$ associated to $\nabla$ as

[TABLE]

We notice now that the tangent bundle of the vector bundle $E\oplus E$ is given by the fibers

[TABLE]

and that the differential of the sum bundle map $sm_{{}_{E}}:E\oplus E\longrightarrow E$ satisfies

[TABLE]

for any $\left(v_{1},v_{2}\right)\in T_{E,\eta_{1}}\oplus T_{E,\eta_{2}}$ such that $d_{\eta_{1}}\pi_{E}\left(v_{1}\right)=d_{\eta_{2}}\pi_{E}\left(v_{2}\right)=0$. We infer that for any sections $\sigma_{j}$ of $E$ such that $\sigma_{j}\left(p\right)=\eta_{j}$, $j=1,2$, hold the equalities

[TABLE]

We conclude the property

[TABLE]

Lemma 15

For any section $\sigma\in C^{1}\left(M,E\right)$ and for any function $u\in C^{1}\left(M,\mathbbm{R}\right)$ the identity holds

[TABLE]

for any point $p\in M$.

Proof.

With the notations in the proof of lemma 14

[TABLE]

thanks to (8.1). Using the identity

[TABLE]

for any $\lambda\in\mathbbm{R}$, we conclude

[TABLE]

∎

We observe also the elementary identity

[TABLE]

for all $\eta\in E$. We show now the identity

[TABLE]

for all $\eta\in E$. Indeed let $\sigma$ be a section such that $\sigma\left(p\right)=\eta$. Using (8.3) and (8.4) we obtain the equalities

[TABLE]

The property (8.5) implies in particular $H_{0_{p}}=d_{p}0_{M}$, where $0_{M}$ is the zero section of $T_{M}$.

Definition 5

A distribution $\mathcal{H}\subset T_{E}$, is called horizontal if the map

[TABLE]

is an isomorphism for all $\eta\in E$.

Lemma 16

Any horizontal distribution $\mathcal{H}\subset T_{E}$, which satisfies the conditions $\left(\ref{aditivDistrib}\right)$ and $\left(\ref{multHoriz}\right)$ with $H_{\eta}:=\left(d_{\eta}\pi_{E\mid\mathcal{H}_{\eta}}\right)^{-1}$, determines a connection $\nabla$ over E with associated horizontal distribution $\mathcal{H}$.

Proof.

The connection $\nabla$ is defined by the formula

[TABLE]

for any $\xi\in T_{M,p}$. The definition is well posed because

[TABLE]

which follows from the identity

[TABLE]

It is obvious that the additive property of $\nabla$ is equivalent to the condition $\left(\ref{aditivDistrib}\right)$. We observe now that with the previous definition, the covariant Leibniz property

[TABLE]

is equivalent to the identity

[TABLE]

We develop the right hand side using (8.4). We infer that the previous identity is equivalent to the following one

[TABLE]

The later hold true thanks to lemma 15 and the assumption (8.5). ∎

The data of a smooth horizontal distribution over $E$ coincides with the one of section

[TABLE]

such that $d\pi_{E}\cdot H=\mathbbm{I}_{\pi^{\ast}_{E}T_{E}}$. (We notice that $d\pi_{E}\in C^{\infty}\left(E,T^{\ast}_{E}\otimes\pi^{\ast}_{E}T_{M}\right)$). Such type of section determines a connection if and only if it satisfies the identity $\left(\ref{multHoriz}\right)$.

For any vector $\Xi\in T_{E,\eta}$ we denote by

[TABLE]

its vertical component with respect to the horizontal distribution $\mathcal{H}$. In particular

[TABLE]

8.2 The induced connection

Let $\psi:N\longrightarrow M$ be a smooth map. We define the vector bundle $\psi^{\ast}E:=N\times_{\psi}E$ over $N$. In explicit terms

[TABLE]

and the projection over $N$ is given by the restriction of the projection to the first factor. We will denote by $\Psi:\psi^{\ast}E\longrightarrow E$ the restriction of the projection to the second factor. The sections of $\psi^{\ast}E$ are identified with the maps $\sigma:N\longrightarrow E$ such that $\pi_{E}\circ\sigma=\psi$. In this way, if $s$ is a section of $E$ then the section $\psi^{\ast}s:=s\circ\psi$ is a section of $\psi^{\ast}E$. More in general if $\alpha$ is a section of $\Lambda^{p}T^{\ast}_{M}\otimes E$, we define the section $\psi^{\ast}\alpha\in\Lambda^{p}T^{\ast}_{N}\otimes\psi^{\ast}E$ as

[TABLE]

We provide a generalization of lemma (15).

Lemma 17

For any section $\sigma\in C^{1}\left(N,\psi^{\ast}E\right)$ and for any function $u\in C^{1}\left(N,\mathbbm{R}\right)$ the identity holds

[TABLE]

for any point $p\in N$.

Proof.

A local frame $e$ of $E$ induces a local frame $\psi^{\ast}e$ of $\psi^{\ast}E$ over the open set $\psi^{-1}\left(U\right)$. Then $\sigma=\psi^{\ast}e\cdot f$ with $f\in C^{1}\left(\psi^{-1}\left(U\right),\mathbbm{R}^{r}\right)$. We denote by $\theta_{e}:U\times\mathbbm{R}^{r}\longrightarrow E_{\mid U}$ the trivialization map induced by the local frame $e$ of $E$. Then the differential of this map at the point $\left(\psi\left(p\right),f\left(p\right)\right)$ provides an isomorphism

[TABLE]

and

[TABLE]

for any $\xi\in T_{N,p}$ we have

[TABLE]

thanks to (8.1). Using the equality

[TABLE]

for any $\lambda\in\mathbbm{R}$, we conclude the required identity

[TABLE]

∎

The induced connection $\nabla^{\psi}$ over $\psi^{\ast}E$ is defined by the formula

[TABLE]

for any $\xi\in T_{N,p}$. It is obvious that the additive property of $\nabla^{\psi}$ follows from the condition $\left(\ref{aditivDistrib}\right)$. We show now that $\nabla^{\psi}$ satisfies the Leibniz property

[TABLE]

Indeed using lemma 17 and the identity (8.5) we have

[TABLE]

[TABLE]

We observe also that for any $s\in C^{\infty}\left(M,E\right)$ and $\xi\in T_{N,p}$ we have the equalities

[TABLE]

in other terms the functorial formula

[TABLE]

holds.

8.2.1 The induced connection (second approach)

We observe that the tangent space of $\psi^{\ast}E$ at the point $\left(y,\eta\right)$ is given by the equality

[TABLE]

Given any horizontal distribution $H\in C^{\infty}\left(E,\pi^{\ast}_{E}T^{\ast}_{M}\otimes T_{E}\right)$ over $E$, we define the horizontal distribution

[TABLE]

In explicit terms

[TABLE]

If $H$ satisfies the identities $\left(\ref{aditivDistrib}\right)$ and $\left(\ref{multHoriz}\right)$ then so does $H^{\psi}$. This follows indeed from the identities

[TABLE]

By definition of  $H^{\psi}$ we infer that the induced connection $\nabla^{\psi}$ over $\psi^{\ast}E$ satisfies the formula

[TABLE]

for any $\xi\in T_{N,y}$.

The local frame $e$ induces a local frame $\eta:=e\circ\psi$ of $\psi^{\ast}E$ over $\psi^{-1}\left(U\right)$. We compute the local connection $A^{\psi}$ form of $\nabla^{\psi}$ with respect to such frame. We notice that $\nabla^{\psi}\eta=\psi^{\ast}\left(e\cdot A\right)=\eta\cdot\psi^{\ast}A$ by the previous remark. We infer the equality $A^{\psi}=\psi^{\ast}A$.

8.2.2 Parallel transport

We consider a smooth curve $\gamma:\left(-\varepsilon,\varepsilon\right)\longrightarrow M$ and a section $\sigma\in C^{1}\left(\left(-\varepsilon,\varepsilon\right),\gamma^{\ast}E\right)$ which satisfies the equation

[TABLE]

over $\left(-\varepsilon,\varepsilon\right)$ with $\sigma\left(0\right)=\eta\in E_{\gamma\left(0\right)}$. If we write $\sigma\left(t\right)=e\left(\gamma\left(t\right)\right)\cdot f\left(t\right)$ then

[TABLE]

We infer that the parallel transport map $\tau_{\gamma,t}:E_{\gamma\left(0\right)}\longrightarrow E_{\gamma\left(t\right)}$, $t\in\left(-\varepsilon,\varepsilon\right)$ given by $\tau_{\gamma,t}\left(\eta\right)=\sigma\left(t\right)$, is linear. We show the following fact.

Lemma 18

For any smooth curve $\gamma:\left(-\varepsilon,\varepsilon\right)\longrightarrow M$ and for any section $\sigma\in C^{1}\left(\left(-\varepsilon,\varepsilon\right),\gamma^{\ast}E\right)$, holds the identity

[TABLE]

Proof.

We notice first that the term $\tau^{-1}_{\gamma,t}\cdot\sigma\left(t\right)$ is given by the intrinsic identities

[TABLE]

Integrating the first equation we infer

[TABLE]

Using the second equation we obtain

[TABLE]

Deriving with respect to the variable $t$ we obtain

[TABLE]

Evaluating at $t=0$ and multiplying both sides with $e\left(\gamma\left(0\right)\right)$ we infer the required conclusion. ∎

We consider now a $C^{1}$-vector field $\xi$ over $M$ and let $\varphi_{\xi,t}$ be the associated $1$-parameter subgroup of transformations of $M$. Let $\Phi_{\xi,t}:E\longrightarrow E$ be the parallel transport map along the flow lines of $\varphi_{\xi,t}$. It is obvious by definition, that the map $\Phi_{\xi,t}$ satisfies $\pi_{E}\circ\Phi_{\xi,t}=\varphi_{\xi,t}\circ\pi_{E}$.

The vector field $\Xi:=\dot{\Phi}_{\xi,0}$ over $E$ satisfies the equality $\Xi\left(\eta\right)=H_{\eta}\left(\xi\right)$, for any $\eta\in E$. This is a direct consequence of the definition of the induced connection along the flow lines of $\xi$.

To any section $\sigma\in C^{1}\left(M,E\right)$ we can associate a $C^{1}$-vector field $\Sigma$ over $E$ defined as $\Sigma\left(\eta\right):=T_{\eta}[\sigma\circ\pi_{E}\left(\eta\right)]$. Let $\Phi_{\Sigma,t}$ be the associated $1$-parameter subgroup of transformations of $E$. In explicit terms it satisfies

[TABLE]

Then

[TABLE]

The fact that the map $\Phi_{\xi,-t}$ is linear on the fibers implies

[TABLE]

Thus for any $\eta\in E_{p}$ holds

[TABLE]

We conclude

[TABLE]

i.e for any $\eta\in E$ the equality holds

[TABLE]

Iterating twice we deduce the identity

[TABLE]

Moreover the fact that by (8.8) the vector fields $[\Xi_{j},\Sigma]$, $j=1,2$ are tangent to the fibers of $E$ and constant along them implies

[TABLE]

8.3 The geometric meaning of the curvature tensor

Lemma 19

Let $R:=\nabla^{2}$ be the curvature tensor of the connection $\nabla$. Then for any vector fields $\xi_{1},\xi_{2}$ over $M$ and for any $\eta\in E$ the identity holds

[TABLE]

Proof.

Let $\sigma$ be a local section of $E$ such that $\sigma\left(p\right)=\eta$. By definition of horizontal lift $\Xi$ of a vector field $\xi$ we have

[TABLE]

We infer by (8.8) the identity

[TABLE]

We infer $\sigma_{\ast}\xi=\Xi+[\Xi,\Sigma]$ over $\operatorname{Im}\sigma$. Thus

[TABLE]

thanks to (8.10). We rewrite the previous equality as

[TABLE]

Using (8.9) we deduce

[TABLE]

We infer the required conclusion. ∎