DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 133–140

ALMOST COMPLEX STRUCTURES

AND CONNECTIONS ON TM

Anton Dekrét

Abstract. Let α be a vector (1,1)-form on T M. In this paper natural operators transforming α into connections on T M, especially when α is an almost complex structure, are studied and some their properties are found.

Introduction

All manifolds and maps are assumed to be infinitely differentiable. Let (x^{i}) be a local chart on a manifold M and (x^{i}, x_{1}^{i}), (x^{i}, x_{1}^{i}, x_{j}^{i}) be induced

charts on T M and on the space JT M of all 1-jets of local cross - section of π : T M → M.

There are different equivalent definitions of connections on a fibre bundle, see [7]. We will use, in the case of T M, the following ones. The connection on T M is

a) a global section Γ : T M → JT M, (x^{i}, x_{1}^{i}) → (x^{i}, x_{1}^{i}, x_{j}^{i} = Γ_{j}^{i}(x, x_{1}),

b) a decomposition Γ : T (T M) = V T M ⊕ HΓ on the vertical subbundle V T M and on the so-called Γ-horizontal subbundle HΓ satisfying an equation dx_{1}^{i} = Γ_{j}^{i}dx^{j},

c) a (1,1)-form h_{Γ} on T M (horizontal form of Γ) such that T π · h_{Γ} = T π, h_{Γ}(V T M) = 0, h_{Γ} = dx^{i} ⊗ ∂/∂x^{i} + Γ_{j}^{i}dx^{j} ⊗ ∂/∂x_{1}^{i},

d) a (1,1)-form v_{Γ} (vertical form of Γ) such that v_{Γ}|_{V T M} = Id_{V T M}, v_{Γ}(T T M) = V T M, v_{Γ} = (dx_{1}^{i} − Γ_{j}^{i}dx^{i}) ⊗ ∂/∂x_{1}^{i}.

A connection Γ on T M is linear if Γ : T M → JT M is linear. Then Γ_{j}^{i}(x, x_{1}) = Γ_{kj}^{i}(x)x_{1}^{k}.

Recall some notions and well known facts: A scalar 1-form ω or a vector (1,1)-form α on T M is semibasic if ω(V T M) = 0

or α(V T M) = 0.

1991 Mathematics Subject Classification. 53C05, 58A20.

Key words and phrases. Tangent bundle, tangent value (1,1)-form, almost complex structure, connection, semispray.

This paper is in final form and will not be submitted for publication elsewhere.

If h_{Γ} is the horizontal form of Γ and β = b_{j}^{i}dx^{j} ⊗∂/∂x_{1}^{i} is a semibasic (1,1)-form with values in V T M then h_{Γ} + β is the horizontal form of the other connection denoted by Γ + β.

On T M there are the canonical semibasic (1,1)-form J = dx^{i}⊗∂/∂x_{1}^{i} induces by the identification V T M = T Mx_{M}T M and the Liouville vector field V = x_{1}^{i}∂/∂x_{1}^{i} generated by the homotheties in the individual fibres of T M.

A semispray is a vector field S : T M → T T M on T M such that J(S) = V, S = x_{1}^{i}∂/∂x^{i} + η^{i}(x, x_{1})∂/∂x_{1}^{i}. If Γ is a connection on T M then there is a unique semispray S_{Γ} which is Γ-horizontal, i.e. S_{Γ} ⊂ HΓ. It is said a Γ-semispray, S_{Γ} = x_{1}^{i}∂/∂x^{i} + Γ_{j}^{i}x_{1}^{j}∂/∂x_{1}^{i}.

A vertical prolongation of a given connection Γ on T M is the so-called connection V Γ on the fibre bundle V T M → M. In a local chart (x^{i}, x_{1}^{i}, η^{i}) on V T M it is of the form dη^{i} = Γ_{jk_{1}}^{i}η^{k}dx^{i}, dx_{1}^{i} = Γ_{j}^{i}dx^{j}, see [6], where we use throughout this paper the denotation \frac{∂f}{∂x^{i}} = f_{i}, \frac{∂f}{∂x_{1}^{i}} = f_{i_{1}}. By the identification V T M = T Mx_{M}T M

a connection \overline{V Γ}, dx_{1}^{i} = Γ_{jk_{1}}^{i}x_{1}^{k}dx^{j}, is determined on T M.

Every semispray X = x_{1}^{i}∂/∂x^{i} + η^{i}∂/∂x_{1}^{i} induces a connection Γ_{X}, dx_{1}^{i} = \frac{1}{2}η_{j_{1}}^{i}dx^{j}, [4].

Remark. Let X = x_{1}^{i}∂/∂x^{i} + Γ_{j}^{i}x_{1}^{j}∂/∂x_{1}^{i} be a Γ-semispray. Then Γ_{X} is given by the equations

dx_{1}^{i} = \frac{1}{2}(Γ_{jk_{1}}^{i}x_{1}^{j} + Γ_{k}^{i})dx^{k}.

This mean that if Γ is without torsion (Γ_{jk_{1}}^{i} = Γ_{kj_{1}}^{i}) then Γ_{X} = \frac{1}{2}(\overline{V Γ} + Γ).

An almost complex structure on a manifold M, dim M=2 m, is a (1,1)-form α on M such that α^{2} = −Id_{M}. Janyška, [3], has proved that there is not any linear connection on T M which can be constructed from α by natural polynomial operators. We are interested in this question when α is an almost complex structure on T M.

Every connection Γ on T M generates the canonical almost complex structure α_{Γ} satisfying α_{Γ}J = h_{Γ}, α_{Γ}h_{Γ} = −J, see for example [5]. In coordinates, if Γ is given by dx_{1}^{i} = Γ_{j}^{i}dx^{j} then

(1) α_{Γ} = (−Γ_{j}^{i}dx^{j} + dx_{1}^{i}) ⊗ ∂/∂x^{i} + [(δ_{j}^{i} − Γ_{k}^{i}Γ_{j}^{k})dx^{j} + Γ_{j}^{i}dx_{1}^{j}] ⊗ ∂/∂x_{1}^{i} .

Connections and almost complex structures on T M

Let α = (a_{j}^{i}dx^{j} + b_{j}^{i}dx_{1}^{j}) ⊗ ∂/∂x^{i} + (c_{j}^{i}dx^{j} + h_{j}^{i}dx_{1}^{j}) ⊗ ∂/∂x_{1}^{i} be an arbitrary (1,1)-form on T M. It is called vertical if b = JαJ = 0, i.e. if α(V T M) ⊂ V T M.

In coordinates α is an almost complex structure on T M iff

(2)  A_{j}^{i} ≡ a_{s}^{i}a_{j}^{s} + b_{s}^{i}c_{j}^{s} = −δ_{j}^{i},  B_{j}^{i} ≡ a_{s}^{i}b_{j}^{s} + b_{s}^{i}h_{j}^{s} = 0,
C_{j}^{i} ≡ c_{s}^{i}a_{j}^{s} + h_{s}^{i}c_{j}^{s} = 0,  H_{j}^{i} ≡ c_{s}^{i}b_{j}^{s} + h_{s}^{i}h_{j}^{s} = −δ_{j}^{i}.

Let us recall that according to the identification V T M ≡ T M ×_{M} T M any semibasic (1,1)-form K = K_{j}^{i}dx^{j} ⊗ ∂/∂x_{1}^{i} with values in V T M can be interpreted as a section T M → T^{∗}M ⊗_{T M} T M or as a section T M → V^{∗}T M ⊗ V T M, i.e. as a vector bundle morphism V T M → V T M over Id_{T M}.

There are two cases of α:

1. b = JαJ = b_{j}^{i}dx^{j} ⊗ ∂/∂x_{1}^{i} ≠ 0. Author, [2], has shown, that only in the case when b is regular there are natural operators Φ of zero order transforming α into connections on T M and Φ are of the form Γ = Φ(α) = Γ_{α} +φ(b, Jα^{2}J, α−αb^{−1}T π ·α), where Γ_{α} is the connection the horizontal subbundle HΓ_{α} of which is spanned on such vectors X that α(X) ∈ V T M, i.e. Γ_{j}^{i} = −b̃_{k}^{i}a_{j}^{k}, b̃_{s}^{i}b_{j}^{s} = δ_{j}^{i}, and φ is a natural operator of zero order determining by the semibasic (1,1)-forms b, Jα^{2}J, α − αb^{−1}T π · α ≡ γ with values in V T M the other semibasic V T M valued (1,1)-form on T M.

Lemma 1. Let α be a (1,1)-form on T M. Then α(V T M) is a horizontal vector bundle of a unique connection Γ iff b = JαJ is regular.

Proof. Let Γ be a connection on T M with the equation dx_{1}^{i} = Γ_{j}^{i}dx^{j}. Then α(Y) = b_{j}^{i}η^{j}∂/∂x^{i} + h_{j}^{i}η^{j}∂/∂x_{1}^{i} is Γ-horizontal for every vertical vector Y = η^{i}∂/∂x_{1}^{i} iff Γ_{k}^{i}b_{j}^{k}η^{j} = h_{j}^{i}η^{j}, i.e. iff Γ_{k}^{i}b_{j}^{k} = h_{j}^{i}. This equation has a unique solution Γ_{k}^{i} iff det b_{j}^{i} ≠ 0. Then Γ_{j}^{i} = h_{k}^{i}b̃_{j}^{k}.

The connection the horizontal subbundle HΓ of which is α(V T M) will be denoted by Γ_{α}^{2}. It is of the form Φ(α), Γ_{α}^{2} = Γ_{α} + b^{−1}Jα^{2}Jb^{−1}. It can be described by the scalar forms. Let ω = y_{i}dx^{i} +z_{i}dx_{1}^{i} be a scalar form on T M. Then the form i_{α}ω = ωα = (y_{i}a_{j}^{i} + z_{i}c_{j}^{i})dx^{j} + (b_{j}^{i}y_{i} + z_{i}h_{j}^{i})dx_{1}^{j} is semibasic iff b_{j}^{i}y_{i} + z_{i}h_{j}^{i} = 0, i.e. y_{i} = −h_{k}^{u}b̃_{i}^{k}z_{u}, i.e. iff ω|_{HΓ_{α}^{2}} = 0.

Lemma 2. Let S_{Γ_{α}} or S_{Γ_{α}^{2}} be the horizontal semispray of the connection Γ_{α} or Γ_{α}^{2}. Then α(S_{Γ_{α}}) = γ(V) and α(b^{−1}(V)) = S_{Γ_{α}^{2}}.

Proof. In coordinates α(S_{Γ_{α}}) = (c_{j}^{i} − h_{k}^{i}b̃_{s}^{k}a_{j}^{s})x_{1}^{j}∂/∂x_{1}^{i} = γ(V), α(b^{−1}(V)) = x_{1}^{i}∂/∂x^{i} + h_{s}^{i}b̃_{j}^{s}x^{j}∂/∂x_{1}^{i} = S_{Γ_{α}^{2}}.

A (1,1)-form α is called a Lagrange form if α(V) is a semispray.

Lemma 3. A (1,1)-form α on T M which is not vertical is the Lagrange form iff b(V) = V.

Proof follows from the coordinate form of the image α(V) = b_{j}^{i}x_{1}^{j}∂/∂x^{i}+h_{j}^{i}x_{1}^{j}∂/∂x_{1}^{i}.

Proposition 1. The canonical almost complex structure α_{Γ} of a connection Γ is a Lagrange form and α_{Γ}(V) is Γ-semispray.

Proof is clair from the relation (1).

Proposition 2. If α is a Lagrange (1,1)-form and b is regular then α(V) is the Γ_{α}^{2}-semispray.

Proof is the consequence of Lemma 2.

Lemma 4. If b = JαJ is regular then there exists a unique such semispray S_{α} that α(S_{α}) is a semispray.

Proof. Let S = x_{1}^{i}∂/∂x^{i} + η^{i}∂/∂x_{1}^{i} be an arbitrary semispray. Then α(S) = (a_{j}^{i}x_{1}^{j} + b_{j}^{i}η^{j})∂/∂x^{i} + (c_{j}^{i}x_{1}^{j} + d_{j}^{i}η^{j})∂/∂x_{1}^{i} is a semispray iff

a_{j}^{i}x_{1}^{j} + b_{j}^{i}η^{j} = x_{1}^{i}.

Because of regularity of b this system has a unique solution

(3)  η^{i} = b̃_{k}^{i}x_{1}^{k} − b̃_{k}^{i}a_{j}^{k}x_{1}^{j} = (b̃_{j}^{i} − b̃_{k}^{i}a_{j}^{k})x_{1}^{j}.

Definition. If b is regular then S_{α} is called the semispray of α.

Lemma 5. If S_{α} is the semispray of α then S_{α} = b^{−1}(V) + S_{Γ_{α}}, i.e. S_{α} is the (Γ_{α} + b^{−1})-semispray and α(S_{α}) is the (Γ_{α}^{2} + γ)-semispray.

Proof. Our first assertion follows immediately from (3). The second one is the consequence of the relations (4) and (5):

α(S_{α}) = x_{1}^{i}∂/∂x^{i} + (c_{s}^{i} + h_{k}^{i}b̃_{s}^{k} − h_{k}^{i}b̃_{u}^{k}a_{s}^{u})x_{1}^{s}∂/∂x_{1}^{i} (4)

γ = α − αb^{−1}T π · α = (c_{j}^{i} − h_{s}^{i}b̃_{k}^{s}a_{j}^{k})dx^{j} ⊗ ∂/∂x_{1}^{i}. (5)

Proposition 3. Let b be regular. Then α^{2} is a vertical (1,1)-form on T M iff Γ_{α} = Γ_{α}^{2}.

Proof. By the relation (2) α^{2} is vertical, Jα^{2}J = 0, iff

B_{j}^{i} ≡ a_{k}^{i}b_{j}^{k} + b_{k}^{i}h_{j}^{k} = 0, i.e. iff h_{t}^{i} = −b̃_{s}^{i}a_{k}^{s}b_{t}^{k}.

Then ^{2}Γ_{j}^{i} = h_{t}^{i}b̃_{j}^{t} = −b̃_{s}^{i}a_{j}^{s} are the Christoffels of the connection Γ_{α}. Conversely, if h_{k}^{i}b̃_{j}^{k} = −b̃_{s}^{i}a_{j}^{s} then b_{s}^{i}h_{k}^{s}b̃_{j}^{k} = −a_{j}^{i}, i.e. b_{s}^{i}h_{j}^{s} + a_{s}^{i}b_{j}^{s} = 0, i.e. α^{2} is vertical.

Proposition 4. Let b be regular and α^{2} be vertical. Then α is an almost complex structure iff Jα^{2} = −Id|_{V T M}.

Proof. In coordinates

Jα^{2} = ((a_{s}^{i}a_{j}^{s} + b_{s}^{i}c_{j}^{s})dx^{j} + (a_{s}^{i}b_{j}^{s} + b_{s}^{i}h_{j}^{s})dx^{j}) ⊗ ∂/∂x_{1}^{i}.

The form α^{2} is vertical iff a_{s}^{i}b_{j}^{s} + b_{s}^{i}h_{j}^{s} = 0. It means that if α^{2} is vertical then Jα^{2} is a semibasic (1,1)-form with values in V T M and determines a vector bundle morphism on V T M over Id_{T M} which is equal to −Id_{V T M} iff a_{s}^{i}a_{j}^{s} + b_{s}^{i}c_{j}^{s} = −δ_{j}^{i}. Then h_{j}^{i} = −b̃_{s}^{i}a_{k}^{s}b_{j}^{k}, c_{j}^{i} = −b̃_{j}^{i} − b̃_{s}^{i}a_{k}^{s}a_{j}^{k}. Then

C_{j}^{i} = c_{s}^{i}a_{j}^{s} + h_{s}^{i}c_{j}^{s} = (−b̃_{s}^{i} − b̃_{t}^{i}a_{k}^{t}a_{s}^{k})a_{j}^{s} − b̃_{t}^{i}a_{k}^{t}b_{s}^{k}(−b̃_{j}^{s} − b̃_{v}^{s}a_{u}^{v}a_{j}^{u}) = 0
H_{j}^{i} = c_{s}^{i}b_{j}^{s} + h_{s}^{i}h_{j}^{s} = (−b̃_{s}^{i} − b̃_{t}^{i}a_{k}^{t}a_{s}^{k})b_{j}^{s} − b̃_{t}^{i}a_{k}^{t}b_{s}^{k}(−b̃_{u}^{s}a_{r}^{u}b_{j}^{r}) = −δ_{j}^{i}.

The relations (2) are satisfied and α^{2} = −Id_{T T M}. The converse is clear.

Corollary. If α is a such almost complex structure that b is regular then Γ_{α} = Γ_{α}^{2}. Lemma 6. If b is regular and α is an almost complex structure then γ = −b^{−1}. Proof. Using (2) and (5) we get γ = −b̃_{j}^{i} − b̃_{k}^{i}a_{s}^{k}a_{j}^{s} + b̃_{k}^{i}a_{u}^{k}b_{r}^{u}b̃_{s}^{r}a_{j}^{s} = −b̃_{j}^{i}. Proposition 5. Let b be regular and α be an almost complex structure. Let S_{α} be the semispray of α. Then α(S_{α}) is the semispray of the connection Γ_{α} − b^{−1} and S_{α} − α(S_{α}) = 2b^{−1}(V). Proof. It immediately follow from Lemmas 5 and 6 that α(S_{α}) is the (Γ_{α} − b^{−1})- semispray. It can be written in the form α(S_{α}) = S_{Γ_{α}} −b^{−1}(V). Then S_{α}−α(S_{α}) = b^{−1}(V) + S_{Γ_{α}} − S_{Γ_{α}} + b^{−1}(V) = 2b^{−1}(V). Proposition 6. Let α^{2} be vertical and b be regular. Then the connection Γ_{α} = Γ_{α}^{2} is without torsion iff the Γ_{α}-horizontalization of the Nijenhuis-Frolicher bracket [α, J] is a (1,2)-form with values in V T M. Proof. Using the induced coordinates we get

h_{Γ_{α}} = dx^{i} ⊗ ∂/∂x^{i} + h_{k}^{i}b̃_{j}^{k}dx^{j} ⊗ ∂/∂x_{1}^{i},

[h_{Γ_{α}}, J] = (h_{uj_{1}}^{i}b̃_{k}^{u} + h_{u}^{i}b̃_{kj_{1}}^{u})dx^{j} ∧ dx^{k} ⊗ ∂/∂x_{1}^{i}, torsion of Γ_{α},

[α, J] = (a_{sj_{1}}^{i}dx^{j} ∧ dx^{s} − b_{js_{1}}^{i}dx_{1}^{j} ∧ dx^{s}) ⊗ ∂/∂x^{i} + [(c_{sj_{1}}^{i} + a_{js}^{i})dx^{j} ∧ dx^{s}+

+ (b_{js}^{i} − d_{js_{1}}^{i} − a_{sj_{1}}^{i})dx_{1}^{j} ∧ dx^{s} + b_{js_{1}}^{i}dx_{1}^{j} ∧ dx_{1}^{s}] ⊗ ∂/∂x_{1}^{i}.

Then [α, J] · h_{Γ_{α}} is a 2-form with values in V T M iff

ω ≡ (a_{sj_{1}}^{i} + b_{uj_{1}}^{i}h_{k}^{u}b̃_{s}^{k})dx^{j} ∧ dx^{s} = 0.

If α^{2} is vertical then a_{j}^{i} = −b_{u}^{i}h_{r}^{u}b̃_{s}^{r}, and ω = −b_{u}^{i}(h_{rj_{1}}^{u}b̃_{s}^{r} + h_{r}^{u}b̃_{sj_{1}}^{r})dx^{j} ∧ dx^{s}. Now the regularity of b completes our proof. Proposition 7. Let α be an almost complex structure on T M with the regular form b. Let α_{Γ_{α}} be the canonical almost complex structure determined by the connection Γ_{α}. Then α_{Γ_{α}} = α iff b = Id|_{V T M}. Proof. If b = Id_{V T M}, b_{j}^{i} = δ_{j}^{i}, then the relations (2) imply h_{j}^{i} = −a_{j}^{i}, a_{s}^{i}a_{j}^{s} = −δ_{j}^{i} − c_{j}^{i}. As Γ_{j}^{i} = −a_{j}^{i} then using (1) we get α_{Γ_{α}} = α. Conversely, if α_{Γ_{α}} = α then the relation (1) immediately gives b^{i} = δ_{j}^{i}. Remark. If α, \overline{α} are arbitrary almost complex structure then (α · \overline{α})^{−1} = \overline{α} · α. Proposition 8. Let α be an almost complex structure on T M the form b of which is regular. Then α · α_{Γ_{α}} is vertical, J · (α · α_{Γ_{α}}) = −b and (α · α_{Γ_{α}}) · J = −b^{−1}. Proof. In the case of the connection Γ_{α}, Γ_{j}^{i} = −b̃_{k}^{i}a_{j}^{k}, the relation (1) gives

α_{Γ_{α}} = (b̃_{k}^{i}a_{j}^{k}dx^{j} + dx_{1}^{i}) ⊗ ∂/∂x^{i} + [(−δ_{j}^{i} − b̃_{s}^{i}a_{k}^{s}b̃_{u}^{k}a_{j}^{u})dx^{j} − b̃_{k}^{i}a_{j}^{k}dx_{1}^{j}] ⊗ ∂/∂x_{1}^{i}. Using the relations (2) we calculate

α · α_{Γ_{α}} = −b_{j}^{i}dx^{j} ⊗ ∂/∂x^{i} + [(−b̃_{r}^{i}b̃_{k}^{r}a_{j}^{k} + h_{j}^{i})dx^{j} − b̃_{j}^{i}dx_{1}^{j}] ⊗ ∂/∂x_{1}^{i}. Then J(α · α_{Γ_{α}}) · J = 0, J(α · α_{Γ_{α}}) = −b_{j}^{i}dx^{j} ⊗ ∂/∂x_{1}^{i} = −b, (α · α_{Γ_{α}}) · J = −b̃_{j}^{i}dx^{j} ⊗ ∂/∂x_{1}^{i} = −b^{−1}.

Corollary. If Y is a vertical vector on T M then α · α_{Γ_{α}}(Y) = −b^{−1}(Y) and α_{Γ_{α}} · α(Y) = −b(Y).

2. b = JαJ = 0, i.e. α is vertical, i.e. α(V T M) ⊂ V T M. The forms a ≡ Jα = a_{j}^{i}dx^{j} ⊗ ∂/∂x_{1}^{i}, h = αJ = h_{j}^{i}dx^{j} ⊗ ∂/∂x_{1}^{i} are semibasic with values in V T M. Then H_{α} : y → ya − hy, (y_{k}^{s}) → (y_{u}^{s}a_{k}^{u} − h_{u}^{s}y_{k}^{u}), is a vector bundle morphism on T^{∗}M ⊗_{T M} T M over Id_{T M}. Author, [2], has proved that only in the case when H_{α} is regular there are natural operators Φ of zero order such that Γ = Φ(α) are connections on T M. These operators are of the form

Φ(α) = Γ_{α} + B(a, h),

where Γ_{α} is the connection the Christoffel functions of which are Γ_{j}^{i} = A_{jv}^{iu}c_{u}^{v}, where A_{jv}^{iu} are the coordinates of the inverse map to H_{α} and B is a natural operator of zero order such that B(a, h) is a semibasic (1,1)-form on T M with values in V T M.

Proposition 9. Let Γ be a connection on T M. Let a, h be semibasic (1,1)-forms on T M with values in V T M. Then there exists a unique vertical (1,1)-form on T M such that Jα = a, αJ = h and α preserves the horizontal subbundle HΓ of the connection Γ, i.e. α(HΓ) ⊂ HΓ.

Proof. Let X = ξ^{i}∂/∂x^{i} + Γ_{j}^{i}ξ^{j}∂/∂x_{1}^{i} be a Γ-horizontal vector of Γ. Let a = a_{j}^{i}dx^{j}⊗∂/∂x_{1}^{i}, h = h_{j}^{i}dx^{j}⊗∂/∂x_{1}^{i}. Let α = a_{j}^{i}dx^{j}⊗∂/∂x^{i}+(c_{j}^{i}dx^{j}+h_{j}^{i}dx_{1}^{j})⊗∂/∂x_{1}^{i} be a vertical (1,1)-form on T M such that Jα = a, αJ = h. The vector α(X) = a_{j}^{i}ξ^{j}∂/∂x^{i} + (c_{j}^{i}ξ^{j} + h_{k}^{i}Γ_{j}^{k}ξ^{j})∂/∂x_{1}^{i} is Γ-horizontal iff

(c_{j}^{i} + h_{k}^{i}Γ_{j}^{k})ξ^{j} = Γ_{k}^{i}a_{j}^{k}ξ^{j}.

It is true for any Γ-horizontal vector X iff

(6)  (Γ_{k}^{i}a_{j}^{k} − h_{k}^{i}Γ_{j}^{k}) = c_{j}^{i}.

It completes our proof.

The relation (6) gives

Corollary. If α is a such vertical (1,1)-form on T M that the map H_{α} is regular then Γ_{α} is a unique connection Γ on T M such that α(HΓ) ⊂ HΓ.

It is clear from the relations (2) that if α is a vertical almost complex structure on T M then a = Jα, h = αJ are almost complex structures on V T M and hence dim M is even.

Proposition 10. Let α be a vertical (1,1)-form on T M and Γ be a such connection on T M that α(HΓ) ⊂ HΓ. Let a = Jα, h = αJ be almost complex structures on V T M. Then α is an almost complex structure on T M.

Proof. It is the consequence of Proposition 9 that the Christoffel functions Γ_{j}^{i} of the connection Γ satisfy the relation (6). Then using a_{j}^{i}a_{k}^{j} = −δ_{k}^{i}, h_{j}^{i}h_{k}^{j} = −δ_{k}^{i} we get c_{s}^{i}a_{j}^{s} + h_{s}^{i}c_{j}^{s} = (Γ_{k}^{i}a_{p}^{k} − h_{k}^{i}Γ_{p}^{k})a_{j}^{p} + h_{s}^{i}(Γ_{k}^{s}a_{j}^{k} − h_{k}^{s}Γ_{j}^{k}) = −Γ_{j}^{i} + Γ_{j}^{i} = 0. It means that α satisfies the relations (2) and then α is an almost complex structure on T M.

Proposition 11. Let α be a vertical almost complex structure. Then the map H_{α} : y → ya − hy on T^{∗}M ⊗_{T M} T M is not regular.

Proof. Denote by h^{−}, h^{+} the maps h^{−} : y → hy, h^{+} : y → yh on T^{∗}M ⊗_{T M} T M. The maps a^{−}, a^{+} are defined analogously. Then H_{α} = a^{+} − h^{−} and using hh = −Id = aa we get

h^{−}H(y) = hya − hhy = hya + y
a^{+}H(y) = (ya − hy)a = −y − hya.

Then h^{−}H + a^{+}H = 0, i.e. (h^{−} + a^{+})H = 0. If H is regular then h^{−} + a^{+} = 0, i.e. hy + ya = 0 for all y ∈ T^{∗}M ⊗_{T M} T M. It is possible iff h = −a = kId|_{V T M}. Then k^{2} = −1. It is contradictory with the fact that h and a are real forms.

Corollary. If α is a vertical almost complex structure then there is not a such natural operator Φ of zero order that Φ(α) is a connection on T M. It can exist an operator of higher order.

Proposition 12. If α is a vertical (1,1)-form on T M such that α^{2} = Id|_{T T M} then the maps H_{α} : y → ya − hy on T^{∗}Mx_{T M}T M is regular iff a = Id = −h or a = −Id = −h.

Proof. If α^{2} = Id then a^{2} = Id = h^{2} and h^{−}H(y) = hya − y, a^{+}H(y) = y − hya. Therefore (h^{−} + a^{+})H = 0. If H is regular then h^{−} + a^{+} = 0. It is possible iff h = a = kId. Then k = ±1. Conversely, if h = −a = ±Id then H = ±2Id is regular.

Corollary. If α is such a vertical (1,1)-form on T M that α^{2} = Id,H is regular, a = Id, then \frac{1}{2}(α + Id) is the horizontal form of the connection Γ_{α} with Γ_{j}^{i} = \frac{1}{2}c_{j}^{i}.

Remarks.

1. If α is a vertical (1,1)-form on T M then the Lie derivative L_{V} α is again vertical. If α is an vertical almost complex structure then the map H_{L_{V} α}y → yJ · L_{V} α − L_{V} α · Jy on T^{∗}M ⊗_{T M} T M can be regular and hence natural operators of first order transforming α into connections φ(α) in T M can exist.

2. If S is a semispray and α is a vertical (1,1)-form on T M then the Lie derivative L_{S}α is a such (1,1)-form on T M that JL_{S}α · J = Jα − αJ = a − h.

3. Let X = A^{i}∂/∂x^{i} + B^{i}∂/∂x_{1}^{i} be a vector field and ω = ω_{j}^{i}dx^{j} ⊗ ∂/∂x_{1}^{i} be a semibasic (1,1)-form with values in V T M. Then

L_{X}ω = −A_{k_{1}}^{i}ω_{j}^{k}dx^{j} ⊗ ∂/∂x^{i} + [(ω_{jk}^{i}A^{k} + ω_{jk_{1}}^{i}B^{k} + ω_{k}^{i}A_{j}^{k} − B_{k_{1}}^{i}ω_{j}^{k})dx^{j}+

+ ω_{k}^{i}A_{j_{1}}^{k}dx_{1}^{j}] ⊗ ∂/∂x_{1}^{i}

is vertical. If X is a semispray then JL_{X}ω = −L_{S}ωJ = −kω and H_{L_{S}ω} is regular iff ω has not eingenvalues ±λ. The vector field α(X) is a semispray iff a(X) = V. If α is a vertical almost complex structure and S is a semispray then α(S) is not a semispray.

Let α be vertical and S be a semispray. Then Lie bracket [α(S), S] is vertical iff

(7)  (h_{j}^{i} − a_{j}^{i} − a_{kj_{1}}^{i}x_{1}^{k})η^{j} = a_{jk}^{i}x_{1}^{j}x_{1}^{k} − c_{j}^{i}x_{1}^{j}.

Because a(V) = a_{j}^{i}x_{1}^{j}∂/∂x_{1}^{i} and L_{a(V)}J = −(a_{j}^{i} + a_{kj_{1}}^{i}x_{1}^{k})dx^{j} ⊗ ∂/∂x_{1}^{i} then (7) immediately yields

Proposition 13. If α is a such vertical (1,1)-form on T M that the semibasic (1,1)-form h + L_{a(V)}J is regular then there is a unique semispray S_{α} such that the Lie bracket [α(S_{α}), S_{α}] is vertical.

Corollary. Under the conditions of Proposition 13 there is a connection Γ_{S_{α}} determined by the semispray S_{α} and the operator α → Γ_{S_{α}} is natural of second order.

Example. Let α be a such vertical (1,1)-form that a = Jα = −αJ = −h and let α be projectable (Jα is a (1,1)-form on M). Then h + L_{a(V)}J = 2h. If h is regular then S_{α} exists and if α is also a V B-form, i.e. α(X) is a projectable linear vector field for every projectable linear vector field X, see [1], then S_{α} is a spray and the connection Γ_{S_{α}} is linear. Such a form α can be an almost complex structure on T M.

Remark. Let A = a_{j}^{i}dx^{j} ⊗ ∂/∂x^{i} be a (1,1)-form on M. Using the theory of natural operators, [6], it is easy to see that all natural operators of zero order transforming a (1,1)-form A on M into (1,1)-forms on T M are of the form

(8)  α = kdx^{i} ⊗ ∂/∂x^{i} + (C_{j}^{i}(a)dx^{j} + kdx_{1}^{i}) ⊗ ∂/∂x_{1}^{i}, k ∈ R

where C = C_{j}^{i}(a) is a natural operator of zero order transforming a (1,1)-form A into a (1,1)-form C(A) on M. By the Janyška result, [3], there is not any linear connection Γ(α) polynomial in α. It is clear that α given by (8) cannot be an almost complex structure on T M.

References

[1] Cabras, A., Kolář, I., Special tangent valued forms and the Frölicher-Nijenuis bracket, Arch. Mathematicum (Brno) Tom. 29 (1993), 71–82.

[2] Dekrét, A., (1,1)-forms and connections on a tangent bundle T M, Acta Univ. M. Bellii, Math. 2 (1994), 3 – 8.

[3] Janyška, J., Remarks on the Nijenhuis tensor and almost complex connections, Arch. Math. (Brno) 26 No. 4 (1990), 229 – 240.

[4] Griffone, J., Structure presque-tangent et connections I., Ann. Inst. Fourier (Grenoble) 22 (1972), 287 – 334.

[5] Klein, J., On variational second order differential equations, Polynomial case, Proc. Conf. ”Diff. Geometry and its Applications 1992” (1993), 449 – 459, Silesian University, Opava.

[6] Kolář, I., Michor, P.W., Slovak, J., Natural operations in differential geometry, Springer - Verlag, 1993.

[7] Modugno, M., New results on the theory of connections, systems, overconnections and prolongations, Proc. of the conference ”Diff. geom and its Appl. 1986”, 245 – 269, Brno.

Department of Mathematics, TU Zvolen Masarykova 24, 960 53 Zvolen, SLOVAKIA E-mail: dekret@vsld.tuzvo.sk