DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 329–335

PROJECTORS OF A 3–WEB

Alena Vanžurová

Abstract. Our aim is to investigate in details projectors of a 3-web, to find a correspondence between Nijenhuis brackets (denoted by [ , ]) of the projector fields on a 3-web manifold and properties of the Chern connections corresponding to the web.

1. Three-webs. A 3-web on an even-dimensional manifold M_{2n} can be regarded as a triple (D_{1}, D_{2}, D_{3}) of smooth n-dimensional integrable distributions which are pairwise complementary (T M = D_{α} ⊕ D_{β} for α ≠ β). Web equivalences are local diffeomorphisms of the underlying manifolds f : M → M′ which respect web distributions, T f(D_{α}) = D′_{α} (another speaking, the tangent map T f is interchangeable with projector tensor fields).

The web distributions determine three couples of complementary projectors P_{α}^{β}, ker P_{β}^{α} = im P_{α}^{β} = D_{α}, P_{α}^{β} + P_{β}^{α} = I, (P_{α}^{β})^{2} = P_{α}^{β}, and (1, 1)-tensor fields B_{γ} = P_{γ}^{α} − P_{β}^{γ} which are involutory, (B_{γ} − I)(B_{γ} + I) = 0. Since all web distributions are integrable they form integrable almost product structures [D_{α}, D_{β}] for which [P_{α}^{β}, P_{α}^{β}] = 0.

In the following we will use local coordinates (U; x^{1}, …, x^{2n}) on M which are adapted to the almost product structure [D_{1}, D_{2}]. This means that the holonomic frame (∂_{i} | ∂_{n+i}) satisfies at any point x ∈ U:

(∂_{1}, …, ∂_{n})_{x} is a basis of D_{1}_{x}, (∂_{n+1}, …, ∂_{2n})_{x} is a basis of D_{2}_{x}.

We use the notation ∂_{i} = ∂/∂x^{i}, ∂_{n+i} = ∂/∂x^{n+i}, and similarly for derivatives of

functions, \frac{∂f^{i}}{∂x^{j}} = f_{x^{j}}^{i} etc.

2. Projectors. Let (D_{1}, D_{2}, D_{3}) be a 3-web on M. About any point p ∈ M, choose a fixed local adapted chart (U; x^{i}), U ∋ p. In adapted coordinates, the projectors associated with the web are represented by (2n, 2n)-matrices as follows:

1991 Mathematics Subject Classification. 53A60.

Key words and phrases. Manifold, three-web, distribution, projector, Nijenhuis tensor.

Supported by grant No. 201/95/1631 of The Grant Agency of Czech Republic.

This paper is in final form and no version of it will be submitted for publication elsewhere.

(1)  \begin{matrix}P_{1}^{2} = ( \begin{matrix}I & 0 \\ 0 & 0 \end{matrix} )  P_{2}^{1} = ( \begin{matrix}0 & 0 \\ 0 & I \end{matrix} ) \\ P_{1}^{3} = ( \begin{matrix}I & 0 \\ −P & 0 \end{matrix} )  P_{3}^{1} = ( \begin{matrix}0 & 0 \\ P & I \end{matrix} ) \end{matrix}

where P = (p_{i}^{k}), p_{i}^{k} = p_{i}^{k}(x^{1}, …, x^{2n}) is a regular (n, n)-matrix at any point, I and 0 denote zero and unit (n, n)-matrices respectively. It can be verified that

P_{2}^{3} = id_{D_{2}}, P_{2}^{3} | D_{1} = (P_{1}^{3} | D_{2})^{−1}.

Denoting by P̃ = (q_{j}^{k}) the inverse matrix of P = (p_{i}^{k}), p_{i}^{k} · q_{k}^{j} = δ_{i}^{j}, we can write

P_{3}^{2} = ( \begin{matrix}I & P̃ \\ 0 & 0 \end{matrix} ) , P_{2}^{3} = ( \begin{matrix}0 & −P̃ \\ 0 & I \end{matrix} ) .

The coordinate expressions of the web-associated fields B_{γ} in adapted coordinates are

B_{1} = P_{1}^{2} − P_{3}^{1} = ( \begin{matrix}I & 0 \\ −P & I \end{matrix} ) , B_{2} = P_{2}^{3} − P_{1}^{2} = ( \begin{matrix}−I & −P̃ \\ 0 & I \end{matrix} ) ,

B_{3} = P_{3}^{1} − P_{2}^{3} = ( \begin{matrix}0 & P̃ \\ P & 0 \end{matrix} ) .

Further,

P_{1}^{2}B_{3} = P_{3}^{1} − P_{2}^{1} = ( \begin{matrix}0 & 0 \\ P & 0 \end{matrix} ) , B_{3}P_{1}^{2} = P_{3}^{2} − P_{1}^{2} = ( \begin{matrix}0 & P̃ \\ 0 & 0 \end{matrix} ) .

3. Regular webs. A 3-web is regular^{1} if it is locally equivalent with a 3web formed by tangent bundles to three foliations of parallel n-planes (in general position) of an affine space A_{2n}. Locally, it can be realized as a Cartesian product of a commutative Lie group and its opposite group, (L, ∘)×(L, ∘^{op}) where x∘^{op}y = y ∘ x.

Let us consider a change of local adapted coordinates

Φ : \begin{matrix}y^{i} = & f^{i}(x^{1}, …, x^{n}), \\ y^{n+i} = & f^{n+i}(x^{n+1}, …, x^{2n}), \end{matrix} \begin{matrix}f_{x^{n+j}}^{i} = 0, \\ f_{x^{j}}^{n+i} = 0. \end{matrix}

Now a 3-web is regular if and only if there is a local [D_{1}, D_{2}]-adapted coordi-

nate system (y^{i}) such that D_{3} is a span of \{\frac{∂}{∂y^{i}} + \frac{∂}{∂y^{n+i}} \}_{i=1,...,n}. The vector

fields (\frac{∂}{∂y^{i}} + \frac{∂}{∂y^{n+i}}) constitue a basis of D_{3} if and only if the system of equations f_{n+i}^{n+j} − p_{i}^{s}f_{x^{s}}^{j} = 0, i, j = 1, …, n is satisfied where p_{i}^{s} are components of the projector P_{3}^{1} introduced in (1), [10].

1More frequently, parallelizable.

Lemma 1. In the above notation, the system of partial differential equations

f_{x^{n+j}}^{i} = 0, f_{x^{j}}^{n+i} = 0, f_{n+i}^{n+j} − p_{i}^{s}f_{x^{s}}^{j} = 0, i, j, s = 1, …, n

is solvable if and only if the following conditions are satisfied [10, 12] :

(2)  (P_{x^{n+i}})_{j}^{s} = (P_{x^{n+j}})_{i}^{s}, (P̃_{x^{i}})_{j}^{s} = (P̃_{x^{j}})_{i}^{s},
(3)  P_{x^{n+j}x^{s}} = P_{x^{n+j}} · P̃ · P_{x^{s}} = −P_{x^{n+j}} · P̃_{x^{s}} · P.

The condition (3) can be written as (P_{x^{n+j}} · P̃)_{x^{s}} = 0. In components, (2) takes the form

I. integrability condition: \frac{∂p_{i}^{s}}{∂x^{n+j}} − \frac{∂p_{j}^{s}}{∂x^{n+i}} = 0,
II. integrability condition: \frac{∂q_{i}^{s}}{∂x^{j}} − \frac{∂q_{j}^{s}}{∂x^{i}} = 0.

In the following, we will re-formulate (2) as a condition on Nijenhuis brackets of projectors.

4. Evaluations in adapted coordinates. Let us evaluate the Nijenhuis brackets of projectors on vector fields of the holonomic frame.^{2}

{P_{1}^{2}, P_{1}^{2}} = 0 follows by integrability of [D_{1}, D_{2}].

[P_{1}^{3}, P_{1}^{2}](∂_{i}, ∂_{j}) = [P_{1}^{3}, P_{1}^{2}](∂_{i}, ∂_{n+j}) = 0,

[P_{1}^{3}, P_{1}^{2}](∂_{n+i}, ∂_{n+j}) = (\frac{∂p_{j}^{k}}{∂x^{n+i}} − \frac{∂p_{i}^{k}}{∂x^{n+j}} ) ∂_{k} = ((P_{x^{n+i}})_{j}^{k} − (P_{x^{n+j}})_{i}^{k}) ∂_{k};

[P_{2}^{3}, P_{1}^{2}](∂_{i}, ∂_{n+j}) = [P_{2}^{3}, P_{1}^{2}](∂_{n+i}, ∂_{n+j}) = 0,

[P_{2}^{3}, P_{1}^{2}](∂_{i}, ∂_{j}) = (\frac{∂q_{i}^{k}}{∂x^{j}} − \frac{∂q_{j}^{k}}{∂x^{i}} ) ∂_{n+k} = ((P̃_{x^{j}})_{i}^{k} − (P̃_{x^{i}})_{j}^{k}) ∂_{n+k}.

Further,

\begin{matrix}[P_{1}^{2}, B_{3}] & (∂_{i}, ∂_{n+j}) = 0, \\ [P_{1}^{2}, B_{3}] & (∂_{i}, ∂_{j}) = ((P̃_{x^{j}})_{i}^{k} − (P̃_{x^{i}})_{j}^{k}) ∂_{n+k}, \\ [P_{1}^{2}, B_{3}] & (∂_{n+i}, ∂_{n+j}) = ((P_{x^{n+i}})_{j}^{k} − (P_{x^{n+j}})_{i}^{k}) ∂_{k}. \end{matrix}

So (2) is equivalent with a couple of conditions [P_{1}^{3}, P_{1}^{2}] = [P_{2}^{3}, P_{1}^{2}] = 0, or with a single condition [P_{1}^{2}, B_{3}] = 0. We will verify that if (2) holds then also the remaining couples of projectors vanish.

[P_{1}^{3}, P_{2}^{3}](∂_{i}, ∂_{j}) = ((P̃_{x_{j}})_{i}^{s} − (P̃_{x_{i}})_{j}^{s})(P)_{s}^{k}∂_{k} + ((P̃_{x^{j}})_{i}^{k} − (P̃_{x^{i}})_{j}^{k}) ∂_{n+k} = 0,
[P_{1}^{3}, P_{2}^{3}](∂_{n+i}, ∂_{n+j}) = ((P_{x^{n+j}})_{i}^{k} − (P_{x^{n+i}})_{j}^{k}) ∂_{k}
+ ((P_{x^{n+i}})_{j}^{s} − (P_{x^{n+j}})_{i}^{s})(P̃)_{s}^{k} ∂_{n+k} = 0,
[P_{1}^{3}, P_{2}^{3}](∂_{i}, ∂_{n+j}) = ((P̃_{x^{n+j}})_{i}^{s}(P)_{s}^{k} + (P̃)_{i}^{s}(P_{x^{n+s}})_{j}^{k}) ∂_{k}
− ((P)_{j}^{s}(P̃_{x^{s}})_{i}^{k} + (P_{x^{i}})_{j}^{s}(P̃)_{s}^{k}) ∂_{n+k}.

2We introduce \{P, Q\} = \frac{1}{2} [P, Q] for commuting P and Q.

By (2) and (P̃ · P)_{i}^{j} = δ_{i}^{j} we obtain

\begin{aligned}(P̃_{x^{n+j}})_{i}^{s}(P)_{s}^{k} + (P̃)_{i}^{s}(P_{x^{n+s}})_{j}^{k} & = (P̃_{x^{n+i}})_{j}^{s}(P)_{s}^{k} + (P̃)_{i}^{s}(P_{x^{n+j}})_{s}^{k} \\ & = \frac{∂}{∂x^{n+j}} (P̃ · P)_{i}^{k} = 0. \end{aligned}

Similarly,

(P)_{j}^{s}(P̃_{x^{s}})_{i}^{k} + (P_{x^{i}})_{j}^{s}(P̃)_{s}^{k} = \frac{∂}{∂x^{i}} (P̃ · P)_{j}^{k} = 0.

Since web distributions are integrable the almost product structures [D_{1}, D_{3}] and [D_{2}, D_{3}] are also integrable, that is, [P_{1}^{3}, P_{1}^{3}] = [P_{2}^{3}, P_{2}^{3}] = 0. This can be obtained also by the direct evaluation:

\begin{matrix}\{P_{1}^{3}, P_{1}^{3}\} & (∂_{i}, ∂_{j}) = \{P_{1}^{3}, P_{1}^{3}\}(∂_{i}, ∂_{n+j}) = 0, \\ \{P_{1}^{3}, P_{1}^{3}\} & (∂_{n+i}, ∂_{n+j}) = ((P)_{i}^{s}(P_{x^{s}})_{j}^{k} − (P)_{j}^{s}(P_{x^{s}})_{i}^{k} + (P_{x^{n+i}})_{j}^{k} − (P_{x^{n+j}})_{i}^{k}) ∂_{k}; \\ \{P_{2}^{3}, P_{2}^{3}\} & (∂_{i}, ∂_{n+j}) = \{P_{2}^{3}, P_{2}^{3}\}(∂_{n+i}, ∂_{n+j}) = 0, \\ \{P_{2}^{3}, P_{2}^{3}\} & (∂_{i}, ∂_{j}) = ((P̃)_{i}^{s}(P̃_{x^{n+s}})_{j}^{k} − (P̃)_{j}^{s}(P̃_{x^{n+s}})_{i}^{k} + (P̃_{x^{i}})_{j}^{k} − (P̃_{x^{j}})_{i}^{k}) ∂_{n+k}. \end{matrix}

By the second equality in (2), (P̃_{x^{t}}P)_{i}^{l} = (P̃_{x^{i}}P)_{t}^{l} which we can rewrite in the form (P̃P_{x^{t}})_{i}^{l} = (P̃P_{x^{i}})_{t}^{l}. This is equivalent (using multiplicaton by P twice) with the equality (P)_{k}^{t}(P_{x^{t}})_{m}^{l} = (P)_{m}^{i}(P_{x^{i}})_{k}^{l}. Similarly, by the first part of (2), (P_{x^{n+i}} P̃)_{j}^{k} = (P_{x^{n+j}} P̃)_{i}^{k} holds. We can re-write this equality as (P P̃_{x^{n+i}})_{j}^{k} = (P P̃_{x^{n+j}})_{i}^{k} which is equivalent with (P̃)_{m}^{i}(P̃_{x^{n+i}})_{t}^{k} = (P̃)_{t}^{j}(P̃_{x^{n+j}})_{m}^{k}. So we have proved:

Proposition 1. The following conditions are equivalent:

(i) [P_{1}^{2}, B_{3}] = 0,

(ii) [P_{1}^{3}, P_{1}^{2}] = [P_{2}^{3}, P_{1}^{2}] = 0,

(iii) [P_{α}^{β}, P_{γ}^{δ}] = 0 (α ≠ β, γ ≠ δ).

5. Chern connections. A three-web manifold is endowed with three Chern connections ∇^{γ}, γ = 1, 2, 3

∇^{γ}_{X}Y = ∑_{(α,β)∈S_{2}}(P_{γ}^{α} − P_{β}^{α})[P_{α}^{β}X, P_{γ}^{β}Y] + P_{α}^{β}[P_{β}^{α}X, P_{α}^{β}Y]

such that

(i) ∇^{γ}_{X}P_{α}^{β}Y = 0 for any vector fields X, Y and α, β, γ ∈ {1, 2, 3} (all projectors of the web are covariantly constant),

(ii) for any permutation of indexis (α, β, γ), T^{γ} (P_{α}^{β}X, P_{β}^{α}Y) = 0 (all couples of “ homogeneous ” vectors are conjugated with respect to the torsion tensor).

The torsion is given by the formula

\begin{aligned}T^{γ} = ∑_{(α,β)∈S_{2}} & (P_{γ}^{α} − P_{β}^{α})([P_{α}^{β}X, P_{γ}^{β}Y] − [P_{γ}^{β}X, P_{α}^{β}Y]) \\ & + [P_{β}^{α}X, P_{α}^{β}Y] + [P_{α}^{β}X, P_{β}^{α}Y] − [X, Y], \end{aligned}

and satisfies for (α, β, γ) ∈ S_{3}:

T^{γ} (P_{α}^{β}X, P_{β}^{α}Y) = 0,  P_{α}^{β}T^{γ} (P_{β}^{α}X, P_{β}^{α}Y) = 0,
B_{γ} T^{γ} (P_{α}^{β}X, P_{α}^{β}Y) = −P_{β}^{α}B_{γ}{B_{γ}, B_{γ}}(P_{α}^{β}X, P_{α}^{β}Y)

= T^{γ} (B_{γ}P_{α}^{β}X, B_{γ}P_{α}^{β}Y).

On D_{α}, the torsion T^{γ} coincides with B_{γ}[P_{α}^{β}, B_{γ}], and on D_{β}, we obtain T^{γ} = −B_{γ}[P_{α}^{β}, B_{γ}]. All web distributions are parallel with respect to the Chern connections, any distribution is autoparallel and parallel relative to any other. Any Chern connection is reducible to any leaf of the integral foliations of web distributions [10, 7]. With respect to [D_{α}, D_{β}]-adapted coordinates, the only non-zero

Christoffels of the Chern connection Γ^{γ} are

Γ^{γ}_{ij}^{s} = \frac{∂q_{j}^{k}}{∂x^{i}} p_{k}^{s},  Γ^{γ}_{n+i,n+j}^{n+s} = \frac{∂p_{j}^{k}}{∂x^{n+i}} q_{k}^{s}

where P = (p_{i}^{j}) is the (n, n)-matrix from the representation of the projector P_{γ}^{α}

in [D_{α}, D_{β}]- adapted coordinates, P_{γ}^{α} = ( \begin{matrix}0 & 0 \\ P & I \end{matrix} ), P̃ = (q_{i}^{j}) denotes its inverse.

Therefore the only non-zero components of the torsion tensor are

T^{γ} _{ij}^{s} = (\frac{∂q_{j}^{k}}{∂x^{i}} − \frac{∂q_{i}^{k}}{∂x^{j}} ) p_{k}^{s},  T^{γ} _{n+i,n+j}^{n+s} = (\frac{∂p_{j}^{k}}{∂x^{n+i}} − \frac{∂p_{i}^{k}}{∂x^{n+j}} ) q_{k}^{s}.

The curvature tensor satisfies

R^{γ}P_{α}^{β} = P_{α}^{β}R^{γ}, B_{γ} R^{γ} = R^{γ}B_{γ},

R^{γ}(P_{α}^{β}X, P_{α}^{β}Y) = 0 for any vector fields X, Y ,

R^{γ}(P_{α}^{β}X, P_{β}^{α}Y)P_{α}^{β}Z =

− P_{α}^{β}[P_{β}^{α}Y, P_{α}^{β}B_{γ}[P_{α}^{β}X, B_{γ}P_{α}^{β}Z]] − P_{α}^{β}[P_{β}^{α}[P_{α}^{β}X, P_{β}^{α}Y], P_{α}^{β}Z]

− P_{α}^{β}B_{γ}[P_{α}^{β}[P_{α}^{β}X, P_{β}^{α}Y], B_{γ}P_{α}^{β}Z] + P_{α}^{β}B_{γ}[P_{α}^{β}X, B_{γ}P_{α}^{β}[P_{β}^{α}Y, P_{α}^{β}Z]],

R^{γ}(P_{α}^{β}X, P_{β}^{α}Y)P_{β}^{α}Z =

P_{β}^{α}[P_{α}^{β}X, P_{β}^{α}B_{γ}[P_{β}^{α}Y, B_{γ}P_{β}^{α}Z]] − P_{β}^{α}[P_{α}^{β}[P_{α}^{β}X, P_{β}^{α}Y], P_{β}^{α}Z]

− P_{β}^{α}B_{γ}[P_{β}^{α}[P_{α}^{β}X, P_{β}^{α}Y], B_{γ}P_{β}^{α}Z] − P_{β}^{α}B_{γ}[P_{β}^{α}Y, B_{γ}P_{β}^{α}[P_{α}^{β}X, P_{β}^{α}Z]].

If we evaluate the components of R^{γ} in [D_{α}, D_{β}]-adapted coordinates we obtain

\begin{matrix}R^{γ}(∂_{i}, ∂_{n+j})∂_{k} & = −P_{α}^{β}[∂_{n+j}, (P_{γ}^{α} − P_{β}^{α})[∂_{i}, (P_{γ}^{β} − P_{α}^{β})∂_{k}]], \\ R^{γ}(∂_{i}, ∂_{n+j})∂_{n+k} & [∂_{i}, (P_{γ}^{β} − P_{α}^{β})[∂_{n+j}, (P_{γ}^{α} − P_{β}^{α})∂_{n+k}]]. \end{matrix}

It follows that the non-zero components of the curvature tensor are

R^{γ}_{i,n+j,n+k}^{n+s} = (C(x^{i}, x^{n+j}))_{k}^{s}  R^{γ}_{i,n+j,k}^{s} = (P̃ · C(x^{i}, x^{n+j}) · P)_{k}^{s}

where the matrices C are given by C(x^{i}, x^{n+j}) = (P_{x^{n+j}} P̃)_{x^{i}}, P = P_{α}^{β}. Therefore

the condition R^{γ} = 0 is equivalent with (3).

The curvature and torsion tensors play a central role by classification of webs.

Especially, a 3-web is regular if and only if T^{γ} = R^{γ} = 0 [1, 3]. The explicit formula

for evaluation of R^{γ} through projectors can be also established but is complicated.

6. Paratactical webs. A 3-web for which T^{γ} vanish is called paratactical [1]. Paratactical webs are characterized through projectors as follows.

Proposition 2. The following conditions are equivalent for any fixed permutation (α, β, γ) of indexis:

(i) T^{γ} = 0,

(ii) [P_{α}^{β}, B_{γ}] = 0, (iii) [P_{α}^{γ}, P_{α}^{β}] = [P_{β}^{γ}, P_{β}^{α}] = 0, (iv) the Nijenhuis brackets vanish for all couples of projectors.

Theorem. A 3-web is paratactical if and only if Nijenhuis tensors vanish for all couples of projectors.

As a consequence we obtain:

Proposition 3. If one of the Chern connections is symmetric (=torsion-free) so are the other.

References

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[11] A. Vanžurová, On torsion of a three-web, submitted.

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Department of Algebra and Geometry, Fac. Sci., Palackého universita Tomkova 40, 779 00 Olomouc, CZECH REPUBLIC