Lobachevskii Journal of Mathematics Vol. 4, 1999, 163–175 \begin{matrix}c \\ ◯ \end{matrix}D.V. Tunitsky

D.V. Tunitsky

ON CONTACT EQUIVALENCE OF HOLOMORPHIC MONGE-AMPÉRE EQUATIONS.

(submitted by B.N.Shapukov)

Abstract. This paper deals with holomorphic Monge-Ampère equations on 5-dimensional complex contact manifolds, i.e., Monge-Ampère equations with two complex independent variables. If a Monge-Ampère equation is in general position,then a complex affine connection can be put in correspondence to this equation in natural manner. This correspondence allows to formulate and prove a number of results on contact equivalence of Monge-Ampère equations using suitable properties of affine connections.

All objects below are supposed to be complex and holomorphic. All manifolds are Hausdorff paracompact and connected. For any fiber bundle Ω denote by Γ(Ω) the set of all sections to this bundle. In sheaf theory’s constructions we use terminology of the book [1].

1. Monge-Ampère equations.

Let M be a five-dimensional complex manifold with fixed contact structure and C^{ω} the sheaf of commutative groups over M such that C^{ω}(U) is the set of all holomorphic functions on an open set U ⊆ M. By λ denote the sheaf of local differential two-forms on M,

λ(U)Γ (Λ^{2}(U)) ,

and by l the sheaf of local differential linear forms on M,

l(U)Γ (T^{∗}(U)) .

The sheafs λ and l are the sheafs of C^{ω}-modules. A contact structure on M is determined by a locally free one-dimensional subsheaf

I_{0} ⊂ l (1) that is non-degenerate. It means that for any point m ∈ M and germ

1991 Mathematical Subject Classification. 53B05, 53C07, 53C30, 58G03, 58G16, 58G07.

Key words and phrases. Monge-Ampère equation, characteristic bundle, characteristic connection, contact equivalence, contact symmetry, homogeneous equation.

ω_{0} ∈ I_{0}(m)

such that ω_{0}(m) ≠ 0 the inequality

dω_{0}(m) ∧ dω_{0}(m) ∧ ω_{0}(m) ≠ 0

holds. Here I_{0}(m) is the stalk of the sheaf I_{0} at a point m:

I_{0}(m) lim_{\overrightarrow{U∋m}} I_{0} (U) .

It’s clear that the definition of non-degeneracy doesn’t depend on choice of germ ω_{0}.

By J_{0} denote the following sheaf of C^{ω}-modules over M:

J_{0}(U) {α ∧ ω_{0} + bdω_{0} : ω_{0} ∈ I_{0}(U), α ∈ l(U), b ∈ C^{ω}(U)} .

Define the factor sheaf

λ^{ε}λ/J_{0}.

The factor-sheaf λ^{ε} is said to be the sheaf of effective two-forms on M (cf. [2]). By

π: λ ⟶ λ^{ε} (2)

denote the corresponding natural projection. Let

μ ⊂ λ^{ε} (3)

be a locally free one-dimensional subsheaf of λ^{ε}. By I_{μ} denote the subsheaf of λ such that

I_{μ}(U) {ω ∈ λ(U) : π ∘ ω ∈ μ(U)} . (4)

Definition 1. The Monge-Ampère equation that is determined by a locally free one-dimensional subsheaf (3) is the system of exterior differential equations

I_{0}0, I_{μ}0. (5)

An immersion

σ: S ⟶ M

of a two-dimensional manifold S is a solution of the Monge-Ampère equation (5) if

σ^{∗}(I_{0}(U))0, σ^{∗}(I_{μ}(U))0

for any open set U ⊆ M.

Let M̃ be a contact five-dimensional manifold, which contact structure is determined by a locally free one-dimensional non-degenerate subsheaf

Ĩ_{0} ⊂ l̃,

where l̃ is the sheaf of local linear differential forms on M̃. A biholomorphic map

f: M̃ ⟶ M (6) is a contact transformation if

f^{∗}(I_{0})Ĩ_{0},

i.e.,

f^{∗}(I_{0}(f(U)))Ĩ_{0}(U).

for any open set U ⊆ M̃.

Let λ̃^{ε} be the sheaf of effective two-forms on M̃,

μ̃ ⊂ λ̃^{ε}

a locally free one-dimensional subsheaf of λ̃^{ε}, and

Ĩ_{0}0, Ĩμ̃0 (7)

the corresponding Monge-Ampère equation. The Monge-Ampère equations (5) and (7) are contact equivalent if there exists a contact transformation (6) such that

f^{∗} (I_{μ}) Ĩμ̃.

By definition (4), this condition holds if and only if

f^{∗}(μ)μ̃. (8)

If (8) holds, then a contact transformation (6) is said to be a contact equivalence of equations (5) and (7).

Construct the category MA of the Monge-Ampère equations.

The objects Ob(MA) of this category are all Monge-Ampère equations, i.e., triples (M, I_{0}, μ), where M is a five-dimensional manifold, I_{0} is a locally free one-dimensional non-degenerate subsheaf (1), and μ is a locally free onedimensional subsheaf (3). For any Monge-Ampère equations Q(M, I_{0}, μ) and Q̃(M̃, Ĩ_{0}, μ̃) the set Hom_{MA}(Q̃, Q) consists of all contact equivalences (6) of these equations.

2. Characteristic bundles.

For any point m ∈ M and tangent vector ξ ∈ T_{m}M put

ξ⌟I_{0}(m) {ξ⌟α(m) : α ∈ I_{0}(m)}

Similarly,

ξ⌟J_{0}(m) {ξ⌟β(m) : β ∈ J_{0}(m)} , ξ⌟I_{μ}(m) {ξ⌟ω(m) : ω ∈ I_{μ}(m)} .

A tangent vector ξ ∈ T_{m}M is a characteristic vector at a point m ∈ M if

ξ⌟I_{0}(m)0, ξ⌟I_{μ}(m)ξ⌟J_{0}(m).

Denote by H the set of all characteristic vectors. Thus

H ⋃_{m∈M} H(m), (9)

where

H(m) {ξ ∈ T_{m}M : ξ⌟I_{0}(m)0, ξ⌟I_{μ}(m)ξ⌟J_{0}(m)} . (10)

The Monge-Ampère equation (5) is non-degenerate at m if the span of H(m) is equal to the fiber of the contact bundle at m, i.e.,

span(H(m)) {ξ ∈ T_{m}M : ξ⌟I_{0}(m)0} .

Consider an example of Monge-Ampère equation and it’s characteristic bundle.

Example 1. Traditionally by holomorphic Monge-Ampère equation with two complex independent variables we call a relation of the following kind

A + Bz_{xx} + 2Cz_{xy} + Dz_{yy} + E(z_{xx}z_{yy} − z_{xy}^{2})0,

where coefficients A, B, C, D and E are fixed holomorphic functions that depend on x, y, z, z_{x} and z_{y}. Here x and y are complex independent variables,

zz(x, y)

is an unknown holomorphic function, and Monge’s notations

pz_{x}, qz_{y}

are used.Let M be the complex space ℂ^{5} of parameters x, y, p, q and z. The linear differential form

ω_{0}dz − pdx − qdy

determines the standard contact structure on M. Put

ωAdx ∧ dy + Bdp ∧ dy + C(dx ∧ dp + dq ∧ dy) + Ddx ∧ dq + Edp ∧ dq.

This differential two-form on M is associated with the equation under consideration in the obvious way. The form ω_{0} uniquely defines the sheaf I_{0} on M:

I_{0}(U) {bω_{0} : b ∈ C^{ω}(U)} .

Similarly, the form π ∘ ω uniquely defines the free one-dimensional subsheaf (3), where π is the corresponding natural projection (2). By definition (4),

I_{μ}(U) {aω + α ∧ ω_{0} + bdω_{0} : ω_{0} ∈ I_{0}(U); α ∈ l(U); a, b ∈ C^{ω}(U)} .

Therefore, a traditional Monge-Ampère equation uniquely determines the corresponding Monge-Ampère equation (5) on M = ℂ^{5}. In this case exterior differential equations (5) are equivalent to the system

ω_{0}0, ω0. (11)

Let m ∈ ℂ^{5}. In case of the Monge-Ampère equation (11) the stalk I_{0}(m) of the sheaf I_{0} is generated by the form ω_{0} and the stalk I_{μ}(m) of the sheaf I_{μ} is generated by the forms ω_{0}, dω_{0}, and ω. Hence a tangent vector ξ belongs to the set (10) if and only if

ξ⌟ω_{0}(m)0, (ξ⌟ω(m)) ∧ (ξ⌟dω_{0}(m)) 0.

Let λ_{1} and λ_{2} be the roots of the following second order algebraic equation

λ^{2} − AE + BC − C^{2}0.

Direct computations show that

H(m)H_{1}(m) ∪ H_{2}(m),

where H_{j}(m) is the two-dimensional linear subspace of T_{m}ℂ^{5} that is generated by the following tangent vectors

e_{j1}E (\frac{∂}{∂x} + p\frac{∂}{∂z} ) − D\frac{∂}{∂p} + (C − λ_{j}) \frac{∂}{∂q} ,

e_{j2}E (\frac{∂}{∂y} + q \frac{∂}{∂z} ) + (C + λ_{j}) \frac{∂}{∂p} − B \frac{∂}{∂q} ,

e_{j3}B (\frac{∂}{∂x} + p\frac{∂}{∂z} ) + (C − λ_{j}) (\frac{∂}{∂y} + q \frac{∂}{∂z} ) − A\frac{∂}{∂p},

e_{j4} (C + λ_{j}) (\frac{∂}{∂x} + p\frac{∂}{∂z} ) + D (\frac{∂}{∂y} + q \frac{∂}{∂z} ) − A\frac{∂}{∂q} ,

j1, 2, among which only two are linearly independent. For example, e_{j1} and e_{j2} are linearly independent when E ≠ 0, and e_{j3} and e_{j2} are linearly independent when B ≠ 0. It’s easy to see that

H_{1}(m) ∩ H_{2}(m)0

if and only if

λ_{1} ≠ λ_{2}.

Thus the Monge-Ampère equation (11) is non-degenerate at m ∈ M if and only if

AE(m) − BD(m) + C^{2}(m) ≠ 0.

The considered above properties of the set (9) of Monge-Ampère equation’s (11) characteristic vectors are correct in general case too (cf. [3]).

Proposition 1. Let (5) be a non-degenerate Monge-Ampère equation and H the set of this equation’s characteristic vectors.

(a) The set H is a locally trivial fiber bundle over M with typical fiber ℂ^{2}∨ℂ^{2}, where ℂ^{2} ∨ ℂ^{2} is the union of corresponding linear spaces with marked zeros.

(b) Let (7) be a Monge-Ampère equation and H̃ the set of this equation’s characteristic vectors. A contact transformation

f: M̃ ⟶ M

is a contact equivalence of the Monge-Ampère equations (5) and (7) if and only if

f_{∗}(H̃)H.

Proof. The proof is straightforward. For technic details see [4], [5], and [7].

The set (9) is said to be the characteristic bundle of the non-degenerate equation (5). Let U be an open subset of M such that the restriction of characteristic bundle H to U is trivial. Thus H |_{U} is isomorphic to U × (ℂ^{2} ∨ ℂ^{2}). In other words,

H(m)H_{1}(m) ∨ H_{2}(m)

for all m ∈ U, where H_{1} and H_{2} are trivial two-dimensional vector subbundles of T U. Let ξ_{j1} and ξ_{j2} be basic sections of the bundle H_{j}, j1, 2. Remind that the first derivative H_{j}^{(1)}(m) of subbundle H_{j} at a point m is the linear subspace of the tangent space T_{m}M that is generated by the vectors

ξ_{j1}(m), ξ_{j2}(m), [ξ_{j1}, ξ_{j2}] (m).

Similarly, the second derivative H_{j}^{(2)}(m) is generated by the vectors

ξ_{j1}(m), ξ_{j2}(m), [ξ_{j1}, ξ_{j2}] (m), [ξ_{j1}, [ξ_{j1}, ξ_{j2}]] (m), [ξ_{j2}, [ξ_{j1}, ξ_{j2}]] (m).

Definition 2. The non-degenerate equation (5) is in general position at m ∈ U if

dim_{ℂ} H_{j}^{(2)}(m)5

both for j1 and j2.

The correctness of this definition is obvious.

Construct the category MA^{×} of the Monge-Ampère equations in general position.

The objects Ob(MA^{×}) of this category are all Monge-Ampère equations

Q ∈ Ob(MA)

that are in general position at all points of the corresponding manifold. Hence

Ob(MA^{×}) ⊂ Ob(MA).

For any

Q, Q̃ ∈ Ob(MA^{×})

the set Hom_{MA}^{×}(Q̃, Q) consists of all contact equivalences

f: M̃ ⟶ M (12)

of these equations. It’s easy to see that

Hom_{MA}^{×}(Q̃, Q)Hom_{MA}(Q̃, Q),

i.e., MA^{×} is a complete subcategory of MA.

Remark 1. The class MA^{×} was considered by T. Morimoto in the work [3]. He denoted it by H_{(4,4)}.

Consider the category AC of the affine connections on five-dimensional manifolds. The objects Ob (AC) of this category are the affine connections, i.e., pairs (M, Σ), where M is a five-dimensional manifold and Σ is an affine connection on M. For any affine connections (M, Σ) and (M̃, Σ̃) the set Hom_{AC} ((M̃, Σ̃), (M, Σ)) of homomorphisms consists of all biholomorphic affine maps (12).

Definition 3. A functor

Ξ: MA^{×} ⟶ AC

such that

Ξ: Ob (MA^{×}) ∋ Q (M, I_{0}, μ) ⟼ (M, Σ) ∈ Ob (AC)

is a characteristic connection functor if the following two properties hold. (a) This functor

Ξ: Hom_{MA}^{×}(Q̃, Q) ∋ f ⟼ f ∈ Hom_{AC} (Ξ(Q̃), Ξ(Q))

for any Monge-Ampère equations Q, Q̃ and morphism f ∈ Hom_{MA}^{×}(Q̃, Q). In other words, any contact equivalence of Q and Q̃ is an affine equivalence of Ξ(Q) and Ξ(Q̃).

(b) Let m, m̃ ∈ M be two arbitrary points and

F: Tm̃M ⟶ T_{m}M

the parallel displacement with respect to the affine connection Ξ(Q) along some piecewizse smooth curve that connects points m and m̃. Then the linear isomorphism F takes the equation’s Q set H(m̃) of all characteristic vectors at m̃ onto the set H(m) of all characteristic vectors at m:

F (H(m̃))H(m).

Proposition 2. Let Ξ be a characteristic connection functor,

Q, Q̃ ∈ Ob(MA^{×}),

and

f ∈ Hom_{AC} (Ξ(Q̃), Ξ(Q)) .

Then

f ∈ Hom_{MA}^{×}(Q̃, Q)

if and only if there exists at least one pair m, m̃ of points m ∈ M and m̃ ∈ M̃ such that mf(m̃) and f_{∗}(m̃) takes the equation’s Q̃ set H̃(m̃) of all characteristic vectors at m̃ to the equation’s Q set H(m) of all characteristic vectors at m:

f_{∗}(H̃(m̃))H(m).

Proof. Follows immediately from definition 3 and proposition 1(b). For technic details see [7].

The following proposition is an existence theorem.

Theorem 1. There exists a characteristic connection functor Ξ from the category MA^{×} to the category AC.

Proof. We follow the procedure that is described in [6].

Consider a Monge-Ampère equation Q(M, I_{0}, μ) in general position and it’s characteristic bundle H. By proposition 1(a), there exists an open domain U ⊆ M and the vector subbundles H_{1} and H_{2} of the tangent bundle T U such that the representation

H(m)H_{1}(m) ∨ H_{2}(m)

holds for all m ∈ U. Since the Monge-Ampère equation under consideration is in general position, we see that the first derivatives H_{1}^{(1)} and H_{2}^{(1)} of the bundles H_{1} and H_{2} are three-dimensional vector subbundles of the tangent bundle T U. Consequently we have the following representation

T UH_{0} ⊕ H_{1} ⊕ H_{2}, (13)

where

H_{0}H_{1}^{(1)} ⋂ H_{2}^{(1)}

is a linear bundle, i.e., one-dimensional vector bundle. This representation determines projections

χ^{l}: T U ⟶ H_{l}, (14)

l0, 1, 2, onto the corresponding summands at the right side of the representation (13).

Construct an affine connection Σ^{U} over U. This construction is based on the representation (13). The connection Σ^{U} is the direct sum

Σ^{U}Σ_{0} ⊕ Σ_{1} ⊕ Σ_{2} (15)

of linear connections Σ_{0}, Σ_{1} and Σ_{2} on the vector bundles H_{0}, H_{1} and H_{2} respectively. The definitions of these connections are the following.

The covariant differentiation operator ∇^{j} of the linear connection Σ_{j}, j1, 2, is uniquely defined by the following properties:

∇_{X}^{j}Y χ^{j} ([X, Y])

for X ∈ Γ(H_{0} ⊕ H_{3−j}), Y ∈ Γ(H_{j}) and

R^{j}(X, Y)0

for X ∈ Γ(H_{0}), Y ∈ Γ(H_{3−j}), where R^{j} is the curvature of Σ_{j} and χ^{j} is the projection operator (14).

The linear connection Σ_{0} on the bundle H_{0} is the mean of linear connections Σ_{0,1} and Σ_{0,2}:

Σ_{0} \frac{1}{2}(Σ_{0,1} + Σ_{0,2}). (16)

The covariant differentiation operators ∇^{0,1} and ∇^{0,2} of the linear connections Σ_{0,1} and Σ_{0,2} are uniquely defined by the following properties:

∇_{X}^{0,j}Y χ^{0} ([X, Y]) (17)

for X ∈ Γ(H_{1} ⊕ H_{2}), Y ∈ Γ(H_{0}) and

R^{0,j}(X, Y)0 (18) for X, Y ∈ Γ(H_{j}), where R^{0,j} is the curvature of Σ_{0,j}, j1, 2.

Let {U} be the set of all open domains such that the restriction of characteristic bundle H to U is trivial. Construct the set of the corresponding affine connections {Σ^{U}} in accordance with the procedure described above. It follows from the construction of the affine connection Σ^{U} that

Σ^{U_{1}} |_{U_{1}∩U_{2}} Σ^{U_{2}} |_{U_{1}∩U_{2}}

for all U_{1}, U_{2} ∈ {U}. By proposition 1(a),

⋃_{U∈{U}} UM.

Hence the relation

Σ |_{U} Σ^{U},

U ∈ {U}, defines an affine connection Σ on M correctly. Put

Ξ(Q)Σ

for all Q ∈ MA^{×}.

The proof of the properties (a) and (b) of definition 3 follows immediately from the construction of the affine connection Σ. For technic details see [7].

3. Contact equivalence.

Fix a characteristic connection functor Ξ. Let

Q(M, I_{0}, I_{μ}), Q̃(M̃, Ĩ_{0}, Ĩμ̃) ∈ Ob (MA^{×})

be Monge-Ampère equations in general position and H, H̃ their characteristic bundles. For any open set U ⊆ M by Q |_{U} denote the restriction of Q to U, i.e.,

Q |_{U} (U, I_{0} |_{U}, I_{μ} |_{U}).

By ∇, R, T and ˜∇, R̃, T̃ denote the covariant differentiations, tensors of curvature and torsion of the connections Ξ(Q) and Ξ(Q̃) respectively.

Theorem 2. Let m ∈ M and m̃ ∈ M̃ be arbitrary fixed points. The following propositions are equivalent.

(a) There exist neighborhoods U of m, Ũ of m̃, and a contact transformation

f: Ũ ⟶ U

such that f(m̃)m and

f ∈ Hom_{MA}^{×} (Q̃ |Ũ, Q |_{U}) .

(b) There exists a linear isomorphism

F: Tm̃M̃ ⟶ T_{m}M

that satisfies the condition

F (H̃(m̃))H(m)

and takes the connection’s Ξ(Q̃) covariant derivatives of the curvature ˜∇^{k}R̃(m̃) and torsion ˜∇^{k}T̃(m̃) to the connection’s Ξ(Q) covariant derivatives of the curvature ∇^{k}R(m) and torsion ∇^{k}T (m) respectively for k0, 1, 2, ...

Proof. Follows immediately from the well known properties of affine connections [8, ch. 6, par. 7] and proposition 2.

Theorem 3. Let all conditions of theorem 2 hold. Moreover, let the manifolds M and M̃ be simple connected and the connections Ξ(Q), Ξ(Q̃) complete. Then the following propositions are equivalent.

(a) There exists a contact transformation

f: M̃ ⟶ M

such that f(m̃)m and

f ∈ Hom_{MA}^{×} (Q̃, Q) .

(b) There exists a linear isomorphism

F: Tm̃M̃ ⟶ T_{m}M

that satisfies the condition

F (H̃(m̃))H(m)

and takes the connection’s Ξ(Q̃) covariant derivatives of the curvature ˜∇^{k}R̃(m̃) and torsion ˜∇^{k}T̃(m̃) to the connection’s Ξ(Q) covariant derivatives of the curvature ∇^{k}R(m) and torsion ∇^{k}T (m) respectively for k0, 1, 2, ...

Proof. Follows immediately from the well known properties of affine connections [8, ch. 6, par. 7] and proposition 2.

Definition 4. A Monge-Ampère equation Q ∈ Ob (MA^{×}) is invariant with respect to parallelism if the following property holds for the characteristic affine connection Ξ(Q) of this equation and two arbitrary points m, m̃ ∈ M. For any continuous curve that connects these points and the parallel displacement

F: Tm̃M ⟶ T_{m}M

along this curve there exist neighborhoods U of m, Ũ of m̃, and a contact transformation

f: Ũ ⟶ U

such that

f ∈ Hom_{MA}^{×} (Q |Ũ, Q |_{U})

and the equalities

f(m̃)m, f_{∗}(m̃)F

hold.

Theorem 4. An equation Q ∈ Ob (MA^{×}) is invariant with respect to parallelism if and only if the equalities

∇R0, ∇T 0 (19)

hold at all points of manifold M.

Proof. Follows immediately from the well known properties of affine connections [8, ch. 6, par. 7] and proposition 2.

Theorem 5. Let Monge-Ampère equations Q, Q̃ ∈ Ob (MA^{×}) be invariant with respect to parallelism. Then the following propositions are equivalent.

(a) For any points m ∈ M and m̃ ∈ M̃ there exist neighborhoods U of m, Ũ of m̃, and a contact transformation

f: Ũ ⟶ U

such that f(m̃)m and

f ∈ Hom_{MA}^{×} (Q̃ |Ũ, Q |_{U}) .

(b) There exists at least one point m ∈ M and one point m̃ ∈ M̃ such that there is a linear isomorphism

F: Tm̃M̃ ⟶ T_{m}M

that satisfies the condition

F (H̃(m̃))H(m)

and takes the connection’s Ξ(Q̃) tensors of curvature R̃(m̃) and torsion T̃(m̃) to the connection’s Ξ(Q̃) tensors of curvature R(m) and torsion T (m) respectively.

Proof. Follows immediately from the well known properties of affine connections [8, ch. 6, par. 7] and proposition 2.

Remind that a contact transformation

f: M ⟶ M

is a contact symmetry of Monge-Ampère equation Q if

f ∈ Hom_{MA}(Q, Q).

An equation Q, Q ∈ Ob(MA^{×}), is called homogeneous if this equation’s group of contact symmetries is a Lie group that acts transitively on M. Thus M is homogeneous in this case. If the homogeneous manifold M is reductive, then an equation Q is called a homogeneous reductive Monge-Ampère equation.

Theorem 6. Let Q ∈ Ob (MA^{×}) be a Monge-Ampère equation on a simply connected manifold M. The equation Q is a homogeneous reductive Monge-Ampère equation and it’s characteristic connection Ξ(Q) is the canonical connection of the manifold M if and only if the connection Ξ(Q) is complete and equalities (19) hold at all points of M.

D.V. Tunitsky 174

Proof. Follows immediately from the well known properties of invariant connections on homogeneous reductive manifolds [9, ch. 10, par. 2] and proposition 2.

Theorem 7. Let Q, Q̃ ∈ Ob (MA^{×}). Suppose these equations are homogeneous reductive Monge-Ampère equations and Ξ(Q), Ξ(Q̃) are the canonical connections of the manifolds M and M̃ respectively. Then the following propositions are equivalent.

(a) For any points m ∈ M and m̃ ∈ M̃ there exists a contact transformation

f: M̃ ⟶ M

such that f(m̃)m and

f ∈ Hom_{MA}^{×} (Q̃, Q) .

(b) There exists at least one point m ∈ M and one point m̃ ∈ M̃ such that there is a linear isomorphism

F: Tm̃M̃ ⟶ T_{m}M

that satisfies the condition

F (H̃(m̃))H(m)

and takes the connection’s Ξ(Q̃) tensors of curvature R̃(m̃) and torsion T̃(m̃) to the connection’s Ξ(Q̃) tensors of curvature R(m) and torsion T (m) respectively.

Proof. Follows immediately from theorem 6, the well known properties of affine connections [8, ch. 6, par. 7], and proposition 2.

References

[1] R.O. Wells. Differential analysis on complex manifolds, Prentice-Hall, Englewood Cliffs, N.J., 1973.

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[3] T. Morimoto. La géométrie des équations de Monge-Ampère. C.R. Acad. Sc. Paris. T. 289 (1979), no. 1, pp. 25-28.

[4] D.V. Tunitsky. On contact linearization of Monge-Ampère. Izvestia RAN. Seria Matematicheskaya. V. 60 (1996), no. 2, p. 195-220. (Russian. English translation in Izvestia RAN: Mathematics. V. 60 (1996), no. 2, p. 425-451.)

[5] D.V. Tunitsky. On Levi’s conditions for Monge-Ampère equations. Differentsialnye Uravnenia. V. 32 (1996), no. 6, p. 1-11. (Russian. English translation in Differential Equations. V. 32 (1996), no. 8, p. 1083-1092.)

[6] D.V. Tunitsky. Contact equivalence of Monge-Ampère equations with transitive symmetries. Differential Geometry and Applications. Proc. Conf., Aug. 28 — Sept. 1, 1995, p. 479-487. Czech Republic, Brno, Masaryk University, 1996.

[7] D.V. Tunitsky. Equivalence and characteristic connections of Monge-Ampère equations. Matematicheskii Sbornik. V. 186 (1997), no. 5, p. 1069-1111. (Russian. English translation in Sbornik: Mathematics. V. 186 (1997), no. 5, p. 771-797.)

[8] S. Kobayashi, K. Nomizu. Foundations of differential geometry, V. 1, Interscience, New York–London, 1963.

175 HOLOMORPHIC MONGE-AMPÉRE EQUATIONS

[9] S. Kobayashi, K. Nomizu. Foundations of differential geometry, V. 2, Interscience, New York–London, 1969.

International Sophus Lie Center. 3–9–19 Pluscheva st., Moscow 111524 Russia. E-mail address: tunitsky@glasnet.ru; http://www.glasnet.ru/˜ tunitsky

Received July 27, 1999