On maximal totally real embeddings
-
Nefton Pali
Abstract
We 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 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.
1 Introduction and statement of the main result
Let
at the point 0. Composing
We denote by
We notice also the obvious equalities
Definition 1
A real sub-manifold
1.1
M
-totally real almost complex structures over
T
M
We consider
Any almost complex structure that is a continuous extension of
Over an arbitrary small neighborhood of
We remind that the data of a smooth complex horizontal distribution over
such that
For any complex vector field
We notice that we can write
such that
if and only if
This condition is equivalent to the property:
implies
We notice that with respect to the canonical complex structure of
the canonical section which at the point
Definition 2
Let
and
such that
for all
Every almost complex smooth extension of the canonical complex structure
We provide below an explicit formula for the almost complex structure
Indeed
This shows that for any
In equivalent terms,
for any
A well-known theorem by Bruhat and Whitney [6] states that for any real-analytic manifold
Moreover, Bruhat and Whitney [6] show that if
In other words, the structure
We state below our results on the integrability conditions for
1.2 The integrability equations for
M
-totally real almost complex structures
Let
by the formula
for any
where the composition operator
Theorem 1
Let M be a smooth manifold and let
for any point
We notice that
Notation for the statement of the main theorem.
For any
We will denote for notation simplicity
acting on the first three entries of any
For any covariant derivative
with
with
we define the exterior product
as
with
With these notations, we can state our main theorem.
Theorem 2
(Integrability in the fiberwise real analytic case) Let M be smooth manifold equipped with a torsion free covariant derivative operator
with
with
Then
where
In more explicit terms,
The assumption that the complex structure tensor is real-analytic along the fibers of the tangent bundle is quite natural. Indeed if
In this article, we request from the readers very good knowledge of the geometric theory of linear connections. Basics of such theory can be found in the Appendix.
1.3 Application of the main integrability result
Over a Riemannian manifold
the geodesic flow, where
Corollary 1
Let
with
and with
The almost complex structure J is integrable over U and for any
The components
for all
for all
It has been 64 years since the existence of complex structures on Grauert Tubes was proven for the first time by Bruhat–Whitney [6]. Still, up to now, the explicit form of the Taylor expansion has remained mysterious. This is finally clarified in the main Theorem 1.5 in [18], which is based on the statement of Corollary 1. Indeed in the study by Pali and Salvy [18], Theorem 1.5, we obtain a rather simple and explicit global expression for the complex structure on Grauert tubes.
The expression in Theorem 1.5 in the study by Pali and Salvy [18] (see also Theorem 1.6 there for a more general statement) is important for applications to analytic micro local analysis over manifolds. It allows indeed an explicit global construction of the complex extension of a given global Fourier integral operator defined on a real analytic manifold.
The expression in Theorem 1.5 in the study by Pali and Salvy [18] allows also to perform useful explicit global intrinsic operator computations in the sense of Pali [17]. In more explicit terms, given a global intrinsic section over the Grauert tube, an explicit formula for the complex structure such as the one in Theorem 1.5 in [18], allows to determine if the section is holomorphic or not.
The proof of Corollary 1 will be given in the Section 6.2. In the case
in the statement of Corollary 1 are satisfied for all
with
Proposition 1
Let
The proof will be provided in Section 7. In subsection 5.1 in the study by Pali and Salvy [18], we provide a shorter proof of the vanishing of
Conjecture 1
Let M be a smooth manifold and let
with
for all
for all
A general mathematical proof for the vanishing of all the integrability conditions
Corollary 2
(Canonical maximal totally real embeddings). Let M be a real analytic manifold and let
for any
with
for all
is an M-totally real complex structure over U, which is real-analytic over U.
Indeed in the statement of the main Theorem 2, we set
We notice that the notation
The expression of
We obtain in this more general setting a canonical
We wish to point out that in the general setting of a torsion free complex covariant derivative operator
Therefore there exist no geodesic flow associated with
We wish also to point out that in mathematics and in theoretical physics there are many important natural complex differential operators that are defined via complex connections as above.
The set up of Corollary 1 is inspired by the articles [10,13–15,20,21]. The genesis of their approach will be reminded in Sub-section 6.1 and is needed for the proof of Corollary 1.
The long series of articles due to Guillemin and Stenzel [10], Lempert [13], Lempert and Szöke [14,15]
Szöke [20,21], Burns [2,3], Burns et al. [4] as well as Aslam et al. [1] are inspired by the fundamental work of Grauert [9].
Their existence results are needed in a crucial way in analytic micro-local analysis, in pluri-potential theory (see the work by Zelditch [22]) as well as in Hamiltonian dynamics and in geometric quantization (see the work by Morao and Nunes [16], Hall and Kirwin [11]).
2 General connections over vector bundles
2.1 Basic definitions
Definition 3
Let
We will denote by
Lemma 1
For any connection
is an isomorphism for all
Proof
The assumption
This equality shows that the map (2.1) is surjective. The injectivity follows from the fact that if
We denote by
such that
and the smooth vector bundle decomposition
The data of a connection form
where
Definition 4
The curvature form
for all
The definition is tensorial. Indeed if
The conclusion follows from the fact that
and such element is uniquely determined by the curvature field
for all
is called the curvature operator. The terminology is consistent with the fact that if we denote by
Parallel transport. Given any horizontal form
over
We consider now a
with initial condition
Moreover the vector field
We deduce that
2.2 The geometric meaning of the curvature field
The following result provides a clear geometric meaning of the curvature field.
Lemma 2
Let
for any
Proof
We observe first that if we have a family of transformations
By applying the last equality to
and thus,
In a similar way,
with
for all parameters
2.3 Comparison of the curvature fields of two connections
We consider now two connection forms
which satisfies
We want to compare the curvature fields
Lemma 3
In the aforementioned set up, the identity
holds for any
Proof
We notice first the equalities
In the last line, we use the well-known identity
we deduce that this vector field is vertical. In the same way,
The required formula (2.3) follows from the identity
which we show now. We first remind the reader that for any vector space
then
The fact that the bilinear form
On the other hand, by definition,
We conclude the required identity
which shows (2.4).□
We notice now that for any covariant derivative
for any vector field
Lemma 4
Let
for any vector field
We observe that (2.6) implies (2.5), since
Proof
To show the identity (2.6) we notice first that the assumption
The fact that
We infer
for any
We remind now that if
thanks to formula (A7). We apply the previous identity to the curve
Moreover,
We conclude the equality
which represents the required formula (2.6).□
We can show now the following result.
Lemma 5
Let
Then for any section
where
Proof
In the case
By using functorial properties of the pull-back, we have (with no abuse of notation)
We conclude by (2.3) that if
We infer the required formula (2.7).□
3 First reduction of the integrability equations
Proof of Theorem 1
Proof
Let
for all smooth complex vector fields
Let
We express the second equation of the system (3.2) as follows:
By using formula (2.6), we infer
which can be expressed as follows:
We conclude that the system (3.2) is equivalent to the system
It follows that, using the identification
Remark 1
We notice that in the case
In this way, we re-obtain the statement of Lemma 13.
Lemma 6
Under the assumptions of the Theorem
2, the
(i.e.,
for all
Proof
Let
The identification
and
We remind the reader of the formula
for any vector field
Let now
Then
since
We notice now the equalities
and
We infer the equality
Let
By using (3.8), we can write the equation (3.6) under the form
over
thanks to the equality (3.9). If we denote by
Thus, by homogeneity, equation (3.10) is equivalent to the countable system
The first equation in the system means
where
since
Remark 2
In the case
The equation (3.12) means that the complex connection
with
4 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
for all
We first remind the reader that for any complex connection
where
and
We conclude that if a connection is torsion free then then its curvature operator satisfies the algebraic Bianchi identity.
We denote by
Lemma 7
Let V be a vector space over a field
is exact.
Proof
The equality
is obvious. We show now the equality
We show first the inclusion
Let now
we obtain
which rewrites as follows:
i.e.,
and with
and
By using the circular identity
This combined with the fact that
which shows the required identity.□
A direct consequence of the proof of Lemma 7 is the following fact.
Corollary 3
Let
We infer by Corollary 3 that equation (3.4) is satisfied by
and with
The fact that the tensor
is symmetric in the last two variables implies that equation (4.3) is equivalent to the equation
which we can rewrite under the form
with
Then by using expression (4.2), we can rewrite equation (4.4) in the explicit form
We notice that the fact that the complex connection
Lemma 8
Let
if and only if
with
Proof
We observe first that the assumptions on
and
where we denote by
By using the circular assumptions on
Then by the proof of Lemma 7 in the case
with
The fact that
which shows the required expressions for
By equation (4.5), we can apply Lemma 8 to the tensor
We deduce that equation (4.4) is equivalent to equation (4.7). This concludes the proof of the Proposition 2 thanks to Lemma 6.
5 Third reduction of the integrability equations and proof of the main theorem
In this section, we will prove the following result.
Lemma 9
Under the assumptions of the Theorem
2, the M-totally real almost complex structure
and for all
Proof
We show that the statement of proposition 2 is equivalent to the statement of Lemma 9. We show indeed by induction on
Statement 1
The tensors
for all
with
with
with
Statement 1 follows directly from the following fact.
Fact 1
Let
To show the fact 1 we observe first that (5.1) rewrites as follows:
By using expression (5.2) for
By the proof of Lemma 7, we deduce
Therefore, identity (5.3) is equivalent to
Proof of the main theorem
Proof
We show that the recursive definition of
for all
and we write
By using the inductive assumption (5.4), we infer the expressions
This combined with the identity
Putting the terms together, we obtain (5.4) for
We remind first the following elementary and well-known fact.
Lemma 10
For any covariant derivative operator
6 The symplectic approach
6.1 General facts
Let
for all
for any
For any curve
We consider now two curves
With the previous notation hold the following well-known lemma (see also Klingenberg’s book [12] for a proof using local coordinates).
Lemma 11
The formula
hold for any
Proof
With respect to a local coordinate trivialization of the tangent bundle, we can extend in a linear way the vectors
We distinguish two cases.
by the linear nature of the local extension. Then
The case
which implies the required conclusion.□
We need to remind in detail also the following very well-known lemma [12].
Lemma 12
Let
Proof
For any
and thus,
by the definition of the vector field
In the case
and thus,
and thus,
Thus, the flow line
We deduce
and
We provide now a proof of the following well-known result due to Lempert and Szöke [14]. See also, Guillemin and Stenzel [10], Burns [2,3], and Burns et al. [4].
Corollary 4
Let
if and only if for any
Proof
We define the Reeb vector field
holds for any
and thus,
for any
for all
and is also equivalent to the identity
thanks to the fact that
i.e.,
This combined with (6.7) and with (6.2) implies that (6.5) is equivalent to the identity
We show now that the later combined with (6.4) is equivalent to the
But
thanks to (6.2). Then by using the property (A5) of the linear connection
The smooth map
thus, if and only if
For
The condition (6.4) implies that
6.2 Proof of Corollary 1
Proof
If we write
We set
By using (1.7), we infer that the previous identity is equivalent to
Taking
Then the system (6.12) rewrites as
and thus as
implies
for all
By applying the
we infer from the statement of Theorem 2 and with the notation there
for
for all
If we set
On the other hand, if the expansion in the statement of the lemma under consideration hold, then
For
7 Proof of the proposition 1
Proof
We expand first the term
thanks to formula (5.5). By using the differential Bianchi identity, we infer
To simplify the notation in the computations that will follow we will use from now on the identification
for any tensor
We let
and we observe the identities
Summing up, we obtain
where we denote by
where as mentioned earlier we denote by
We set now for notation simplicity
We observe that, by definition, the tensor
satisfies the circular identity with respect to its last three entries. We expand now the term
We observe the identities
Summing up we obtain
where we denote by
We conclude the expression
We expand now the term
From now on, we will denote for notation simplicity
We observe the identities
Summing up, using the symmetries of
We combine now the terms
By expanding further, we obtain the complete expansion:
By expanding the terms
given by
where as mentioned earlier we denote by
For
For
For
For
For
We infer the required identity (1.16) in the statement of Proposition 1.□
8 The almost complex structure associated with a connection over the tangent bundle
This section is not needed for the proof of the results in the article. We include it to clarify the integrability of an
It is well-known [7] that we can construct an
where
If we decompose any vector
We infer
with
for any any
Lemma 13
The torsion form
where
Proof
Let
are local vector field extensions of
The last equality follows from to the computation at the end of the proof of Lemma A6 and thanks to the identity (A8) in the Appendix. (We notice that
The required formula follows from the identity
The fact that that the distribution
vanishes for all
We observe that a connection over
-
Author contribution: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
-
Conflict of interest: The authors declare no potential conflict of interests.
A Appendix
In this Appendix, we provide some well-known basic facts about the geometric theory of linear connections needed for the reading of the article [8]. We strongly recommend its reading even to experts.
A.1 The horizontal distribution associated with a linear connection
We start with the following fact.
Lemma A1
Let
is independent of the sections
Proof
Let
With respect to it, the equality hold
We observe now the linear identity
and
Thus,
i.e., if
which shows the required conclusion.□
Let
We deduce the identity
We notice now that the tangent bundle of the vector bundle
and that the differential of the sum bundle map
for any
hold. We conclude that
Lemma A2
For any section
for any point
Proof
With the notation in the proof of Lemma A1
thanks to (A1). Using the identity
for any
We observe also the elementary identity
for all
for all
The property (A5) implies in particular
Definition A1
A distribution
is an isomorphism for all
Lemma A3
Any horizontal distribution
Proof
The connection
for any
which follows from the identity
It is obvious that the additive property of
is equivalent to the identity
We develop the right-hand side using (A4). We infer that the previous identity is equivalent to the following one:
The later hold true thanks to Lemma A2 and Assumption (A5).□
The data of a smooth horizontal distribution over
such that
For any vector
its vertical component with respect to the horizontal distribution
A.2 The induced connection
Let
and the projection over
We provide a generalization of Lemma (A2).
Lemma A4
For any section
for any point
Proof
A local frame
and
For any
thanks to (A1). Using the equality
for any
The induced connection
for any
Indeed using Lemma A4 and the identity (A5), we have
We observe also that for any
and in other terms, the functorial formula
holds.
A.2.1 The induced connection (second approach)
We observe that the tangent space of
Given any horizontal distribution
In explicit terms
If
By definition of
for any
The local frame
A.2.2 Parallel transport
We consider a smooth curve
over
We infer that the parallel transport map
Lemma A5
For any smooth curve
holds.
Proof
We notice first that the term
Integrating the first equation we infer
Using the second equation, we obtain
Deriving with respect to the variable
Evaluating at
We consider now a
The vector field
To any section
Then
The fact that the map
Thus, for any
We conclude
i.e., for any
Iterating twice, we deduce the identity
Moreover, the fact that by (A8) the vector fields
A.3 The geometric meaning of the curvature tensor
Lemma A6
Let
Proof
Let
We infer by (A8), the identity
We infer
thanks to (A10). We rewrite the previous equality as
By using (A9), we deduce
We infer the required conclusion.□
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This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Research Articles
- Geodesics and magnetic curves in the 4-dim almost Kähler model space F4
- On maximal totally real embeddings
- Geometry of analytic continuation on complex manifolds – history, survey, and report
- On line bundles arising from the LCK structure over locally conformal Kähler solvmanifolds
- Remarks on a result of Chen-Cheng
- Geometry of transcendental singularities of complex analytic functions and vector fields
- Real structures on primary Hopf surfaces
Articles in the same Issue
- Research Articles
- Geodesics and magnetic curves in the 4-dim almost Kähler model space F4
- On maximal totally real embeddings
- Geometry of analytic continuation on complex manifolds – history, survey, and report
- On line bundles arising from the LCK structure over locally conformal Kähler solvmanifolds
- Remarks on a result of Chen-Cheng
- Geometry of transcendental singularities of complex analytic functions and vector fields
- Real structures on primary Hopf surfaces
