Introduction
The methods of contact geometry have evolved from the mathematical formalism of classical mechanics and play an important role in modern mathematics and theoretical physics. For a review of the theory of contact metric structure and related structures, see [
1,
2,
3,
4]. There are inclusions
and
of well-known classes of metric structures on a manifold: almost contact
, contact
, Sasakian
and cosymplectic
, respectively.
We introduce new metric structures on a smooth manifold (called “weak" structures, see particular case in [
5] and ([
6], Section 5.3.8)) that generalize the almost contact, Sasakian, cosymplectic, etc. metric structures
and allow us to take a fresh look at the theory of these classical structures. To demonstrate this statement, we investigate the geometry of such weak structures and generalize several well-known results on Sasakian and cosymplectic metric structures and contact vector fields. Our metric structures: weak almost contact
, weak contact
, weak Sasakian
and weak cosymplectic
, form wider classes than the classical structures, i.e., the complex structure on
is replaced by a nonsingular skew-symmetric tensor, see (
1) in what follows. Using the homothetical equivalence relation, we define the classes
of our structures with natural inclusions
and
. Any metric structure
mentioned above belongs to a family of weak structures that are homothetically equivalent to the classical structure; thus,
. A natural question arises:
How rich are new structures compared to the classical ones ? In this article, we answer this question for weak Sasakian and weak cosymplectic structures.
The study of this question for weak contact metric manifolds and further generalization of classical results on the curvature and topology of contact metric manifolds and submanifolds in them and their connection with such geometrical structures as warped products and Ricci solitons is postponed to the future. The theory presented here can deepen our knowledge of the geometry of contact and symplectic manifolds as well as find new applications in theoretical physics.
This article consists of an introduction and six sections. In
Section 1, we introduce new metric structures
, define their homothetical equivalence classes, and discuss the properties of our structures in order to demonstrate their similarity with the corresponding classical structures. In
Section 2 and
Section 3, we introduce a new tensor
to calculate the covariant derivative of
for weak contact metric manifolds and study the tensor
h. In
Section 4 we show that any weak Sasakian structure is homothetically equivalent to a Sasakian structure, i.e., Sasakian structures are rigid in some sense. In
Section 5, we study weak almost contact metric structure and show that such metric structure with parallel tensor
is a weak cosymplectic structure, we also give an example of such a structure on the product of a manifold with a line or a circle. In
Section 6, we find conditions for a vector field to be a weak contact vector field. The proofs use the properties of new tensors, as well as the constructions required in the classical case.
1. Preliminaries
Here, we define new metric structures that generalize the almost contact metric structure. A
weak almost contact structure on a smooth odd-dimensional manifold
is a set
, where
is a
-tensor,
is the characteristic vector field and
is a dual 1-form, satisfying
and
Q is a nonsingular
-tensor field such that
for a constant
, see [
5] and ([
6], Section 5.3.8) where
. By
, the form
determines a smooth
-dimensional contact distribution
, defined by the subspaces
for
. We assume that the distribution
is
-invariant,
as in the classical theory of almost contact structure [
1], where
. By (
1) and (
3), the distribution
is invariant for
Q:
. If there is a Riemannian metric
g on
M such that
then
is called a
weak almost contact metric structure on
M and
g is called a
compatible metric. A weak almost contact manifold
endowed with a compatible Riemannian metric
g is said to be a
weak almost contact metric manifold and is denoted by
.
Putting
in (
4) and using
and
, we get, as in the classical theory,
where
(“ iota") is the interior product operation. In particular,
is
g-orthogonal to
for any compatible metric
g.
The following statement (a) generalizes ([
1], Theorem 4.1).
Proposition 1. (a)
For a weak almost contact structure on M, the tensor φ has rank and (b)
For a weak almost contact metric structure, φ is skew-symmetric and Q is self-adjoint,
Proof. (a) By (
1),
, hence, either
or
is a nontrivial vector of
. Applying (
1) to
, we get
. If
for a nonzero function
, then
– a contradiction. Assuming
for some
and nonzero
, by (
1) we get
– a contradiction to non-singularity of
Q. Thus,
.
Next, since
everywhere,
. If a vector field
satisfies
, then (
1) gives
. One may write
for some
and
. This yields
, i.e.,
, hence,
is collinear with
, and so
.
To show
, note that
, and, using (
3), we get
for
. Using (
1) and
, we get
for any
, that proves
.
(b) By (
4), the restriction
is self-adjoint. This and (
2) provide (
6)
. For any
there is
such that
. Thus, (
6)
follows from (
4) and (
6)
for
and
. □
Remark 1. According to [
5], a weak almost contact structure admits a compatible metric if
in (
1)–(
3) has a skew-symmetric representation, i.e., for any
there exist a neighborhood
and a frame
on
, for which
has a skew-symmetric matrix. By (
4), we get
for any nonzero vector
; thus,
Q is positive definite.
The fundamental 2-form
of
is defined (as in the classical case) by
We introduce a
weak contact metric structure as a weak almost contact metric structure satisfying
where
Recall the following formulas (with
):
where
is the Lie derivative in the
Z-direction. A weak contact metric structure
, for which
is Killing, i.e.,
, is called a
weak K-contact structure.
A weak almost contact structure
on a manifold
M will be called
normal if the following tensor (which is well-known for
, e.g., [
1]) is identically zero:
The Nijenhuis torsion
of
is given by
Remark 2. The Levi-Civita connection ∇ of a Riemannian metric
g is given by, e.g. [
6],
and has the properties
and
. Thus, (
12) can be written in terms of
as
in particular, since
,
A normal weak contact metric manifold will be called a weak Sasakian manifold.
Next, we define an equivalence relation on the set of weak almost contact structures and on its important subsets of weak contact metric structures and weak Sasakian structures . The factor-spaces are denoted by for . One can easily verify that the following definition is correct (see the proof of Proposition 2).
Definition 1. (i) Two weak almost contact structures
and
on
M are said to be
homothetically equivalent if
for some positive numbers
and
.
(ii) Two weak contact metric structures
and
on
M are said to be
homothetically equivalent if they satisfy conditions (
16a,b) and
(iii) Two weak Sasakian structures
and
on
M are
homothetically equivalent if they satisfy conditions (
16a-c) and
In our modification of classical conditions we are motivated by the following.
Proposition 2.
Let be a weak almost contact structure on M such that
for some . Then the following is true: (i)
is an almost contact structure on M, where is given by (ii)
if is a weak contact metric structure on M with conditions (17a) and (16c), then is a contact metric structure on M.
(iii)
if is a weak Sasakian structure on M with conditions (17a), (16c) and then is a Sasakian structure on M.
Proof. (i) Substituting
in (
1), for
we get identity, and for
we get
which reduces to the classical equality when
. Thus,
is an almost contact structure on
M.
(ii) Let
and
for some functions
and
on
M. Using (
4), we get
– no restrictions for
and
. The same is when
or/and
. Next, we calculate
which gives
. Next, substituting the values of
in (
7), we get
For
, (16e) is valid for
, i.e.,
, and for
or/and
we find
, see Proposition 3 below. Hence,
is a contact metric structure on
M.
(iii) Substituting the values of
in the equality
and using the property of Lie bracket, we get
for any
, which reduces to the classical case when
. Hence,
is a Sasakian structure on
M. □
The three tensors
and
are well known in the classical theory, see [
1]:
Using (
8) and
, one can calculate (19) explicitly as
2. Weak contact metric manifolds
Here, we study the weak contact metric structure. Define a “small" (1,1)-tensor
by
and note that
and
, where
, see (
2). We also obtain
Note that is the projection of the vector onto .
Remark 3. The notion of almost paracontact structure is an analog of that one of almost contact structure and is closely related to the almost product structure. Similarly to (
1), we can define a weak para-contact structure by
, (see [
5] and ([
6], Section 5.3.8), where
), which allows us to extend some results (in
Section 2,
Section 3,
Section 4,
Section 5 and
Section 6) for the case of para-contact metric structures introduced in [
7].
The following theorem generalizes ([
1], Theorem 6.1), i.e.,
.
Theorem 1.
For a weak almost contact metric structure , the vanishing of implies that and vanish and
Proof. By assumption:
. Thus, taking
instead of
Y and using the expression of Nijenhuis tensor (
12), we obtain
Taking the scalar product of (24) with
and using skew-symmetry of
,
and
, we get
hence, (21) yields
. Next, combining (24) and (25), we get
Applying
and using (
1) and
, we achieve
Further, (25) and (
8) yield
Since Q is non-singular, from (26) and (27) we get , i.e, .
Replacing
X by
in our assumption
and using (
12) and (
8), we acquire
Using (
1) and equality
, we rewrite (28) as
Equation (27) gives
. So, (29) becomes
Finally, combining (30) with (22), we get
from which and
the required expression of
follows. □
Proposition 3.
For either(i)a weak contact metric manifold, or(ii)a normal weak almost contact metric manifold, we get (25); moreover, the integral curves of ξ are geodesics.
Proof. (i) Since
, then
, see Theorem 1. Thus, (21) provides the required (25). (ii) Equation (
7) with
yields
for any
; thus, we get (25). Using the identity
, from the above we also have
As in the proof of ([
1], Theorem 4.5), we obtain
for any
. From this and (31) we get
. □
Theorem 2. For a weak contact metric structure , the tensors and vanish; moreover, vanishes if and only if is a Killing vector field.
Proof. Applying (
7) in (22) and using skew-symmetry of
we get
. We prove that
, taking into account (21) as well as Proposition 3. Next, we find
Then, invoking the formula (
7) in (33) and using (32), we obtain
Since
, the exterior derivative
d commutes with the Lie-derivative, i.e.,
, and using Proposition 3, we get that
is invariant under the action of
, i.e.,
. Therefore, (34) implies that
is a Killing vector field if and only if
. □
The following result generalizes ([
1], Lemma 6.1).
Lemma 1.
For a weak almost contact metric structure , the covariant derivative of φ is given by
where we supplement the traditional sequence of tensors , with a new skew-symmetric with respect to Y and Z tensor defined by
For particular values of we get
Proof. Using (
13) and the skew-symmetry of
, one can compute
Using (
4), we obtain
Thus, and in view of the skew-symmetry of
and applying (37) six times, (36) can be written as
Recall the co-boundary formula for exterior derivative
d on a 2-form
,
We also have
From this and (38) we get the required result. □
According to Theorem 2, on a weak contact metric manifold, we get
Thus, invoking (
7) and using (
7) and Proposition 3 in Lemma 1, we obtain
Corollary 1.
For a weak contact metric structure , the covariant derivative of φ is given by
In particular, we have
Example 1. Consider a weak almost contact metric structure
with
for a constant
. Using
, we obtain
Using (40), we find expressions of the tensor
(see Lemma 1)
3. The tensor field h
The tensor field
plays an important role for contact metric manifolds because of its remarkable properties, e.g., [
1]. Here, we study its generalization. We define the tensor field
h on a weak contact metric manifold similarly as for a contact metric manifold,
We compute
Taking
in (42) and using
, see (39) and (35) and
(see Proposition 3), we get
From (42), using
we conclude that the distribution
is invariant with respect to
h.
On a contact manifold,
h is a self-adjoint linear operator that anticommutes with
, see ([
1], Lemma 6.2). The following lemma generalizes this results.
Lemma 2.
On a weak contact metric manifold, the tensor h satisfies
where is the conjugate operator to h and the tensor is given in (35).
Proof. (i) The scalar product of (42) with
Y for
, using (43) and Corollary 1, gives
Similarly,
The difference of (47) and (48), in view of
(see Theorem 2), gives (44).
(ii) Using
(see Proposition 3),
and
, we obtain
By this and (42), we get
that provides (45). Note that
.
(iii) From Corollary 1 with
, we find
From (
12) with
, we get
Using (
4), (50) and (
9), we calculate
Since
and
, from (42) we get
Since
, we find
Thus, combining (49), (51) and (52), we deduce
Since
Q is symmetric and
is skew-symmetric, replacing
Z by
in (54) and using (
1),
and
, we get
From this, using (41) and (35), we achieve (46). □
4. Weak Sasakian manifolds
We are based on some classical results (see ([
1], Chapter 6); [
4]), and show the equality
.
Lemma 3.
Let be a weak Sasakian manifold. Then
Proof. Since
is a weak contact metric structure with
, by Corollary 1, we get
Using (
7) and (
4) in (56), we get (55). □
Since on a weak contact metric manifold, the tensor
vanishes if and only if
is Killing (Theorem 2), then a weak almost contact metric structure is weak
K-contact if and only if
. Thus we have the following generalization of ([
1], Corollary 6.3).
Corollary 2. A weak Sasakian manifold is weak K-contact.
Proof. In view of (53), Equation (55) with
becomes
Combining (54) and (57), we achieve
, which implies
. □
The main result in this section is the following rigidity of a Sasakian structure.
Theorem 3. A weak almost contact metric structure on is weak Sasakian if and only if it is homothetically equivalent to a Sasakian structure on .
Proof. Let
be a weak Sasakian manifold. Since
, by Theorem 1, we get
. By (35), we then obtain
. Take
such that
and
. Since
is skew-symmetric, applying (55) with
in (
14), we obtain
Recall that
and
. Thus, (58) yields
From this, using (23) and definition of
, we get
By (59), since
holds, the following condition is valid for all
:
In view of
and
, equation (60) reduces to
thus,
. By Proposition 2(iii), we conclude that our structure is homothetically equivalent to a Sasakian structure. □
One can get Corollary 2 as a consequence of Theorem 3 and Proposition 2.
Remark 4. A 3-Sasakian structure is based on a set of three Sasakian structures, see [
2]. Similarly, we get a
weak 3-Sasakian structure if each of the structures
is weak Sasakian with the same tensor
Q and compatible metric satisfying
for any cyclic permutation
of
, see [
5]. Thus, the results of this section on weak Sasakian manifolds yield certain results for weak 3-Sasakian manifolds.
5. Weak cosymplectic manifolds
Cosymplectic manifolds constitute an important class of almost contact manifolds, e.g., [
3,
8]. Here, we study wider classes of weak (almost) cosymplectic manifolds and characterize the class
in
by the condition
.
Definition 2. A weak almost cosymplectic (or a weak almost coKähler) manifold is a weak almost contact metric manifold , whose fundamental 2-form and the 1-form are closed: . If a weak almost cosymplectic structure is normal, we say that M is a weak cosymplectic (or a weak coKähler) manifold.
Theorem 4. For a weak almost cosymplectic structure , the tensors and vanish and . Moreover, vanishes if and only if is a Killing vector field.
Proof. By (22) and (21) and since
, the tensors
and
vanish on a weak almost cosymplectic structure. Moreover, by (
11) and (34), respectively, the tensor
coincides with the Nijenhuis tensor of
, and
vanishes if and only if
is a Killing vector field. □
By application of the above theorem, we make the following corollary to point out some important attributes of a weak almost cosymplectic structure.
Corollary 3. In any weak almost cosymplectic manifold the integral curves of are geodesics.
Proof. By Theorem 4,
; thus, from (21) and
we get for any
,
Therefore, . □
Proposition 4.
Let be a weak cosymplectic structure. Then
Proof. For a weak almost cosymplectic structure
, we get
From (64), using condition
we get (61). Using (38) and (61), we can write
hence, (62). Using (
14), (61) and the skew-symmetry of
, we obtain
This and (62) with
X replaced by
provide (63). □
Remark 5. For a weak cosymplectic structure, using (61), we obtain (compare with (46))
Recall that an almost contact metric structure
is cosymplectic if and only if
is parallel, e.g., ([
1], Theorem 6.8). The following our theorem completes this result.
Theorem 5. Any weak almost contact structure with the property is a weak cosymplectic structure with vanishing tensor .
Proof. Using condition
, from (
14) we obtain
. Hence, from (
11) we get
, and from (
15) we obtain
From (38), we calculate
hence, using condition
again, we get
. Next,
Thus, setting
in Lemma 1 and using the condition
and the properties
,
and
, we find
. By (35) and (65), we get
hence,
. By the above,
. Thus,
is a weak cosymplectic structure. Finally, from (61) and condition
we get
. □
Example 2. Let
M be a
-dimensional smooth manifold and
an endomorphism of rank
such that
. To construct a weak cosymplectic structure on
or
, take any point
of either space and set
,
and
where
,
or
and
. Then (
1) holds and Theorem 5 can be applied.
6. Weak contact vector fields
Contact vector fields (and contact infinitesimal transformations) is a fruitful tool in contact manifold geometry. Here, we are based on some classical results, e.g., ([
1], Section 5.2).
Definition 3. A vector field
X on a weak contact metric manifold
is called a
weak contact vector field (or, a
weak contact infinitesimal transformation), if the flow of
X preserves the form
, i.e., there exists a smooth function
such that
and if
, then the vector field
X is said to be
strict weak contact vector field.
The following our result generalizes ([
1], Theorem 5.7).
Theorem 6.
A vector field X on a weak contact metric manifold is a weak contact infinitesimal transformation if and only if there exists a function f on M such that
Moreover, a weak contact vector field X is strict if and only if .
Proof. (a) One can explicitly write
. Using (
8), we rewrite (66) as
where
. Using
, see (
7), in (68), we get
, which provides
Applying
and using (
1), we obtain (67). Multiplying (69) by
gives
, that proves the assertion about the “strict" property of a vector field
X.
(b) Conversely, assume the condition (67) for
X. The scalar product of (67) with
gives
therefore,
. Multiplying (67) by
and using (
1), gives
Then, in view of non-degeneracy of
Q, we get
Thus, using (
4), (
7) and (
8), one can calculate
By this, (66) with is valid. □
7. Concluding remarks
The theory of contact and symplectic manifolds have been significantly developed and successfully applied to theoretical physics. In this article, we investigated new metric structures on a smooth manifold, which allow us to take a fresh look at the classical theory and find new applications. We expect weak contact manifolds and submanifolds in them to be fruitful in theoretical physics as well. We have shown that
(i) a weak Sasakian structure is homothetically equivalent to a Sasakian structure,
(ii) a weak almost contact structure with parallel structural tensor is weak cosymplectic,
(iii) conditions are found under which a vector field is a weak contact vector field.
We propose a further study of the geometry of weak contact manifolds, in particular, when such manifolds equipped with a warped products or Ricci-type soliton structure carry a canonical (e.g., with constant sectional curvature or Einstein type) metric.
Author Contributions
All authors contributed equally in writing this article.
Conflicts of Interest
The authors declare no conflict of interest.
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