3.1. Algebraic Schouten Solitons Associated with Levi-Civita Connections on Three-Dimensional Lorentzian Lie Groups
Throughout this paper, by
we shall denote the connected, simply connected three-dimensional Lie group equipped with a left-invariant Lorentzian metric
g, and having Lie algebra
. Let ∇ be the Levi-Civita connection of
and let
R be its curvature tensor, taken with the convention
The Ricci tensor of
is defined by
where
is a pseudo-orthonormal basis, with
being time-like and the Ricci operator (Ric) is given by
The Schouten tensor is defined by
where
s denotes the scalar curvature. We generalize the definition of the Schouten tensor to
where
is a real number. Refer to [
16], we have
Definition 1. is called the algebraic Schouten soliton associated with the connection ∇
if it satisfieswhere c is a real number, and D is a derivation of i.e., Theorem 3. If and , then this case corresponds to being the algebraic Schouten soliton associated with the connection ∇.
Proof of Theorem 1. Therefore,
We can write
D as
Hence, by (
8), there exists an algebraic Schouten soliton associated with the connection ∇ if and only if the following system of equations is satisfied
Since we have and □
Theorem 4. If and are satisfied, is the algebraic Schouten soliton associated with the connection
Proof of Theorem 2. According to [
3], we have
Consequently, the scalar curvature is given by
We have
Equation (
8) is satisfied if and only if
The first and second equations of system (
14) imply that
Since
we have
In this case, system (
14) reduces to
If
then we have
If
we have
According to [
3], this is a contradiction. □
Theorem 5. If one of the following conditions is satisfied, is the algebraic Schouten soliton associated with the connection :
- (i)
for all
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- (vii)
- (viii)
- (ix)
- (x)
- (xi)
- (xii)
Proof of Theorem 3. The Ricci operator is given by
where
Moreover, we have
So
Therefore, (
8) now becomes
Suppose that
we have
If
we have two cases (i)–(ii). If
for cases (iii)–(v), system (
21) holds. Now, we assume that
then
Meanwhile, we have
If
cases (vi)–(viii) hold. If
for cases (ix)–(xii), system (
22) holds. □
Theorem 6. When and is the algebraic Schouten soliton associated with the connection
Proof of Theorem 4. Ref. [
3] makes it obvious that
A direct computation shows that the value of the scalar curvature is
We have
By applying the formula shown in (
8), we can calculate
Via simple calculations, we can obtain
Let
we have
If
then
we have
This is a contradiction. □
Theorem 7. If one of the following two conditions is satisfied
- (i)
- (ii)
then is the algebraic Schouten soliton associated with the connection ∇.
Proof of Theorem 5. By [
3], it is immediate that
Then we have
and
By using (
8) and making tedious calculations, we have the following:
We assume that
Since
and
we have
Consider
then case (i) is true. If
we have case (ii). Now, we assume that
then
for case (ii), system (
29) holds. □
Theorem 8. is the algebraic Schouten soliton associated with the connection ∇ if and only if
- (i)
- (ii)
- (iii)
- (iv)
Proof of Theorem 6. In [
3], the Ricci operator is given by
So
A simple calculation shows that
Thus, Equation (
8) is satisfied if and only if
Suppose that
by taking into account
and
we have
Set
we have case (i). If
we have case (ii). Let
then
Consequently,
Consider
then case (iii) is true. If
for case (iv), system (
33) holds. □
Theorem 9. If is the algebraic Schouten soliton associated with the connection then we have
Proof of Theorem 7. Then
Computations show that
Since we have and □
3.2. Algebraic Schouten Solitons Associated with Canonical Connections and Kobayashi–Nomizu Connections on Three-Dimensional Lorentzian Lie Groups
We define a product structure
J on
by
then
and
. By [
5], we define the canonical connection and the Kobayashi–Nomizu connection is as follows:
The Ricci tensors of
associated with the canonical connection and the Kobayashi–Nomizu connection are defined by
The Ricci operators
and
are given by
Similar to (
5) and (
6), we have
and
Definition 2. is called the algebraic Schouten soliton associated with the connection if it satisfieswhere c is a real number, and D is a derivation of ; that is is called the algebraic Schouten soliton associated with the connection if it satisfies Theorem 10. When is the algebraic Schouten soliton associated with the connection
Proof of Theorem 8. From [
7], it is obvious that
Moreover,
We obtain that
Then, Equation (
52) becomes
Taking into account that, we have and □
Theorem 11. If then this case corresponds to being the algebraic Schouten soliton associated with the connection
Proof of Theorem 9. Therefore,
D is described by
Note that then we have and □
Theorem 12. When and is the algebraic Schouten soliton associated with the connection
Proof of Theorem 10. According to [
7], we have
Obviously,
From Equation (
52) it is easy to obtain
The second and third equations in (
62) transform into
Note that We have and then □
Theorem 13. If are satisfied, then is the algebraic Schouten soliton associated with the connection
Proof of Theorem 11. We have
this can be found in [
7]. Moreover,
From this,
D is given by
In this way, (
52) is satisfied if and only if
Since
we have
The second equation in (
67) transforms into
We have
□
Theorem 14. If one of the following conditions is satisfied, then is the algebraic Schouten soliton associated with the connection
- (i)
for all
- (ii)
- (iii)
or
- (iv)
or
Proof of Theorem 12. A direct computation for the scalar curvature shows that
It is easy to obtain
If then cases (i)–(iii) hold. Choose and , we obtain two cases (iii)–(iv). □
Theorem 15. is the algebraic Schouten soliton associated with the connection if and only if
- (i)
- (ii)
- (iii)
- (iv)
Proof of Theorem 13. We have
which is clear from [
7]. By definition, we have
Hence,
Equation (
52) now becomes
We consider . In this case, cases (i)–(iii) hold. If we consider then and case (iv) holds. □
Theorem 16. If is the algebraic Schouten soliton associated with the connection then we have
Proof of Theorem 14. Then
According to the condition
we calculate that
For
and
a straightforward calculation shows that
Solving (
81), we have
□
Theorem 17. is not the algebraic Schouten soliton associated with the connection
Proof of Theorem 15. So we have
If
is the algebraic Schouten soliton associated with the connection
, then
so
For this reason, Equation (
52) now becomes
Equation (
85) has no solutions, we find that
is not the algebraic Schouten soliton associated with the connection
□
Theorem 18. If , then this case corresponds to being the algebraic Schouten soliton associated with the connection
Proof of Theorem 16. By the analysis above, we have
On the basis of we have □
Theorem 19. If is satisfied, is the algebraic Schouten soliton associated with the connection
Proof of Theorem 17. From
we have
It follows that
Note that if then we have □
Theorem 20. If one of the following two conditions is satisfied
- (i)
- (ii)
then is the algebraic Schouten soliton associated with the connection
Proof of Theorem 18. We recall the following result:
Moreover, we have
Therefore, for
we have
According to the condition
we calculate that
We choose then we have and We set and By the calculation, we have and then □
Theorem 21. If one of the following two conditions is satisfied
- (i)
- (ii)
is the algebraic Schouten soliton associated with the connection
Proof of Theorem 19. It is a simple matter of
It follows that
An easy computation shows that
The first and fourth equations of system (
98) imply that
Because then we have and Let then If then □
Theorem 22. is the algebraic Schouten soliton associated with the connection if and only if
- (i)
- (ii)
Proof of Theorem 20. Clearly,
It follows that
A long but straightforward calculation shows that
The first and third equations of system (
102) yield
for
we have
Let us regard
We have
Since
we have
Then
We assume that in this case, we obtain If then □
Theorem 23. When and is the algebraic Schouten soliton associated with the connection
Proof of Theorem 21. Of course
It follows that
Therefore, Equation (
52) now becomes
Throughout the proof, recall that
and
Assume first that
In this case,
Next suppose that
Then, we have
that is,
□