1. Introduction
Nonlinear differential equations arise in wide areas of research in pure and applied sciences. In this note, we study some non-linear differential equations whose solutions are quasi-Einstein metrics that appear in mathematical physics [
1,
2]. In our approach, we will use the techniques of mathematical analysis, more suitable in this case, as in article [
3]. The quasi-Einstein metric is a generalization of the Einstein metric, it contains gradient Ricci solitons and it is also closely related to the construction of the warped product Einstein metrics. The study of quasi-Einstein metrics was initiated by Chaki and Maity in [
4]. In [
5], Chaki and Ghoshal studied some global properties of quasi-Einstein manifolds, while in [
6], De and Ghosh gave some examples, proven their existence, and underline some properties. As applications, gravitational instantons are defined to be solutions of the quasi-Einstein equations [
7,
8].
To set forth our research in this article, we organized our discussion in two parts.
In the first part of this research, we consider a pseudo-Riemannian manifold
, which has constant sectional curvature
K, [
9]. Given a smooth real valued mapping
F, defined on
M and
a real constant, suppose that the tensor having the components
is non-degenerate and has constant signature. With these assumptions, we establish a link between the Christoffel symbols
of
h and the Christoffel symbols
of the initial metric
.
When
has constant sectional curvature
K, it is known that
. The relation which defines the quasi-Einstein pseudo-Riemannian metrics:
can be written as:
This means that the Christoffel symbols
of
h and the Christoffel symbols
of
have to be identical. The equalities:
imply necessary conditions such that the initial metric
, with constant curvature
K, becomes a quasi-Einstein pseudo-Riemannian metric.
As application, we shall show that , endowed with the pseudo-Riemann metric of Kruskal-type, having null sectional curvature, is a quasi-Einstein manifold.
The second part of this research aims to continue the study of A. Pitea, recently published in [
10,
11]. Following this research, we introduce the explicit form of the sectional curvature under the necessary conditions that a generalized Poincaré metric
becomes quasi-Einstein. We show that if the sectional curvature
K is constant, then the metric
becomes a quasi-Einstein one.
As application, we consider the warped product Riemannian metrics , with constant sectional curvature, and we obtain a complete classification of those metrics which are quasi-Einstein too.
2. A New Class of Quasi-Einstein Metrics
Finding quasi-Einstein metrics classes has attracted the attention of much research, since there were studied in different forms in the recent past [
2,
7,
10,
11,
12,
13].
Consider a smooth n-dimensional pseudo-Riemannian manifold and fix a smooth real valued function on M. One denotes with its Hessian with respect to .
A natural extension of the Ricci tensor
of
is the
-Bakry–Emery Ricci tensor:
and hence, if
F is constant then
. For more details on the
-Bakry–Emery Ricci tensor see [
14,
15].
The pseudo-Riemannian manifold
is called
quasi-Einstein if the
-Bakry–Emery Ricci tensor is a constant multiple of the metric tensor:
In local coordinates, the relation (
1) becomes a system of differential equations:
for some real constants
and
, where [
13]:
For
the relation (
1) defines
gradient Ricci solitons, [
16,
17].
Suppose that the tensor of components
is non-degenerate and has constant signature. Then
h is a new pseudo-Riemannian metric, which has the Levi–Civita connection
and the Christoffel symbols
, see [
18,
19]. We will derive the symbols
in terms of the
of
:
Theorem 1. Let be the contravariant components of the pseudo-Riemannian metric h and the curvature tensor field of . Then the Christoffel symbols are given by: Proof. From:
and the similar formulas for
and
, we have:
Substituting
by
and using the formula for
we obtain:
We reduce the terms
with
and
with
and we find:
Finally, we obtain the formula:
and the proof is complete. □
In the following, we consider the case when
has constant sectional curvature
K and then:
The relation (
2) takes the form:
From Theorem 1 we get that
, for all
if and only if:
Example 1. We consider an open subset M of endowed with the pseudo-Riemannian metric of Kruskal-type , where are smooth functions. In addition we assume that: , and . From [20] we have and the Christoffel symbols: The system of differential Equation (3) becomes: By integrating with respect to y the Equation (5), we obtain where a is a function which depends on x only. Substituting in Equation (4) and then integrating with respect to x, we get , where b is a function, which depends on y only. The last equation leads us to , hence must be a positive constant. Then , and , so In a similar manner, from the last two equations we obtain From the Equation (6) we have and from the Equation (7) we deduce . The last equalities lead us to , so these expressions must be a positive constant r. Therefore , and . Turning back to the initial system (2) which defines quasi-Einstein metrics, we have The Equation (10) leads us to The Equation (12) is equivalent to The Equation (11)becomes . If , then F is a constant, a trivial case. So we have to impose and . Then we choose and . The Equation (10) becomeshence The relation (11) can be writtenhence These last two equations lead us to Since the left-hand side is a function with one only variable x, and the right-hand side is a function with one only variable y, then Finally, the potential function is which corresponds to and .
3. Generalizing a Question of Besse
An important question was stated by Besse [
21], which was how to determine examples of Einstein manifolds, which are warped products [
16]. A natural generalization asks to finding examples of quasi-Einstein manifolds of warped product type. In the following, we continue the study started by Pitea [
10], by finding new classes of explicit quasi-Einstein Riemannian manifolds, endowed with generalized Poincaré metrics, which also have constant sectional curvature. We consider the manifold
M as an open subset of
, endowed with metric of diagonal type
, where
g and
f are strictly positive smooth functions [
16,
22,
23]. In order to find
g,
f and
F which satisfy (
2), Pitea [
10] shows that in the case when the potential function
F depends only on
y,
F has the form:
Also, Pitea [
10] proves that if we introduce new functions
,
, then from (
2)
h and
p must satisfy the equation:
Now, we introduce the expression of the sectional curvature:
in the Equation (
13). It follows
, hence
Substituting in (
13) we obtain:
which is equivalent to:
Substituting again
by
we find:
or:
But
and
. Finally, we get:
The sectional curvature
K must be either
or
. Therefore the use of
K from (
14) seems be the most suitable way to find quasi-Einstein manifolds with constant sectional curvature, endowed with generalized Poincaré metrics.
Hence we expressed the relation (
13) according to sectional curvature
K, obtaining the relation (
14) and the following theorem.
Theorem 2. An open subset M of with the metric , having the constant sectional curvature or is a quasi-Einstein Riemannian manifold corresponding to the real constants ρ and μ.
Case study. In the following, we shall determine all metrics of the form
with constant sectional curvature. Taking
in the above formula we obtain the sectional curvature:
and the expression of potential function:
The relation
yields:
The solutions of Equation (
15), obtained in the general case, enable us to classify the two dimensional quasi-Einstein Riemannian metrics
with constant sectional curvature
K.
Class 1. If then . Hence a real constant, the metric component and the potential function is . If then and is trivial. If then .
Therefore, we just proved the following proposition.
Proposition 1. The Riemannian manifold with sectional curvature is a quasi-Einstein manifold corresponding to the real constants ρ and μ, with and the potential function: Class 2. If then we may write or .
Class 2.1. If
, then we integrate
and we have
,
and the potential function
If then is trivial. If , then .
Therefore, we just proved the following proposition.
Proposition 2. The Riemannian manifold with the sectional curvature is a quasi-Einstein manifold corresponding to the real constants ρ and μ with and the potential function: Class 2.2. If
then
hence
,
,
and the potential function:
If then is trivial. If then we choose and in this case we obtain a gradient Ricci soliton.
Hence we just demonstrated the following proposition.
Proposition 3. The Riemannian manifold , with the sectional curvature , is quasi-Einstein corresponding to the real constants ρ and and with the potential function: In fact, is a gradient Ricci soliton; more precisely, is the Euclidean metric in polar coordinates.
Class 2.3. If then we have , hence or .
Subclass 2.3.1. If
then:
and the potential function:
If then is trivial. If then .
We just showed the following proposition.
Proposition 4. The Riemannian manifold , with sectional curvature is a quasi-Einstein corresponding to the real constants ρ and μ with and the potential function: Subclass 2.3.2. If
, then:
and the potential function:
If , then is trivial. If then .
Thus, we obtain the following result.
Proposition 5. The Riemannian manifold , with the sectional curvature is quasi-Einstein corresponding to the real constants ρ and μ, with and the potential function: Remark 1. The pseudo-Riemannian metric corresponding to is exactly the two-dimensional version of anti de Sitter metric of General Relativity, a very important model in that theory.