Einstein solitons with unit geodesic potential vector field

1.   Introduction

An $n$-dimensional Riemannian manifold $(M, g)$ ($n > 2$) is an Einstein soliton if there exist a vector field $\xi$ and a real constant $\lambda$ such that

where $£_{\xi}$ stands for the Lie derivative operator in the direction of $\xi$, $\operatorname{Ric}$ is the Ricci curvature and $r$ is the scalar curvature of $g$. Remark that if $r$ is constant, then the notions of Ricci and Einstein soliton coincide. Generalizing these notions by allowing $\lambda$ to be a function, we talk about an almost Einstein soliton. In the particular case when $\xi$ is of gradient type and $\lambda$ is constant, G. Catino and L. Mazzieri introduced ^[1] the gradient Einstein soliton as a self-similar solution $(g, \xi, \lambda)$ of the Einstein flow

This notion was generalized in the same paper to gradient $\rho$-Einstein soliton as being a data $(g, \xi = \operatorname{grad}(f), \lambda)$ satisfying

for $\rho$ a nonzero real number. Properties of gradient Einstein and $\rho$-Einstein solitons can be found in ^{[2,3,4,5,6,7]}.

In the present paper, following the ideas developed in ^[8], we obtain some results on almost Einstein solitons with unit geodesic potential vector field. Moreover, we provide characterization theorems for trivial solitons, which are solitons $(g, \xi, \lambda)$ with Killing potential vector field. Remark that in the trivial case, Schur's lemma implies that $\lambda$ must be a constant, hence the scalar curvature will be constant, too. On the other hand, if $\lambda$ is a constant, then we have an almost Ricci soliton ^[9] with $\operatorname{div}(£_{\xi}g) = 0$.

2.   Preliminaries

We shall briefly present some properties satisfied by the potential vector field of an almost Einstein soliton immediately deduced from the soliton equation.

Let $(M, g)$ be an $n$-dimensional Riemannian manifold ($n > 2$).

For any $(1, 1)$-tensor field $T_1$ and for any symmetric $(0, 2)$-tensor field $T_2$ on $M$, we shall denote by $\Vert \cdot\Vert$ their norms defined respectively by

for $\{E_i\}_{1\leq i\leq n}$ a local orthonormal frame field on $(M, g)$.

Consider $(g, \xi, \lambda)$ an almost Einstein soliton defined by the Riemannian metric $g$, the vector field $\xi$ and the smooth function $\lambda$. Then

In particular, if $£_{\xi}g = 0$, i.e., if $\xi$ is a Killing vector field, then the soliton will be called trivial.

Denote by $\eta: = i_{\xi}g$ the dual $1$-form of $\xi$ and define the $(1, 1)$-tensor field $F$ by

for any $X, Y\in \mathfrak{X}(M)$.

From (2.1) we obtain

where $\nabla$ is the Levi-Civita connection of $g$, $Q$ is the Ricci operator defined by $g(QX, Y): = \operatorname{Ric}(X, Y)$, for $X, Y\in \mathfrak{X}(M)$ and $I$ is the identity endomorphism on the set of vector fields on $M$.

By a direct computation we get the divergence of $\xi$ and $F\xi$, precisely

for $\{E_i\}_{1\leq i\leq n}$ a local orthonormal frame field on $(M, g)$. Also

and in the compact case, by applying the divergence theorem, we conclude

LEMMA 2.1. If $(g, \xi, \lambda)$ is an almost Einstein soliton on the compact $n$-dimensional smooth manifold $M$ ($n > 2$), then:

Remark that the Riemann curvature $R$ of $\nabla$ satisfies

for any $X, Y\in \mathfrak{X}(M)$, which by contraction gives

and

for $\{E_i\}_{1\leq i\leq n}$ a local orthonormal frame field on $(M, g)$.

3.   Almost Einstein solitons with unit geodesic potential vector field

We shall give some properties of almost Einstein solitons with unit geodesic potential vector field and provide, in this case, necessary and sufficient conditions for the soliton to be trivial.

Let $(g, \xi, \lambda)$ be an almost Einstein soliton on the $n$-dimensional smooth manifold $M$ ($n > 2$) and assume that $\xi$ is a unit geodesic vector field, i.e., $\nabla_{\xi}\xi = 0$. Then Eq (2.2) implies

and

By a direct computation, taking into account that $Q$ is symmetric and $F$ is skew-symmetric, we get

LEMMA 3.1. If $(g, \xi, \lambda)$ is an almost Einstein soliton on the $n$-dimensional smooth manifold $M$ ($n > 2$) and $\xi$ is a unit geodesic vector field, then:

In the compact case, from Lemmas 2.1 and 3.1, we obtain

PROPOSITION 3.2. If $(g, \xi, \lambda)$ is an almost Einstein soliton on the compact $n$-dimensional smooth manifold $M$ ($n > 2$) and $\xi$ is a unit geodesic vector field, then:

Also, from the soliton Eq (2.1), we have

PROPOSITION 3.3. If $(g, \xi, \lambda)$ is an almost Einstein soliton on the compact $n$-dimensional smooth manifold $M$ ($n > 2$), then:

We will further deduce necessary and sufficient conditions for an almost Einstein soliton $(g, \xi, \lambda)$ with unit geodesic potential vector field to be trivial, i.e., $£_{\xi}g = 0$. In this case, $2n\lambda+(n-2)r = 0$.

THEOREM 3.4. Let $(g, \xi, \lambda)$ be an almost Einstein soliton on the compact $n$-dimensional smooth manifold $M$ ($n > 2$) with unit geodesic potential vector field $\xi$. If the scalar curvature is nonzero, then the soliton is trivial if and only if $(2n\lambda+(n-2)r)r\geq 0$.

Proof. The direct implication is trivial. For the converse implication, notice that from Schwartz's inequality $\Vert Q\Vert ^2\geq \frac{r^2}{n}$, by using Proposition 3.2 we deduce $\Vert Q\Vert ^2 = \frac{r^2}{n}$ and $2n\lambda+(n-2)r = 0$, provided $r$ is nonzero. From Proposition 3.3 we obtain $£_{\xi}g = 0$, i.e., the soliton is trivial.

THEOREM 3.5. Let $(g, \xi, \lambda)$ be an almost Einstein soliton on the compact and connected $n$-dimensional smooth manifold $M$ ($n > 2$) with unit geodesic potential vector field $\xi$ and nonzero scalar curvature. Then $\xi$ is an eigenvector of the Ricci operator with constant eigenvalue, i.e., $Q\xi = \sigma \xi$, for $\sigma\in \mathbb{R}^*$, satisfying $(n\sigma-r)r\geq 0$, if and only if the soliton is trivial.

Proof. The converse implication is trivial. Assume now that $Q\xi = \sigma \xi$, $\sigma\in \mathbb{R}^*$. Since $\xi$ is a unit geodesic vector field, then $F\xi = \left(\sigma-\lambda-\frac{r}{2}\right)\xi$ which, for $M$ connected, by taking the inner product with $\xi$, implies either $\xi = 0$, hence the soliton is trivial, or $\sigma = \lambda+\frac{r}{2}$. In the second case, $(g, \xi, \lambda)$ is a Ricci soliton and since $M$ is compact, it follows that the soliton is of gradient type ^[10], hence $\eta$ is closed and $F = 0$. Then from (2.5), (2.2) and (2.3), we consequently obtain

In this case, the Bochner formula ^[11]

becomes

But $\Delta(r) = 2\sigma(n\sigma-r)$ and $\operatorname{div}(r \operatorname{grad}(r)) = r\Delta(r)+\Vert \operatorname{grad}(r)\Vert ^2$ imply

which replaced in the previous relation give

Using Schwartz's inequality we deduce $\Vert Q\Vert ^2 = \frac{r^2}{n}$ and since $r$ is nonzero, $n\sigma = r$, therefore, $2n\lambda+(n- 2)r = 0$. From Proposition 3.3 we obtain $£_{\xi}g = 0$, i.e., the soliton is trivial.

THEOREM 3.6. Let $(g, \xi, \lambda)$ be an almost Einstein soliton on the compact and connected $n$-dimensional smooth manifold $M$ ($n > 2$) with unit geodesic potential vector field $\xi$. Then

if and only if the soliton is trivial.

Proof. From (2.2) we get

Using Proposition 3.3 and Bochner formula ^[11]

we obtain

Using Schwartz's inequality we deduce $\Vert Q\Vert ^2 = \frac{r^2}{n}$, hence $Q = \frac{r}{n}I$. Therefore,

which, by taking the inner product with $\xi$, implies either $\xi = 0$ or $2n\lambda+(n-2)r = 0$. In both of the cases we deduce $£_{\xi}g = 0$, i.e., the soliton is trivial.

THEOREM 3.7. Let $(g, \xi, \lambda)$ be an almost Einstein soliton on the connected $n$-dimensional smooth manifold $M$ ($n > 2$) with unit geodesic potential vector field $\xi$. Then the soliton is trivial if and only if $\operatorname{Ric}(\xi, \xi)\geq \Vert F\Vert ^2$ and the function $2n\lambda+(n-2)r$ is constant on the integral curves of $\xi$.

Proof. From (2.4) we get

If the soliton is trivial, from (2.1) we obtain $Q = (\lambda+\frac{r}{2})I$ and $2n\lambda+(n-2)r = 0$, so we get the conclusion. Conversely, if $2n\lambda+(n-2)r$ is constant on the integral curves of $\xi$, then $\operatorname{Ric}(\xi, \xi) = \Vert F\Vert ^2-\left\Vert Q-\left(\lambda+\frac{r}{2}\right)I\right\Vert ^2\geq \Vert F\Vert ^2$ implies $Q = (\lambda+\frac{r}{2})I$, which from the soliton Eq (2.1) gives $£_{\xi}g = 0$, i.e., the soliton is trivial.

COROLLARY 3.8. Let $(g, \xi, \lambda)$ be an almost Einstein soliton on the compact and connected $n$-dimensional smooth manifold $M$ ($n > 2$) with unit geodesic potential vector field $\xi$. Then the soliton is trivial if and only if

Proof. If the soliton is trivial, from (2.1) we get $Q = (\lambda+\frac{r}{2})I$ and $2n\lambda+(n-2)r = 0$, and from Theorem 3.7 we obtain the conclusion. Conversely, taking into account that $\xi(2n\lambda+(n-2)r) = \operatorname{div}((2n\lambda+(n-2)r)\xi)-\frac{(2n\lambda+(n-2)r)^2}{2}$, from (3.2), by integration we get

hence $Q = (\lambda+\frac{r}{2})I$, which from the soliton Eq (2.1) gives $£_{\xi}g = 0$, i.e., the soliton is trivial.

Let us further assume that $(g, \xi, \lambda)$ is an almost Einstein soliton on a compact and connected $n$-dimensional smooth manifold $M$ ($n > 2$). Then

As $\xi$ is a unit geodesic vector field, we have

where $v = F\xi$. Then $v$ is a closed vector field, as for the smooth function $h = \frac{1}{2}\left\Vert \xi \right\Vert ^{2}$, we get $v = -\frac{1}{2}{ \operatorname{grad}}(h)$. Moreover, we have

for any $X\in \mathfrak{X}(M)$. Since $v$ is closed, we get

for any $X, Y\in \mathfrak{X}(M)$. Taking $Y = \xi$ and using

we conclude

for any $X\in \mathfrak{X}(M)$, therefore

Since, $\xi$ is a unit geodesic vector field, using (3.4), we obtain

Note that $v$ being a closed vector field, we can define a symmetric operator $A$ by

for any $X, Y\in \mathfrak{X}(M)$, which is precisely

and satisfies $\operatorname{trace}(A) = \operatorname{div} (v)$. Also, for any $X, Y\in \mathfrak{X}(M)$, we get

and

Note that

and inserting it into (3.7), we have

Integrating the above equation and using $\operatorname{div} ((\operatorname{div} (v) v) = v(\operatorname{div} (v))+(\operatorname{div} (v))^{2}$, we conclude

Under the assumption $\operatorname{Ric}(v, v)\geq \frac{n-1}{n}(\operatorname{div} (v))^{2}$, the above integral implies

and combining it with (3.6), we conclude

Taking now its inner product with $\xi$ and noticing that $v$ is orthogonal to $\xi$ and that $\xi$ is a unit geodesic vector field, we get

If $\xi = 0$, then the soliton is trivial, so we have $\operatorname{div} (v) = 0$, that is, $A = 0$. Thus, Eq (3.5) implies

Also, equation (3.7) gives $\operatorname{Ric}(Y, v) = 0$, that is $Qv = 0$. Thus, we have $Fv = (\lambda+\frac{r}{2})v$, and taking the inner product with $\xi$, we conclude

Now, the Eq (3.4) takes the form

Taking the covariant derivative in the above equation and using (3.3), we have

which yields

Integrating this relation and using Lemma 2.1, we conclude

Thus, we are ready to prove

THEOREM 3.9. Let $(g, \xi, \lambda)$ be an almost Einstein soliton on the compact and connected $n$-dimensional smooth manifold $M$ ($n > 2$) with unit geodesic potential vector field $\xi$. Then the Ricci curvature satisfies $\operatorname{Ric}(F\xi, F\xi)\geq \frac{n-1}{n}(\operatorname{div} (F\xi)) ^{2}$ and $(2n\lambda +(n-2)r)\lambda\leq 0$, if and only if the soliton is trivial.

Proof. Equation (3.8) gives $Q = (\lambda+\frac{r}{2})I$, that implies $\lambda+\frac{r}{2}$ is a constant (as $n > 2$). Also, we find that $r = n(\lambda+\frac{r}{2})$ is a constant, hence $\lambda = -\frac{n-2}{2n}r$ is a constant. Moreover, by Proposition 3.3, we conclude $£_{\xi}g = 0$, i.e., the soliton is trivial. The converse implication is trivial.

We end these considerations by providing two examples.

EXAMPLE 3.10. Let $S^{2n+1}$ be the odd dimensional unit sphere with the usual Sasakian structure $\left(\varphi, \xi, \eta, g\right)$. Then it follows that the Reeb vector field $\xi$, being a unit Killing vector field, is a geodesic vector field and consequently

holds, where the scalar curvature is $r = 2n(2n+1)$ and $\lambda = -n(2n-1)$. Thus, $\left(g, \xi, \lambda \right)$ is a trivial Einstein soliton on the sphere $S^{2n+1}$ with $\xi$ a unit geodesic vector field.

EXAMPLE 3.11. Let $(M, g)$ be an $n$-dimensional complete quasi-Einstein manifold $n\geq 3$ (cf. ^[12]) whose Ricci curvature is given by

where $a, b$ are smooth functions and $\alpha$ is a $1$-form on $M$. Let $f$ be the distance function on $M$ (cf. ^[13]) and we choose $b = \frac{1}{f}$ and $\alpha = df$. Note that $f$ being the distance function, we have $\left\Vert \operatorname{grad}(f)\right\Vert = 1$ and that integral curves of the vector field $\operatorname{grad}(f)$ are geodesics (cf. ^[13]). Thus, choosing $\xi = \operatorname{grad}(f)$, we get

that is, $\xi$ is a unit geodesic vector field. Also, by using geodesic coordinates on a normal neighborhood, we find that

and that

Thus, we have

and we get

Moreover, the scalar curvature $r = na+\frac{1}{f}$ and in view of this, we have

where

Hence, $\left(g, \xi, \lambda \right)$ is an almost Einstein soliton on the considered quasi-Einstein manifold $(M, g)$ with $\xi$ a unit geodesic vector field.

Acknowledgements

The authors extend their appreciations to the Deanship of Scientific Research, King Saud University for funding this work through research group no. (RG-1441-P182).

Conflict of interest

The authors declare that they have no conflict of interest.