Invariant vector fields on contact metric manifolds under $\mathcal{D}$-homothetic deformation

1.   Introduction

Let $(M^{2n+1}, \phi, \xi, \eta, g)$ be a contact metric manifold of dimension $2n+1$, $n\geq1$. The so called $\mathcal{D}$-homothetic deformation introduced by Tanno in ^[17] is defined by

for certain positive constant $a$. It is easily seen that any $\mathcal{D}$-homothetic deformation deforms a contact metric structure into another contact metric structure. The study of invariants under a $\mathcal{D}$-homothetic deformation on a contact metric manifold is rather interesting and have been investigated by many authors. For example, any $\mathcal{D}$-homothetic deformation preserves $K$-contact, Sasakian (see ^[1]), contact strongly pseudo-convex $CR$ (see ^[11]), the extended contact Bochner curvature tensor (see ^[5]), $(k, \mu)$-contact structure (see ^[2,3]), the Jacobi $(k, \mu)$-contact structure (see ^[8]) and local $\phi$-symmetry (see ^[4]). Very recently, some necessary conditions for a Ricci almost soliton invariant under a $\mathcal{D}$-homothetic deformation on a contact metric manifold is provided in ^[6]. Some other invariant properties and geometric conditions under a $\mathcal{D}$-homothetic deformation can also be seen in ^[10,12].

A vector field $V$ on a Riemanian manifold $(M, g)$ is said to be conformal if $\mathcal{L}_Vg = \rho g$, where $\rho$ denotes a smooth function on $M$ and $\mathcal{L}$ is the Lie differentiation. In particular, a conformal vector field $V$ is said to be homothetic if $\rho\in\mathbb{R}$ and is said to be Killing if $\rho = 0$. The geometry of conformal and Killing vector fields on contact metric manifolds have been studied in ^[14,15]. In this paper, we present a sufficient and necessary condition for a conformal vector field invariant under a $\mathcal{D}$-homothetic deformation. As an application, some conditions for a holomorphically planar conformal vector field being invariant are provided. In addition, a complete $\eta$-Einstein $K$-contact metric manifold admitting a non-trivial generalized Ricci vector field is studied. We also show that a generalized Ricci vector field cannot be invariant under any $\mathcal{D}$-homothetic deformation on a contact metric manifold.

2.   Contact metric manifolds

All notations adopted throughout this paper follow D. E. Blair ^[1]. A smooth manifold $M$ of dimension $2n+1$ is said to be a contact manifold if there exits on it a global $1$-form $\eta$ such that $\eta\wedge(\mathrm{d}\eta)^n\neq0$ everywhere. If on $M$ there exits a Riemannian metric $g$ compatible with the contact structure, $M$ is said to be a contact metric manifold. This is equivalent to that there exist a $(1, 1)$-type and $(1, 0)$-type tensor fields $\phi$ and $\xi$ respectively such that

for any vector fields $X$ and $Y$, where $\Phi$ denotes the fundamental $2$-form defined by $\Phi(X, Y) = g(X, \phi Y)$. A contact metric manifold is said to be $K$-contact if $\xi$ is Killing and a Sasakian manifold if the contact structure is normal (see ^[1]).

On a contact metric manifold $M^{2n+1}$, we denote by $l = R(\cdot, \xi)\xi$ and $h = \frac{1}{2}\mathcal{L}_\xi\phi$ respectively, where $R$ denotes the Riemannian curvature tensor (defined by $R(X, Y)Z = \nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X, Y]}Z$) and $\mathcal{L}$ is the Lie differentiation. One can check that both $l$ and $h$ are symmetric; and $h$ is trace-free and anti-commutes with $\phi$. Using these notations, the following equation holds (see ^[1]):

Applying (2.3) we see that a contact metric manifold is $K$-contact if and only if $h = 0$. Such a condition is sufficient for a contact metric manifold to be Sasakian for dimension three.

3.   Invariant vector fields

If a geometric condition or property is preserved under a $\mathcal{D}$-homothetic deformation, it is said to be invariant. In this section, we give some invariant vector fields on contact metric manifolds under $\mathcal{D}$-homothetic deformation. Notice that when $a = 1$, (1.1) is just the identity transformation. Therefore, throughout this paper, for any $\mathcal{D}$-homothetic deformation, $a$ is assumed to be a positive constant and not equal to $1$.

A vector field $V$ on a contact metric manifold $(M^{2n+1}, \phi, \xi, \eta, g)$ is called an infinitesimal contact transformation if

for certain smooth function $\sigma$ (see ^[16]). In particularly, a vector field $V$ on a contact metric manifold is said to be a strictly infinitesimal contact transformation if $\mathcal{L}_V\eta = 0$. A vector field $V$ on a contact metric manifold is said to be an infinitesimal automorphism if it leaves $\phi$, $\xi$, $\eta$ and $g$ invariant.

Theorem 3.1. A conformal vector field on contact metric manifolds is invariant under a non-identity $\mathcal{D}$-homothetic deformation if and only if it is an infinitesimal automorphism.

Proof. Suppose a vector field $V$ on a contact metric manifold $(M^{2n+1}, \phi, \xi, \eta, g)$ is conformal, we write $\mathcal{L}_Vg = \rho g$ with $\rho$ a smooth function. In view of (1.1), we have

If $V$ is conformal for the metric $\bar{g}$, i.e., $\mathcal{L}_V\bar{g} = \bar{\rho}\bar{g}$, it follows from the above relation and (1.1) that

Since $V$ is conformal for the metric $g$, we get $\rho = (\mathcal{L}_Vg)(\xi, \xi) = 2\eta(\nabla_\xi V)$ because of (2.3). Using (2.3) we also have $(\mathcal{L}_V\eta)\xi = \eta(\nabla_\xi V)$. Therefore, the action of (3.2) on $(\xi, \xi)$ gives $\rho = \bar{\rho}$ due to $a\neq0$. In view of $a\neq1$, now (3.2) becomes

The action of (3.3) on $(\xi, \phi X)$ implies $(\mathcal{L}_V\eta)\phi X = 0$ for any vector field $X$ and this shows $\mathcal{L}_V\eta = \frac{1}{2}\rho\eta$ because of $(\mathcal{L}_V\eta)\xi = \eta(\nabla_\xi V) = \frac{1}{2}\rho$. This means that $V$ is an infinitesimal contact transformation. It has been proved in [15, Theorem 1] that if a conformal vector field on a contact metric manifold is an infinitesimal contact transformation, then it is an infinitesimal automorphism. The application of this result gives $\rho = 0$ and also the "only if" part proof of the theorem.

Conversely, if a conformal vector field on contact metric manifolds is an infinitesimal automorphism, using $\mathcal{L}_V\eta = 0$ in (3.1) we have $\mathcal{L}_V\bar{g} = a\rho g$. Recalling again the result shown in [15, Theorem 1] or ^[16], we also have $\rho = 0$ and this implies $\mathcal{L}_V\bar{g} = 0$.

From proof of the above theorem, we have

Corollary 3.1. If a conformal vector field on contact metric manifolds is invariant under a non-identity $\mathcal{D}$-homothetic deformation, then it is Killing.

On a contact metric manifold, a holomorphically planar conformal vector (for short, HPCV) field (introduced by Sharma in ^[13]) is defined as a vector field $V$ satisfying

for any vector field $X$ and certain two smooth functions $\alpha$ and $\beta$. It has been proved in ^[13] that if a complete and connected $K$-contact metric manifold $M$ admits a non-zero HPCV field $V$, then either $V$ is a constant multiple of $\xi$, or $M$ is isometric to a unit sphere. By skew-symmetry of $\phi$ with respect to $g$, one can check that an HPCV field is necessarily a conformal vector field.

Theorem 3.2. An HPCV field $V$ on contact metric manifolds of dimension $> 3$ is invariant under a non-identity $\mathcal{D}$-homothetic deformation if and only if $V$ is a constant multiple of $\xi$, $a = 0$ and the manifold is $K$-contact.

Proof. Suppose $V$ is a holomorphically planar conformal vector field, we write $\nabla_XV = \alpha X+\beta\phi X$ for any vector field $X$. For any $\mathcal{D}$-homothetic deformation on a contact metric manifold, from (1.1) and the Koszul formula we have

for any vector fields $X, Y$, where $\bar{\nabla}$ is the Levi-Civita connection of the metric $\bar{g}$. Replacing $Y$ by $V$ in the above equation gives

for any vector field $X$.

If $V$ is also a holomorphically planar conformal vector field for the new contact metric structure (1.1), i.e., $\bar{\nabla}_XV = \bar{\alpha}X+\bar{\beta}\bar{\phi}X$, combining this with the previous relation and using (1.1) we have

for any vector field $X$. Replacing $X$ by $\xi$ in (3.5) gives $\alpha\xi-(a-1)\phi V = \bar{\alpha}\xi$ and this implies $\alpha = \bar{\alpha}$ and $\phi V = 0$, where we have used the assumption $a\neq1$.

It has been proved by A. Ghosh in [7, Lemma 3] that if a contact metric manifold of dimension $> 3$ admits a non-zero HPCV field $V$ such that $\phi V = 0$, then $M$ is $K$-contact. Using $h = 0$, $\alpha = \bar{\alpha}$ and $\phi V = 0$, (3.5) becomes $\beta-(a-1)\eta(V) = \bar{\beta}$ because $X$ is an arbitrary vector field. A. Ghosh in ^[7]HY__HY, Lemma 1] proved that for any HPCV field on a contact metric manifold of dimension $> 3$, the associated function $b$ is constant. Thus, $\phi V = 0$ shows $V = \frac{\beta-\bar{\beta}}{a-1}\xi$ with $\frac{\beta-\bar{\beta}}{a-1}$ a constant. Moreover, for any HPCV field $V$, according to (3.4) we have $\mathcal{L}_Vg = 2ag$, i.e., $V$ is conformal. Following Theorem 3.1, if $V$ is invariant under a $\mathcal{D}$-homothetic deformation, then it is Killing and hence we have $a = 0$. The proof for "if" part is easy to check.

Let $(M, g)$ be a Riemannian manifold and $\mathrm{Ric}$ its Ricci tensor which is defined by $\mathrm{Ric}(X, Y) = \mathrm{trace}\{Z\rightarrow R(Z, X)Y\}$. We denote by $Q$ the associated Ricci operator defined by $\mathrm{Ric}(X, Y) = g(QX, Y)$. A vector field $V$ on $M$ is said to be a generalized Ricci vector field (see ^[9]) if

for any vector field $X$ and certain smooth function $\mu$, or equivalently, $g(\nabla_\cdot V, \cdot) = \mu\mathrm{Ric}$. In particularly, $V$ is said to be a Ricci vector field if $\mu$ in (3.6) is assumed to be a constant. If $V = 0$, (3.6) is meaningless and then a generalized Ricci vector filed is always assumed to be non-zero. Notice that on an Einstein manifold, a generalized Ricci vector field reduces to a concircular one and also a conformal one.

A contact metric manifold is said to be $\eta$-Einstein if $\mathrm{Ric} = \alpha g+\beta\eta\otimes\eta$ for some smooth functions $\alpha$ and $\beta$. In particular, on a $K$-contact manifold of dimension $> 3$, both $\alpha$ and $\beta$ are constant (see ^[18]). Moreover, on a $K$-contact manifold, using $h = 0$ in (2.3) we get $\nabla\xi = -\phi$ and hence $l = Id-\eta\otimes\xi$, and we also have $Q\xi = 2n\xi$.

Theorem 3.3. If a complete $\eta$-Einstein $K$-contact manifold $M$ of dimension $> 3$ admits a generalized Ricci vector field, then $M$ is compact and Sasakian.

Proof. On an $\eta$-Einstein $K$-contact manifold $M$ of dimension greater than three, in view of $Q\xi = 2n\xi$, we write

where $r$ is the constant scalar curvature. Let $V$ on $M$ be a generalized Ricci vector field. Taking the covariant derivative of (3.6) implies that $\nabla_X\nabla_YV = X(\mu)QY+\mu\nabla_XQY$ for any vector fields $X, Y$. It follows directly that

for any vector fields $X, Y$. In view of constancy of $r$, contracting $X$ in the above equation and using the formula $\mathrm{div}Q = \frac{1}{2}Dr$ we obtain

where by $Df$ we mean the gradient of a function $f$. Comparing (3.8) with (3.7) gives

Taking the inner product of (3.9) with $\xi$ gives $\eta(V) = (1-\frac{r}{2n})\xi(\mu)$, which is inserted in (3.9) implying

In view of constancy of $r$, taking the derivative of (3.10) and using $h = 0$, (2.3), we obtain

for any vector field $X$. Recalling that $V$ is a generalized Ricci vector field, from (3.6) and (3.7) we get

Submitting the above relation into the previous one gives

for any vector field $X$.

By using the Poincare lemma (i.e., $\mathrm{d}^2 = 0$) we see that $g(\nabla_XD\mu, Y)$ is symmetric with respect to $X$ and $Y$, and hence it follows from (3.11) that

for any vector fields $X, Y$.

In view of (3.12), we discuss the following several cases. First, if the constant scalar curvature $r = 0$, (3.7) becomes $Q = -Id+(2n+1)\eta\otimes\xi$. It was proved by Sharma in [13, Proposition 1] that on a complete $K$-contact $\eta$-Einstein manifold $M$ satisfying $\mathrm{Ric} = \alpha g+\beta\eta\otimes\eta$, if $\alpha > -2$, then $M$ is compact and Sasakian. Next, in view of (3.12) we consider $r = 2n(2n+1)$ and in this case the manifold is Einstein, i.e., $Q = 2nId$. Following again [13, Proposition 1], the manifold $M$ is compact and Sasakian. Third, if $r\neq0$ and $r\neq2n(2n+1)$, from (3.12) we have

for any vector fields $X, Y$. Let $X = \phi Y$ in the above relation be two unit vector fields orthogonal to $\xi$, it follows that $\xi(\mu) = 0$. Using this in (3.11) we get

for any vector field $X$. Taking the inner product of (3.13) with $\xi$, and using $\xi(\mu) = 0$, $h = 0$, and (2.3) we obtain

for any vector field $X$. Replacing $\phi X$ by $X$ in (3.14) we obtain

The above equation gives either $r = \frac{2n}{1-2n}$ or $\phi^2D\mu = 0$. For the former case, as similar with the above situation, applying again [13, Proposition 1], the manifold $M$ is compact and Sasakian. For the later case, in view of $\xi(\mu) = 0$, we see that $\mu$ is a constant. Using this in (3.10) we have

for any vector field $X$. Since we have assumed that $V$ is non-zero, it follows that $r = 2n$. As similar with the above situation, applying again [13, Proposition 1], the manifold $M$ is compact and Sasakian.

Let $M^{2n+1}$ be a $K$-contact metric manifold and $V$ its generalized Ricci vector field, i.e., $\nabla_XV = \mu QX$. Note that $h = 0$ on a $K$-contact metric manifold, thus, using (1.1) for any $\mathcal{D}$-homothetic deformation (1.1) we have

for any vector fields $X, Y$. Using (3.16), we have $\bar{\nabla}_XV = \mu QX-(a-1)(\eta(X)\phi V+\eta(V)\phi X)$ and hence

for any vector fields $X, Y$, where we have used $Q\xi = 2n\xi$. Suppose $V$ is also a generalized Ricci vector field for the metric $\bar{g}$ in (1.1), i.e., $\bar{g}(\bar{\nabla}_XV, Y) = \bar{\mu}\overline{\mathrm{Ric}}(X, Y)$. Recalling that the Ricci tensor for the metric $\bar{g}$ in (1.1) is given by (see also ^[6]):

for any vector fields $X, Y$. Combining this relation with the previous one gives that

for any vector fields $X, Y$. Notice that the Ricci tensor $\mathrm{Ric}$ is symmetric. When the $\mathcal{D}$-homothetic deformation (1.1) is not identity, it follows from the above relation that

for any vector fields $X, Y$. Let $Y$ in the above relation be $\xi$. This shows $\phi V = 0$. Using this back in the above relation we obtain $\eta(V)g(X, \phi Y) = 0$ for any vector fields $X, Y$. Letting $X = \phi Y$ be two unit vector fields orthogonal to $\xi$ we obtain $\eta(V) = 0$. Finally, the operation of $\phi$ on $\phi V = 0$ implies $V = \eta(V)\xi = 0$. Based on these calculations, we have

Theorem 3.4. On a $K$-contact metric manifold, a generalized Ricci vector field can not be invariant under any non-identity $\mathcal{D}$-homothetic deformation.

Acknowledgments

This first author was supported by the Key Scientific Research Program in Universities of Henan Province (No. 20A110023) and the Fostering Foundation of National Foundation in Henan Normal University (No. 2019PL22). The second author was supported by the National Natural Science Foundation of China (No. 11801306) and the Project Funded by China Postdoctoral Science Foundation (No. 2020M672023).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.