A new approach to gradient Ricci solitons and generalizations

TL;DR

This paper explores inequalities relating curvature and scalar fields in gradient Ricci solitons and introduces generalizations to manifolds with linear connections beyond metric types.

Contribution

It presents new inequalities in Ricci soliton geometry and extends the concept to manifolds with non-metric linear connections.

Findings

Derived inequalities linking curvature and scalar functions in Ricci solitons

Established relationships between curvature and scalar field behavior

Proposed generalizations to non-metric linear connection manifolds

Abstract

This short note concerns with two inequalities in the geometry of gradient Ricci solitons $(g, f, \lambda )$ on a smooth manifold $M$. These inequalities provide some relationships between the curvature of the Riemannian metric $g$ and the behavior of the scalar field $f$ through two second order equations satisfied by the scalar $\lambda $. We propose several generalizations of Ricci solitons to the setting of manifolds endowed with linear connections, not necessary of metric type.

Equations

$$h_f:\mathcal{X}(M)\to \mathcal{X}(M),h_f(X)={\nabla }_X\nabla f$$

$$H_f(X,Y):=g(h_f(X),Y),H_f(X,Y)=H_f(Y,X)$$

$$h_f(E_i)={\lambda }_i{E}_i.$$

$$\nabla f=\sum _{i=1}^n{E}_i(f)E_i,\Vert \nabla f{\Vert }_g^2=\sum _{i=1}^n[E_i(f){]}^2,h_f(X)={\nabla }_X\nabla f=\sum _{i=1}^n({\lambda }_i{X}^i)E_i,$$

$$H_f(X,Y)=\sum _{i=1}^n{\lambda }_i(X^i{Y}^i),\Delta f:=T{r}_g{H}_f=\sum _{i=1}^n{\lambda }_i.$$

$$H_f+Ric+\lambda g=0$$

$$Ric(X,Y)=g(QX,Y)$$

$$h_f+Q+\lambda I=0$$

$$Q(X)=-\sum _{i=1}^n({\lambda }_i+\lambda )X^i{E}_i,\Vert Q{\Vert }_g^2=\sum _{i=1}^n({\lambda }_i+\lambda {)}^2.$$

$$\Vert Ric{\Vert }_g^2=\sum _{i=1}^n{\lambda }_i^2+2\lambda \sum _{i=1}^n{\lambda }_i+n{\lambda }^2=\Vert H_f{\Vert }_g^2+2\lambda \Delta f+n{\lambda }^2.$$

$$n{\lambda }^2+2\Delta f\cdot \lambda +(\Vert H_f{\Vert }_g^2-\Vert Ric{\Vert }_g^2)=0$$

$$0\le {\Delta }^{\prime }:=(\Delta f{)}^2-n(\Vert H_f{\Vert }_g^2-\Vert Ric{\Vert }_g^2).$$

$$\Vert Ric{\Vert }_g^2\ge \Vert H_f{\Vert }_g^2-\frac{1}{n}(\Delta f{)}^2.$$

$$n{\lambda }^2+2\Delta f\cdot \lambda +(\Vert H_f{\Vert }_g^2-\Vert Ric{\Vert }_g^2)=n{\lambda }^2+2(-n\lambda )\lambda +(n{\lambda }^2-0)$$

$$n{\lambda }^2+2\Delta f\cdot \lambda +(\Vert H_f{\Vert }_g^2-\Vert Ric{\Vert }_g^2)=n{\lambda }^2+2(\cdot 0)\lambda +(0-n{\lambda }^2)$$

$$H_f=-\lambda g.$$

$$Ri{c}_g=(n-2)g_S=\frac{1}{2}(g-d{t}^2),\Vert Ric{\Vert }^2=\frac{1}{4}.$$

$$\Delta f+R+n\lambda =0$$

$$n{\lambda }^2+2R\lambda +(\Vert Ric{\Vert }_g^2-\Vert H_f{\Vert }_g^2)=0.$$

$$0\le {\Delta }^{\prime }:=R^2-n(\Vert Ric{\Vert }_g^2-\Vert H_f{\Vert }_g^2)$$

$$\Vert H_f{\Vert }_g^2\ge \Vert Ric{\Vert }_g^2-\frac{R^2}{n}.$$

$$\Vert H_f{\Vert }_g^2-\frac{1}{n}(\Delta f{)}^2\le \Vert Ric{\Vert }_g^2\le \Vert H_f{\Vert }_g^2+\frac{R^2}{n}$$

$$\tilde{\nabla }g=\eta \otimes g.$$

$$\tilde{\nabla }\xi +Q+\lambda I=0.$$

$$\tilde{\nabla }\xi +Q+\lambda I+\mu \eta \otimes \xi =0.$$

$$\tilde{\nabla }\xi +F+\lambda I=0$$

$$\tilde{\nabla }\xi +F+\lambda I+\mu \eta \otimes \xi =0.$$

$${\nabla }_{X}V=c[u(X)V-v(X)U-JX].$$

$$\tilde{\nabla }:=\nabla -[u\otimes I+v\otimes J].$$

$$\tilde{\nabla }g=2u\otimes g$$

Full text

Let $(M^{n},g)$ be a $n$-dimensional Riemannian manifold endowed with a smooth function $f\in C^{\infty}(M)$. The scalar field $f$ yields the Hessian endomorphism:

[TABLE]

where $\nabla$ is the Levi-Civita connection of $f$. Then we know the symmetry of the Hessian tensor field of $f$:

[TABLE]

It follows the existence of a $g$-orthonormal frame $E=\{E_{i}\}_{i=1,...,n}\in\mathcal{X}(M)$ and the existence of the eigenvalues $\lambda=\{\lambda_{i}\}_{i=1,...,n}\in C^{\infty}(M)$:

[TABLE]

Hence we express all the geometric objects related to $f$ in terms of the pair $(E,\lambda)$:

[TABLE]

for $X=X^{i}E_{i}$. Also the Hessian and the Laplacian of $f$ are:

[TABLE]

Suppose now that the triple $(g,f,\lambda\in\mathbb{R})$ is a gradient Ricci soliton on $M$, [3, p. 76]:

[TABLE]

where $Ric$ is the Ricci tensor field of $g$. By considering the Ricci endomorphism $Q\in\mathcal{T}^{1}_{1}(M)$ provided by:

[TABLE]

we can express $(6)$ as:

[TABLE]

with $I$ being the Kronecker endomorphism. From $(4)$ we get that $Q$ is also of diagonal form with respect to the frame $E$:

[TABLE]

By developing the second formula above we derive:

[TABLE]

Hence the scalar $\lambda$ is a solution of the second order equation:

[TABLE]

which means the positivity:

[TABLE]

It follows a boundary of the geometry of $g$ in terms of $f$:

[TABLE]

An ”exotic” consequence is provided by the case of strict inequality in $(12)$; more precisely, it follows that the data $(g,f,\lambda)$ is doubled by the triple $(g,f,-\frac{2\Delta f}{n}-\lambda)$.

Examples 1) (Gaussian soliton) We have $(M=\mathbb{R}^{n},g_{can})$ and $f(x)=-\frac{\lambda}{2}\|x\|^{2}$. It results: $h_{f}=-\lambda I_{n}$ and $\Delta f=-n\lambda$. Since $\|H_{f}\|^{2}=n\lambda^{2}$ the left hand side of $(11)$ is:

[TABLE]

which is exactly zero. Also: $\Delta^{\prime}=(n\lambda)^{2}-n(n\lambda^{2}-0)=0$ which means the uniqueness of $\lambda$ and the equality case in $(13)$: $0=n\lambda^{2}-\frac{(n\lambda)^{2}}{n}$.

2. (Einstein manifold) Let $(M,g)$ be an Einstein manifold: $Ric=-\lambda g$. A function $f$ with vanishing Hessian is called Killing potential in [6] since its gradient is a Killing vector field; in [14, p. 283] such a function is called linear. Hence: $\Delta f=\|H_{f}\|^{2}=0$ and $\|Ric\|^{2}=n\lambda^{2}$ which yields the following value of the left hand side of $(11)$:

[TABLE]

which is exactly zero. Also: $\Delta^{\prime}=0^{2}-n(0-n\lambda^{2})=n^{2}\lambda^{2}\geq 0$ which means that $(13)$ is the inequality $n\lambda^{2}\geq 0-\frac{0}{n}$ and the uniqueness of $\lambda$ gives a steady soliton, equivalently $g$ is Ricci-flat. We consider an interesting open problem to find the linear functions of a steady soliton and of a Ricci-flat metric.

3. A generalization of the first example is provided on a Ricci-flat manifold by a smooth function $f$ satisfying a generalization of Hessian structures:

[TABLE]

Then: $\Delta f=-n\lambda$ and $\|H_{f}\|^{2}=n\lambda^{2}$ exactly as for the Gaussian soliton. Using Lemma 4.1. of [5, p. 1540] it results form $(14)$ that $\nabla f$ is a particular concircular vector field: $h_{f}=-\lambda I$; hence $\lambda_{1}=...=\lambda_{n}=-\lambda$. If $\nabla f$ is without zeros it follows from Theorem 3.1. of [5, p. 1539] that $(M,g)$ is locally a warped product with a $1$-dimensional basis: $(M,g)=(I\times_{\varphi}(F^{n-1},g_{F}))$. In fact: $\nabla f=\varphi(s)\frac{\partial}{\partial s}$ with $\varphi^{\prime}(s)=-\lambda$ which means a affine warping function: $\varphi(s)=-\lambda s+C$.

4. (Hamilton’s cigar) The famous Hamilton’s soliton is the steady soliton provided by the complete Riemannian geometry $(\mathbb{R}^{2},g=\frac{1}{1+x^{2}+y^{2}}g_{can})$ and the potential function $f(x,y)=-\ln(1+x^{2}+y^{2})$. The only non-zero components of the Hessian are $(H_{f})_{11}=(H_{f})_{22}=\frac{-2}{(1+x^{2}+y^{2})^{2}}$ which yields the norm $\|H_{f}\|^{2}=\frac{8}{(1+x^{2}+y^{2})^{2}}$ and the Laplacian $\Delta f=\frac{-4}{1+x^{2}+y^{2}}$. The Gaussian curvature of $g$ is $K(x,y)=\frac{2}{1+x^{2}+y^{2}}$ and $\|Ric\|^{2}=2K^{2}$. In conclusion, the inequality $(13)$ becomes strictly: $\frac{8}{(1+x^{2}+y^{2})^{2}}>\frac{8}{(1+x^{2}+y^{2})^{2}}-\frac{16}{2(1+x^{2}+y^{2})^{2}}=0$.

5. (Cylinder shrinking soliton) Consider the Riemannian product $(M,g)=(\mathbb{S}^{n-1}\times\mathbb{R},2(n-2)g_{S}+dt^{2})$ of the shrinking unit sphere for $n\geq 3$ with a line. We have:

[TABLE]

For $f(x,t)=\frac{t^{2}}{4}$ we obtain a shrinking gradient Ricci soliton with $\lambda=-\frac{1}{2}$ from $H_{f}=\frac{1}{2}dt^{2}$; also: $\Delta f=\frac{1}{2}$. Then (13) becomes the strict inequality: $\frac{1}{4}>\frac{1}{4}-\frac{1}{4n}$.  $\Box$

A new second order equation, similar to $(11)$, follows from a well-known formula from the theory of gradient Ricci solitons, [3, p. 79]:

[TABLE]

obtained by tracing $(6)$; here $R$ is the scalar curvature of $g$. Hence the companion equation of $(11)$ is:

[TABLE]

The new inequality is then:

[TABLE]

and it results a boundary of the behavior of $f$ in terms of geometry of $g$:

[TABLE]

We remark that $(13)$ and $(19)$ can be unified in the double inequality:

[TABLE]

and the simultaneous equalities hold if and only if: $H_{f}=\pm Ric$ with $\Delta f=R=0$; hence $f$ is a harmonic map.

Examples revisited 1) (Gaussian soliton) The inequality $(19)$ becomes: $n\lambda^{2}\geq 0$.

2. (Einstein manifold) $(19)$ becomes an equality: $0=n\lambda^{2}-\frac{(n\lambda)^{2}}{n}$.

3. (Hamilton soliton) $(19)$ is the strict inequality: $\frac{8}{(1+x^{2}+y^{2})^{2}}>\frac{2\cdot 4}{(1+x^{2}+y^{2})^{2}}-\frac{4^{2}}{2(1+x^{2}+y^{2})^{2}}=0$.

4. (Cylinder shrinking soliton) Since $R=\frac{n-2}{2}$ we get the following form of $(19)$: $\frac{1}{4}>\frac{1}{4}-\frac{(n-2)^{2}}{4n^{2}}$.  $\Box$

In the second part of this note we connect the above considerations with a study of Academician Radu Miron who was the Adviser of the author’s PhD Thesis in February 1999. More precisely, let $\tilde{\nabla}$ a linear connection different to the Levi-Civita connection of $g$. Let also the $1$-form $\eta=df$ be the $g$-dual of $\xi=\nabla f$. The data $(M,g,\tilde{\nabla},\eta)$ is called Weyl space in [12] if $g$ is $\tilde{\nabla}$-recurrent with the factor $\eta$:

[TABLE]

Hence, we arrive at a generalization of Ricci solitons in the framework of manifolds endowed with a linear connection:

Definition 1 Let $(M,\tilde{\nabla})$ be given. A triple $(g,\xi\in\mathcal{X}(M),\lambda)$ is a $\tilde{\nabla}$-Ricci soliton if:

[TABLE]

Let $\eta=\xi^{\flat}$ be the $g$-dual form of $\xi$. The data $(g,\xi,\lambda,\mu\in\mathbb{R})$ is a $(\tilde{\nabla},\eta)$-Ricci soliton if:

[TABLE]

More generally, let $(M,\tilde{\nabla},F\in\mathcal{T}^{1}_{1}(M))$ be given. The pair $(\xi,\lambda)$ will be a $(\tilde{\nabla},F)$-soliton if:

[TABLE]

and the triple $(\xi,\lambda,\mu)$ is a $(\tilde{\nabla},F,\eta)$-soliton if:

[TABLE]

On this way we propose a study of classes of solitons, maybe more adapted to Hermitian/Kähler geometry ($F^{2}=-I$) and para-Hermitian/para-Kähler geometry ($F^{2}=I$) by using some linear connections adapted to these settings like the Chern and Bismut complex connections, [7].

As first example, we consider the Vaisman geometry following [1]. Let $(M^{2n},J,g)$ be a complex $n$-dimensional Hermitian manifold and $\Omega$ its fundamental $2$-form given by $\Omega(X,Y)=g(X,JY)$ for any vector fields $X,Y\in\Gamma(TM)$. Recall from [9, p. 1] that $(M,J,g,\Omega)$ is a locally conformal Kähler manifold (l.c.K) if there exists a closed $1$-form $\omega\in\Gamma(T^{0}_{1}(M))$ such that: $d\Omega=\omega\wedge\Omega$. In particular, $M$ is called strongly non-Kähler if $\omega$ is without singularities i.e. $\omega\neq 0$ everywhere; hence we consider $2c=\|\omega\|$ and $u=\omega/2c$ the corresponding $1$-form. Since $\omega$ is called the Lee form of $M$ the vector field $U=u^{\sharp}$ will be called the Lee vector field. Consider also the unit vector field $V=JU$, the anti-Lee vector field, as well as its dual form $v=V^{\flat}$, so: $u(V)=v(U)=0,v=-u\circ J,u=v\circ J$.

Our setting is provided by the particular case of strongly non-Kähler l.c.K. manifolds, called Vaisman manifolds, and given by the parallelism of $\omega$ with respect to the Levi-Civita connection $\nabla$ of $g$. Hence $c$ is a positive constant and the Lemma 2 of [13] gives the covariant derivative of $V$ with respect to any $X\in\Gamma(TM)$:

[TABLE]

It follows a class of general solitons provided by:

Proposition 1 Let $(M,J,g,c=1)$ be a Vaisman manifold and the linear connection:

[TABLE]

Then $(V,0\in\mathbb{R})$ is a $(\tilde{\nabla},J)$-soliton. Moreover, $g$ is recurrent with respect to $\tilde{\nabla}$ with the factor $2u$:

[TABLE]

but $\tilde{\nabla}$ is not the corresponding Weyl connection since is not torsion-free:

[TABLE]

As second example we consider $\tilde{\nabla}$ as being exactly the Weyl connection $(21)$ of the pair $(g,\eta)$. Is well-known its expression:

[TABLE]

and we derive:

Proposition 2 Let $(M,g,\eta,\tilde{\nabla})$ be a Weyl geometry with $\eta$ of constant norm and endowed with an endomorphism $F$. Then $(\xi=\eta^{\sharp},\lambda,\mu)$ is a $(\tilde{\nabla},F,\eta)$-soliton if and only if $(\xi,\lambda-\frac{\|\xi\|^{2}_{g}}{2},\mu+\frac{1}{2})$ is a $(\nabla,F,\eta)$-soliton with $\nabla$ the usual Levi-Civita connection of $g$.

As third example, we consider the vector field $\xi$ as being torse-forming on the Riemannian manifold $(M,g)$:

[TABLE]

for a smooth function $f\in C^{\infty}(M)$ and a $1$-form $\gamma\in\Omega^{1}(M)$. Note that torse-forming vector fields appear in many areas of differential geometry and physics as is pointed out in [10] and are natural generalizations of concircular vector fields. We get immediately:

Proposition 3 Suppose that $\xi$ is a special torse-forming vector field having $f$ a constant function and $\gamma=\eta=\xi^{\flat}$. Then $(g,\xi,\lambda)$ is a Ricci soliton if and only if $g$ is an eta-Einstein metric:

[TABLE]

As fourth example, let $(M,g,\nabla)$ be a hypersuface of the Riemannian manifold $(\tilde{M}^{n+1},\tilde{g},\tilde{\nabla})$ and $A$ its shape operator.

Definition 2 The pair $(\xi\in\mathcal{X}(M),\lambda)$ is a shape soliton on $(M,g)$ if:

[TABLE]

Remark 1 For example, if $M$ is eta-umbilical i.e. $A$ is of eta-type which means that it is has two eigenvalues: $A=\sigma I+(\rho-\sigma)\eta\otimes\xi$ then the above condition yields that $\xi$ is torse-forming:

[TABLE]

Hopf hypersurfaces of eta-umbilical type are studied in [5, p. 60]. Also, the CR submanifolds of maximal CR dimension in complex projective space have eta-type shape operators as is pointed out in [8, p. 190].  $\Box$

We go further with a generalization to the setting of statistical structures of [2] provided by data $(M,g,\tilde{\nabla},\tilde{\nabla}^{\ast})$ where $\tilde{\nabla}$, $\tilde{\nabla}^{\ast}$ is a pair of torsion-free dual connections on $(M,g)$:

[TABLE]

for any vector fields $X$, $Y$, $Z$. We introduce:

Definition 3 i) The statistical manifold $(M,g,\tilde{\nabla},\tilde{\nabla}^{\ast})$ is called Ricci-symmetric if the Ricci tensor field $\tilde{Q}$ of $\tilde{\nabla}$ (equivalently, $\tilde{Q}^{\ast}$ of $\tilde{\nabla}^{\ast}$ by Corollary 9.5.3 of [2, p. 267]) is symmetric.

ii) The pair $(\xi,\lambda)$ is a statistical soliton for the Ricci-symmetric statistical manifold $(M,g,\tilde{\nabla},\tilde{\nabla}^{\ast})$ if the triple $(g,\xi,\lambda)$ is both $\tilde{\nabla}$-Ricci and $\tilde{\nabla}^{\ast}$-Ricci soliton.

Remark 2 Since the Levi-Civita connection of $g$ is the arithmetic mean of the pair $(\tilde{\nabla},\tilde{\nabla}^{\ast})$:

[TABLE]

it follows that a statistical soliton is an usual Ricci soliton.  $\Box$

A last generalization eliminates the scalar $\lambda$. Recall that given the pair $(\tilde{\nabla},F)$ as in Definition 1 the exterior covariant derivative of $F$ with respect to $\tilde{\nabla}$ is:

[TABLE]

Since $(24)$ is expressed as:

[TABLE]

we introduce:

Definition 4 Let $(M,\tilde{\nabla},F)$ be given with a symmetric $\tilde{\nabla}$. The vector field $\xi$ will be a $(\tilde{\nabla},F)$-weak soliton if:

[TABLE]

Remark 3 For the Levi-Civita connection $\nabla$:

[TABLE]

while Lemma 2 of [11, p. 182] gives:

[TABLE]

with the right hand side:

[TABLE]

for an arbitrary orthonormal field $e=\{e_{k}\}_{k=1,...,n}$ on $(M,g)$. Hence $\xi$ is a Ricci weak-soliton if and only if:

[TABLE]

$\Box$