Conformal Diffeomorphisms of Gradient Ricci Solitons and Generalized Quasi-Einstein Manifolds

Abstract

In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. We show that a complete GQE metric admits a structure-preserving, non-homothetic complete conformal vector field if and only if it is a round sphere. We also classify the structure-preserving conformal diffeomorphisms. In the compact case, if a GQE metric admits a structure-preserving, non-homothetic conformal diffeomorphism, then the metric is conformal to the sphere, and isometric to the sphere in the case of a gradient Ricci soliton. In the complete case, the only structure-preserving non-homothetic conformal diffeomorphisms from a shrinking or steady gradient Ricci soliton to another soliton are the conformal transformations of spheres and inverse stereographic projection.

Appendix A: Conformal Fields on Warped Products over a One-Dimensional Base

Here we collect some calculations for conformal fields of a Riemannian metric h of the form

$$h = dt^2 + u(t)^2 g_N $$

on M=I×N, where I is an open interval. Let V be a vector field on M. We write

$$V = v_0(t, x) \frac{\partial}{\partial t} +V_t, $$

where v ₀ is a function on M and V _t is the projection of V onto the factor {t}×N. We have the following necessary and sufficient conditions for V to be a conformal field for h.

Proposition A.1

V is a conformal field for h,

$$L_V h = 2 \sigma h, $$

if and only if

• (1) V _t is a conformal field for g _N for each t: $L_{V_{t}} g_{N} = 2 \omega_{t} g_{N}$;

• (2) $\frac{\partial}{\partial t}(v_{0} u^{-1}) = \omega_{t} u^{-1}$;

• (3) $\frac{\partial V_{t}}{\partial t} = -u^{-2} \nabla^{N} v_{0}$,

where ∇^N v ₀ is the gradient of v ₀(t,⋅) on {t}×N. Moreover,

$$\sigma= v_0u^{-1} \frac{\partial u}{\partial t} + \omega_t = \frac{\partial v_0}{\partial t}.$$

Proof

We compute the Lie derivative of h. Let (x ¹,…,x ⁿ⁻¹) be normal coordinates at some p∈N, and let V _t =v _i (t,x)∂ _i , with the Einstein summation convention in effect for i=1 to n−1. Here, $\partial_{i} = \frac{\partial}{\partial x^{i}}$ and we let $\partial_{t} = \frac{\partial}{\partial t}$. To begin, we record the following Lie brackets.

$$\begin{aligned} [V, \partial_t] &= -\frac{\partial v_0}{\partial t} \partial_t - \frac{\partial v_i}{\partial t} \partial_i \\ [V, \partial_j] &= -\frac{\partial v_0}{\partial x^j} \partial_t - \frac{\partial v_i}{\partial x^j} \partial_i. \end{aligned}$$

Now, at the point (t,p),

$$\begin{aligned} (L_V h) (\partial_t, \partial_t) &= D_V h(\partial_t, \partial_t) -2 h\bigl([V, \partial_t],\partial_t\bigr) \\ &= 2 \frac{\partial v_0}{\partial_t}, \\ (L_V h) (\partial_t, \partial_j) &= D_V h(\partial_t, \partial_j)- h\bigl([V, \partial_t], \partial_j\bigr) - h\bigl( \partial_t, [V, \partial_j]\bigr) \\ &= u^2 \frac{\partial v_j}{\partial t} + \frac{\partial v_0}{\partial x^j}, \\ (L_V h) (\partial_j, \partial_k) &= D_V h(\partial_j, \partial_k)- h\bigl([V, \partial_j], \partial_k\bigr) - h\bigl( \partial_j, [V, \partial_k]\bigr) \\ &= 2v_0 uu' \delta_{jk}+ u^2 \biggl(\frac{\partial v_k}{\partial x^j}+\frac{\partial v_j}{\partial x^k} \biggr) \\ &= 2v_0 uu' g_N(\partial_j, \partial_k)+u^2 (L_{V_t} g_N) ( \partial_j, \partial_k). \end{aligned}$$

In particular, for arbitrary vector fields X,Y tangent to {t}×N,

$$\begin{aligned} (L_V h) (\partial_t, X) &= u^2 g_N \biggl(X, \frac{\partial V_t}{\partial t} \biggr) + D_X v_0, \\ (L_V h) (X,Y) &= 2v_0 uu' g_N(X,Y)+u^2 (L_{V_t} g_N) (X,Y). \end{aligned}$$

Then L _V h equals 2σh if and only if

$$\begin{aligned} \sigma &= \frac{\partial v_0}{\partial t}, \\ u^2 \frac{\partial V_t}{\partial t} &= - \nabla^N v_0, \\ L_{V_t} g_N &= 2 \omega_t g_N, \end{aligned}$$

where ω _t :=σ−v ₀ u ⁻¹ u′. The first equation is equivalent to

$$\frac{\partial}{\partial t}\bigl(v_0 u^{-1}\bigr) = \omega u^{-1}. $$

 □

Two consequences of this result are the following.

Corollary A.2

If V is a conformal field for h as above, then v ₀=v ₀(t) if and only if V _t is a fixed homothetic vector field for g _N .

Proof

Equation (3) of the previous proposition shows that v ₀=v ₀(t) if and only if V _t is independent of t. In this case, (2) implies that ω is constant. □

In fact, we can solve for v ₀ and σ explicitly.

Corollary A.3

With notation as above,

$$\begin{aligned} v_0 =& u(t) \biggl(A(x) + \int \frac{\omega_t(x)}{u(t)} dt \biggr) \\ \sigma =&u'(t) \biggl(A(x) + \int \frac{\omega_t(x)}{u(t)} dt \biggr) + \omega_t(x) , \end{aligned}$$

where A(x) is a function on N.

Proof

Integrating equation (2) of the proposition with respect to t gives the formula for v ₀. The formula for σ follows from $\sigma = \frac{\partial v_{0}}{\partial t}$. □