Uniqueness of Warped Product Einstein Metrics and Applications

Abstract

We prove that complete warped product Einstein metrics with isometric bases, simply connected space form fibers, and the same Ricci curvature and dimension are isometric. In the compact case we also prove that the warping functions must be the same up to scaling, while in the non-compact case there are simple examples showing that the warping function is not unique. These results follow from a structure theorem for warped product Einstein spaces which is proven by applying the results in our earlier paper He et al. (Asian J Math 2011) to a vector space of virtual Einstein warping functions. We also use the structure theorem to study gap phenomena for the dimension of the space of warping functions and the isometry group of a warped product Einstein metric.

Appendices

Appendix 1: The Space $W$ for General Warped Product Manifolds

In this Appendix we compute $W(M,g)$, where $(M,g) = \left( B \times F, g_B + u^2 g_F\right)$ is any warped product manifold. This result is referenced a few times in the proof of Theorem 2.2. As in Sect. 3 we let $\pi _1:M \rightarrow B$ and $\pi _2:M \rightarrow F$ denote the projections. $\pi _1$ is a Riemannian submersion, and we let $X,Y,\dots$ denote horizontal vector fields of this submersion and $U,V,\dots$ denote vertical vector fields. We start by recalling a lemma about the splitting of functions on a warped product.

Lemma 6.1

([10]) If $w : M \rightarrow {\mathbb {R}}$ satisfies

$$\begin{aligned} ({{\mathrm{Hess}}}_{g} w)(X,U) = 0 \end{aligned}$$

for all $X \in TB$ and $U\in TF$, then

$$\begin{aligned} w = \pi _1^*(z) + \pi _1^*(u)\cdot \pi _2^*(v), \end{aligned}$$

where $z: B \rightarrow {\mathbb {R}}$, $v: F \rightarrow {\mathbb {R}}$ are smooth functions.

Remark 6.2

Note that this decomposition of $w$ is not unique, as we can replace $z$ by $z + \alpha u$ and $v$ by $v- \alpha$ for a constant $\alpha$ and still get a valid decomposition for $w$.

This allows us to compute the space $W_{\lambda , n+m}(M)$ for a general warped product metric. The computation breaks into a number of cases.

Theorem 6.3

Let $M = B \times _{u} F$ be a warped product.

• (1) Suppose $u \in W_{\lambda , b+(k+m)}(B, g_B)$.

  • (1.a) If $F$ is Einstein with $\mathrm{Ric}^F = \frac{k-1}{m+k-1}\mu _{B}(u)$ and $\mu _{B}(u)\ne 0$, then $W_{\lambda , n+m}(M)$ is the space of functions

    $$\begin{aligned} \pi _1^*(z) + \pi _1^*(u) \pi _2^*(v), \end{aligned}$$

    where $z \in W_{\lambda , b+(k+m)}(B)$ with $\mu _B(u, z) = 0$ and $v \in W_{\mu _{B}(u), k+m}(F)$.

  • (1.b) If $F$ is Einstein with $\mathrm{Ric}^F = \frac{k-1}{m+k-1}\mu _{B}(u)$ and $\mu _{B}(u)=0$, then $W_{\lambda , n+m}(M)$ is the space of functions

    $$\begin{aligned} \pi _1^*(z) + \pi _1^*(u) \pi _2^*(v), \end{aligned}$$

    where $z \in W_{\lambda , b+(k+m)}(B)$ and $v$ satisfies

    $$\begin{aligned} {{\mathrm{Hess}}}_{F} v= - \frac{1}{m+k-1}\mu _{B}(u, z) g_F. \end{aligned}$$

  • (1.c) If $F$ does not satisfy $\mathrm{Ric}^F = \frac{k-1}{m+k-1}\mu _{B}(u)$, then $W_{\lambda , n+m}(M)$ is the space of functions

    $$\begin{aligned} \pi _1^*(u)\pi _2^*(v), \end{aligned}$$

    where $v \in W_{\mu _{B}(u), k+m}(F)$.

• (2) Suppose $u \not \in W_{\lambda , b+(k+m)}(B, g_B)$.

  • (2.a) If $F$ is $\sigma$-Einstein, then $W_{\lambda , n+m}(M)$ consists of functions of the form

    $$\begin{aligned} \pi _1^*(z), \end{aligned}$$

    where $z:B \rightarrow {\mathbb {R}}$ satisfies

    $$\begin{aligned} {{\mathrm{Hess}}}_{B} z&= \frac{z}{m} \left( \mathrm{Ric}^B - \frac{k}{u} {{\mathrm{Hess}}}_{B} u - \lambda g_B\right) \\ g_B(\nabla u, \nabla z )&= \frac{z}{u m} \left( \sigma - ( u \Delta _{B} u + (k-1) |\nabla u|_{B}^2 + \lambda u^2) \right) . \end{aligned}$$

  • (2.b) If $F$ is not Einstein, then $W_{\lambda , n+m}(M) = \{0\}$.

Remark 6.4

In the case where $B$ has boundary, note that a function $\pi _1^*(z)$ is a smooth function on $B \times _u F$ if and only if $z$ satisfies Neumann boundary conditions, i.e., $\frac{\partial z}{\partial \nu }|_{\partial B} = 0$, where $\nu$ is a normal vector field of $\partial B$.

Proof

The Ricci curvatures of a warped product are given by

$$\begin{aligned} (\mathrm{Ric} - \lambda g) (X,Y)&= \mathrm{Ric}^{B}(X,Y) - \frac{k}{u} ({{\mathrm{Hess}}}_{B} u )(X,Y)- \lambda g_B(X,Y) \\ (\mathrm{Ric} - \lambda g) (X,U)&= 0\\ (\mathrm{Ric}- \lambda g) (U,V)&= \mathrm{Ric}^{F}(U,V) - (u \Delta _{B} u + (k-1) |\nabla u|_{B}^2 + \lambda u^2) g_F(U,V). \end{aligned}$$

If $w \in W_{\lambda , n+m}(M)$ we see that the Hessian splits along the warped product and thus, from Lemma 6.1, we have $w = \pi _1^*(z)+ \pi _1^*(u) \cdot \pi _2^*(v)$ for some functions $z$ on $\mathrm{int} (B)$ and $v$ on $F$. We can also assume that $z$ is not a non-zero multiple of $u$. Multiplying the last set of equations by $\frac{w}{m}$, we have

$$\begin{aligned}&\frac{w}{m} \left( \mathrm{Ric} - \lambda g \right) (X,Y)\\&\quad = \frac{z}{m} \left( \mathrm{Ric}^{B}(X,Y) - \frac{k}{u} ({{\mathrm{Hess}}}_{B} u )(X,Y)- \lambda g_B(X,Y) \right) \\&\qquad {}+ \frac{uv}{m}\left( \mathrm{Ric}^{B}(X,Y) - \frac{k}{u} ({{\mathrm{Hess}}}_{B} u )(X,Y)- \lambda g_B(X,Y) \right) \\&\frac{w}{m}\left( \mathrm{Ric} - \lambda g\right) (U,V)\\&\quad = \frac{z}{m}\left( \mathrm{Ric}^{F}(U,V) - (u \Delta _B u + (k-1) |\nabla u|_{B}^2 + \lambda u^2) g_F(U,V) \right) \\&\qquad {}+ \frac{u v}{m}\left( \mathrm{Ric}^{F}(U,V) - (u \Delta _B u + (k-1) |\nabla u|_{B}^2 + \lambda u^2) g_F(U,V)\right) . \end{aligned}$$

The Hessian of $w$ is

$$\begin{aligned} ({{\mathrm{Hess}}}w) (X,Y)&= v({{\mathrm{Hess}}}_{B} u )(X,Y) + ({{\mathrm{Hess}}}_{B} z)(X,Y) \\ ({{\mathrm{Hess}}}w)(U,V)&= u({{\mathrm{Hess}}}_{F} v)(U,V) + uv|\nabla u|_{B}^2 g_F(U,V)\\&\quad +\, u g_B(\nabla u, \nabla z) g_F(U, V). \end{aligned}$$

Equating the horizontal equations gives us that

Note that the condition $u \in W_{\lambda , b + (k+m)}(B)$ is exactly satisfied if the quantity

$$\begin{aligned} \mathrm{Ric}^{B}(X,Y) - \frac{m+ k}{u} ({{\mathrm{Hess}}}_{B} u )(X,Y)- \lambda g_B(X,Y) \end{aligned}$$

inside the parentheses on the last line is identically zero. If there is a point in $\mathrm{int}(B)$ where the quantity is non-zero, we can fix that point and let $y\in F$ vary. The only quantity in the Eq. (5.1) which changes with $y$ is $v$. This shows that if $u \notin W_{\lambda , b + (k+m)}(B)$, then $v$ must be constant. Then we can write $w = \pi _1^*(z)$ for a possibly new function $z$ and thus $v = 0$. The equations on horizontal and vertical directions then become

$$\begin{aligned} {{\mathrm{Hess}}}_{B} z&= \frac{z}{m} \big (\mathrm{Ric}^{B} - \frac{k}{u} {{\mathrm{Hess}}}_{B} u - \lambda g_B\big ) \\ g_B(\nabla u, \nabla z ) g_F&= \frac{z}{u m} \left( \mathrm {Ric}^{F} - ( u \Delta _{B} u + (k-1) |\nabla u|_{B}^2 + \lambda u^2)g_F \right) . \end{aligned}$$

The second equation above tells us that either $\mathrm{Ric}^{F}$ is constant or $z=0$, and we are in cases (2.a) and (2.b).

Next we assume that $u \in W_{\lambda , b + (k+m)}(B)$. Then the horizontal equation (5.1) becomes

$$\begin{aligned} ({{\mathrm{Hess}}}_{B} z)(X,Y) = \frac{z }{u} ({{\mathrm{Hess}}}_{B} u)(X,Y), \end{aligned}$$

which shows that $z \in W_{\lambda , b + (k+m)} (B)$. In this case note that the quadratic form $\mu _B$ on $W_{\lambda , b+(k+m)}(B, g_B)$ is given by

$$\begin{aligned} \mu _B(z) = z\Delta _{B} z +(k+m-1)\left| \nabla z\right| _{B}^2 + \lambda z^2. \end{aligned}$$

Moreover, since $m+k-1 > 0$, we have a well-defined $\bar{\mu }_{B}(z) = \frac{\mu _B(z)}{m+k-1}$. The vertical equation is then

Dividing $u$ on both sides yields

$$\begin{aligned}&\left( {{\mathrm{Hess}}}_{F} v\right) (U,V)\\&\quad = - g_B(\nabla u, \nabla z) g_F(U, V) + \frac{z}{u m}\left( \mathrm{Ric}^{F}(U,V) - \mu _{B}(u) g_F(U,V) \right) \\&\qquad \qquad \quad +\,\, \frac{z}{u} |\nabla u|_{B}^2 g_F(U,V) + \frac{v}{m}\left( \mathrm{Ric}^{F}(U,V) - \mu _{B}(u) g_F(U,V) \right) \\&\quad = - \bar{\mu }_{B}(u, z) g_F(U, V) + \frac{z}{u}\frac{1}{m}\left( \mathrm{Ric}^{F}(U,V) - (k-1)\bar{\mu }_{B}(u) g_F(U,V) \right) \\&\qquad \qquad \quad {}+\,\, \frac{v}{m}\left( \mathrm{Ric}^{F}(U,V) - (k+m-1)\bar{\mu }_{B}(u) g_F(U,V) \right) . \end{aligned}$$

Fixing a point in $F$ and letting this equation vary over $B$ shows that either $z$ is a constant multiple of $u$, or $g_F$ is $(k-1)\bar{\mu }_B(u)$-Einstein. Since we picked $z$ so that it is not a non-zero multiple of $u$, this shows that if $\mathrm{Ric}^{F}$ is not equal to $(k-1)\bar{\mu }_{B}(u)$, then $z = 0$ and

$$\begin{aligned} {{\mathrm{Hess}}}_{F} v= \frac{v}{m}\left( \mathrm{Ric}^{F} - (k+m-1)\bar{\mu }_{B}(u) g_F\right) , \end{aligned}$$

which gives us case (1.c). If $\mathrm{Ric}^F = (k-1)\bar{\mu }_{B}(u) g_F$, then we have

$$\begin{aligned} {{\mathrm{Hess}}}_F v + \bar{\mu }_{B}(u) vg_F = - \bar{\mu }_{B}(u,z) g_F. \end{aligned}$$

If $\bar{\mu }_{B}(u) \ne 0$, then by adding a constant $\alpha$ (with $\bar{\mu }_{B}(u,z)= \alpha \bar{\mu }_{B}(u)$) to $z$ and subtracting $z$ by $\alpha u$ we may assume that $\bar{\mu }_{B}(u, z) = 0$ and then we have

$$\begin{aligned} {{\mathrm{Hess}}}_{F} v = - \bar{\mu }_{B}(u) v g_F, \end{aligned}$$

which gives us case (1.a). Otherwise, we have $\bar{\mu }_B(u) = 0$, i.e., $(F, g_F)$ is Ricci flat and

$$\begin{aligned} {{\mathrm{Hess}}}_F v = - \bar{\mu }_{B}(u,z) g_F, \end{aligned}$$

which is case (1.b). $\square$

Appendix 2: The Quadratic Form $\mu$

In this Appendix we discuss more details about the quadratic form $\mu$. First we deal with the degenerate $m=1$ case.

Proposition 7.1

Let $m=1$ and suppose that $W_{\lambda , n+1}(M) \ne \{ 0 \}$. Then either

• (1) $M$ is $\lambda$-Einstein,

• (2) $\mu$ is positive definite, $\dim W_{\lambda , n+1}(M) = 1$, and $\mathrm{scal} > (n-1) \lambda$ and is non-constant,

• (3) $\mu$ is negative definite, $\dim W_{\lambda , n+1}(M) = 1$, and $\mathrm{scal} < (n-1) \lambda$ and is non-constant, or

• (4) $\mu (w) = 0$ for all $w \in W$ and $\mathrm{scal} = (n-1)\lambda$ is constant.

In cases (1), (2), and (3) the non-zero functions in $W_{\lambda , n+1}(M)$ do not vanish.

In particular, this implies that $\mu$ is completely degenerate when $m=1$ and $\dim W(M)>1$.

Corollary 5.1

If $\dim W_{\lambda , n+1}(M) >1$ then $\mu (w) = 0$ for all $w \in W$.

Remark 7.3

When there is a $w$ with $\mu (w)=0$, $(M,g)$ is called a static metric. Abstractly, the cases with $\mu \ne 0$ can occur, for example, if

$$\begin{aligned} g_M = dr^2 + \cosh ^2(r) g_{F^{n-1}}, \end{aligned}$$

where $F$ is an $(n-1)$-Einstein metric with Ricci curvature $-(n-1)$. Then $\cosh (r) \in W_{-n , n+1}(M, g_M)$ and $\mu$ is negative definite. On the other hand, if $\mu (w)\ne 0$ there is no Einstein metric $E = M \times _{w} F^1$ because there is no one-dimensional fiber with Ricci curvature $\mu (w)$.

Proof of Proposition 7.1

When $m=1$ we have

$$\begin{aligned} \mu (w) = w^2 \left( \mathrm{scal} - (n-1) \lambda \right) . \end{aligned}$$

If $w$ is constant then $M$ is $\lambda$-Einstein. Otherwise, suppose that $\mu (w) \ne 0$. Then we have

$$\begin{aligned} w^2 = \frac{\mu (w)}{ \mathrm{scal} - (n-1) \lambda }, \end{aligned}$$

showing that $\mu (w)$ and $\mathrm{scal} -(n-1) \lambda$ have the same sign and that $w$ never vanishes. This also shows that $w$ is determined up to a multiplicative constant by the scalar curvature. This implies that $\dim W_{\lambda , n+1}(M)= 1$ and the scalar curvature is non-constant. Cases (2) and (3) then correspond to the sign choice of $\mu$.

Finally, if $\mu (w)$ is zero for some non-zero function $w$, then $\mathrm{scal} = (n-1)\lambda$. This implies $\mu (w)=0$ for all $w \in W$. $\square$

When $m\ne 1$ we can also divide $\mu$ by $(m-1)$ and get the rescaled quadratic form

$$\begin{aligned} \bar{\mu }(u) = |\nabla u|^2 + \kappa w^2, \end{aligned}$$

(5.3)

where

We also record here some basic statements about the interplay of $\bar{\mu }$ with the space $W$ in the case where $m \ne 1$; the proofs follow simply from the definition.

Proposition 7.4

Let $\left( M,g\right)$ be a Riemannian manifold and $w\in W_{\lambda ,n+m}\left( M,g\right)$ with $m\ne 1$. Then the following hold.

• (1) If $\bar{\mu }\left( w\right) \le 0$, then either $w$ is trivial or never vanishes.

• (2) If $\bar{\mu }\left( w\right) >0$ and $\kappa \left( p\right) \le 0$ for some $p\in M$, then $\nabla w|_{p}\ne 0$.

• (3) If $\bar{\mu }\left( w\right) \ge 0$ and $\kappa \left( p\right) <0$ for some $p\in M$, then $\nabla w|_{p}\ne 0$.

• (4) If $\kappa \left( p\right) >0$ for some $p\in M$, then $\bar{\mu }$ is elliptic.

• (5) If $\kappa \ge 0$ on $M$ and $\kappa \left( p\right) =0$ for some $p\in M$, then $\bar{\mu }$ is either elliptic or parabolic.

• (6) The quadratic form $\bar{\mu }$ has index $\le 1$, nullity $\le 1$, and they cannot both be $1$.