2. Preliminaries

2.1 Alexandrov spaces

A metric space (X,d) is called a length space if, for any pair of points x, y in X, the distance between x and y is the infimum of the lengths of curves connecting x and y, and a geodesic space if this infimum is always achieved by some shortest curve. An Alexandrov space is a geodesic space with a lower curvature bound in the comparison geometry sense (for more details, see [Reference Burago, Burago and IvanovBBI01, Reference Burago, Gromov and Perel’manBGP92]). Given an Alexandrov space X, a curve $\gamma:I\to X$ which locally minimizes the distance is called a geodesic. If $\gamma_1$ and $\gamma_2$ are two geodesics starting at $x\in X$ , the angle between them is defined as the limit

\[\lim_{y, z\to x}\measuredangle yxz,\]

where y and z are points on $\gamma_1$ and $\gamma_2$ , respectively. Two geodesics starting at x are called equivalent if the angle between them equals zero. Let $\Sigma'_xX$ denote the set of equivalence classes of geodesics starting at x. Then $\Sigma'_xX$ forms a metric space in which the distance is the angle between the geodesics. The metric completion of $\Sigma'_xX$ , denoted by $\Sigma_xX$ , is called the space of directions of X at x. The cone over $\Sigma_xX$ , which is defined by

\[T_xX:=\big(\Sigma_xX\times[0,\infty)\big)/{(x_1,0)\sim (x_2,0)},\]

is called the tangent cone of X at x.

2.3 The topological entropy

Let (X,d) be a compact metric space, and let $\phi_t:X\to X$ be a continuous flow, that is, a continuous map satisfying $\phi_{t+s}=\phi_t\circ\phi_s$ . For any $T>0$ , define a new metric $d_T$ on X by

\[d_T(x,y)=\max_{0\leq t\leq T}d(\phi_t(x),\phi_t(y)).\]

Let $N_T^{\epsilon}$ denote the minimum number of $\epsilon$ -balls in the metric $d_T$ that are needed to cover X. The topological entropy of the flow $\phi_t$ is defined by

\[h_{\text{top}}(\phi_t)=\lim_{\epsilon\to0}\limsup_{T\to\infty}\frac{1}{T}\log(N_T^{\epsilon}).\]

For a closed Riemannian manifold M with tangent bundle TM, the geodesic flow $\phi_t:TM\to TM$ of M is defined by $\phi_t(x,v)=(\gamma_v(t),\gamma'_v(t))$ , where $\gamma_v(t)$ is the unique geodesic satisfying $\gamma_v(0)=x$ and $\gamma'_v(0)=v$ . The geodesic flow leaves invariant the unit tangent bundle SM of M and it thus induces a geodesic flow $\phi_t:SM\to SM$ . The topological entropy of M, denoted by $h_{\text{top}}(M)$ , is by definition the topological entropy of the geodesic flow $\phi_t:SM\to SM$ .

3. The proof of Theorem B

The goal of this section is to prove Theorem B. We begin by analyzing the action of $\pi_1(L)$ on $H_*(\tilde{L},{\mathbb{Q}})$ when L is the connected fiber of a manifold submetry $\pi:M\to X$ , with $\pi_1(M)=0$ and $\pi_1(L)$ finite.

Given a compact Riemannian manifold M, a point $p\in M$ , and a closed submanifold L of M, we denote by $\Omega(L,p)$ the space of piecewise smooth paths starting from some point in L and ending at p. Moreover, for any real number r, $\Omega^r(L,p)$ denotes the subset of $\Omega(L,p)$ consisting of paths of length at most r.

Proof. This proof is an adaptation of Lemma 3.1 of [Reference Paternain and PaternainPP94], where the result was proved in the case $\pi_1(L)=0$ . Introduce simplicial structures on L and its universal cover $\tilde{L}$ such that the covering map $c:\tilde{L}\to L$ is a simplicial map. Up to refinement, we can extend the simplicial structure on L to one in M (cf. [Reference VeronaVer84], § 8.11) and, in particular, we can obtain a simplicial structure on the mapping cylinder

Recall that the inclusion map $M\to M_c$ and the map $\varphi_c:M_c\to M$ are homotopy equivalences, and $\tilde{L}$ can be identified with the subspace $\tilde{L}\times \{0\}\subseteq M_c$ . Finally, we remark that the simplicial structure on $M_c$ can be chosen so that the maps $\tilde{L}\to M_c$ , $M\to M_c$ and $M_c\to M$ are simplicial maps. Notice that the (simplicial) action of $\pi_1(L)$ on $\tilde{L}$ induces an action on $M_c$ by

\[ g\cdot (\tilde{p},t)=(g\cdot \tilde{p},t),\quad g\cdot (p\in M)=p,\]

and, in particular, it induces a free action of $\pi_1(L)$ on $\Omega(\tilde{L},p)$ , $p\in M$ , such that the map $\hat{\varphi}_c:\Omega(\tilde{L},p)\to\Omega(L,p)$ , $(\hat{\varphi}_c(\gamma))(t):=\varphi_c(\gamma(t))$ is $\pi_1(L)$ -invariant, while the evaluation map $\Omega(\tilde{L},p)\to \tilde{L}$ is $\pi_1(L)$ -equivariant. Therefore, in the following diagram with fibration rows, the columns are fibrations as well.

Since $\pi_1(L)$ acts trivially on $H_*(\tilde{L})$ , we can apply the Serre spectral sequence to the fibration $\tilde{L}\to L\to B\pi_1(L)$ and obtain $H_*(\tilde{L},{\mathbb{Q}})\simeq H_*(L,{\mathbb{Q}})$ . Recall, furthermore, that in the fibration $\Omega M\to \Omega(L,p)\to L$ , the action of $\pi_1(L)$ on $\Omega M$ factors through $\pi_1(L)\to \pi_1(M)=0$ (cf. Exercise 4.3.10 in [Reference HatcherHat02]), thus again the Serre spectral sequence can be applied. Finally, since the spectral sequence of $\Omega M_c\to\Omega(\tilde{L},p)\to \tilde{L}$ is the pullback of the spectral sequence for $\Omega M\to\Omega(L,p)\to L$ , we obtain that $\hat{\varphi}_c:H_*(\Omega(\tilde{L},p),{\mathbb{Q}})\to H_*(\Omega(L,p),{\mathbb{Q}})$ is an isomorphism.

Since $\tilde{L}$ is simply connected, the same arguments of Lemma 2.1 in [Reference Paternain and PaternainPP94] go through, and we obtain the existence of a map $\alpha:(M_c,\tilde{L})\to (M_c, \tilde{L})$ with the following properties.

1. (1) The map $\hat{\alpha}:\Omega(\tilde{L},p)\to\Omega(\tilde{L},\alpha(p))$ given by $\hat{\alpha}(\gamma)(t)=\alpha(\gamma(t))$ induces the identity in homology.

2. (2) $\alpha$ sends the 1-skeleton of $M_c$ to a point.

Letting $\beta=\varphi_c\circ\alpha:(M_c,\tilde{L})\to (M,L)$ and $\hat{\beta}=\hat{\pi}_c\circ\hat{\alpha}:\Omega(\tilde{L},p)\to \Omega(L,\beta(p))$ , which from the previous paragraph induce isomorphisms in rational cohomology, we can then apply the arguments of Lemma 3.1 in [Reference Paternain and PaternainPP94] as follows. Define the subspace $\Omega^{pl}(\tilde{L},p)\subset\Omega(\tilde{L},p)$ of paths that are linear on each simplex of $M_c$ , endowed with a cellular structure with cells $\Phi:\Delta_1\times\ldots\Delta _m\to \Omega^{pl}(\tilde{L},p)$ , where $\Delta_1,\ldots, \Delta_m$ is a sequence of simplices in $M_c$ such that $\Delta_1\subseteq\tilde{L}$ and $\Delta_i,\Delta_{i+1}$ are faces of a common simplex, and where $\Phi(q_1,\ldots q_m)$ is the unique piecewise linear curve passing through the $q_i$ . Letting $\Omega^{pl}(\tilde{L},q)^{(k)}$ denote the k-skeleton, the same computations as Lemma 3.1 in [Reference Paternain and PaternainPP94] give that, for $\gamma\in\Omega^{pl}(\tilde{L},q)^{(k)}$ ,

\[\operatorname{length}(\hat{\beta}(\gamma))=\operatorname{length}(\hat{\beta}(\gamma))-\operatorname{length}(\hat{\beta}(\gamma|_{M_c^{(1)}}))\leq Ck,\]

where C depends on the piecewise linear metric in $M_c$ and the Lipschitz constant of $\beta$ . From this we have $\hat{\beta}(\Omega^{pl}(\tilde{L},q)^{(k)})\subseteq\Omega^{Ck}(L,q)$ , and from the commutative diagram in rational cohomology

for all $j\leq k$ , we have the result.

We now proceed to the proof of Theorem B.

Proof of Theorem B. Let $X^{\text{reg}}$ denote the set of manifold points of X, and let $N^{\text{reg}}:=\pi_N^{-1}(X^{\text{reg}})$ . Fix $x\in X^{\text{reg}}$ . By Theorem $3.27$ in [Reference PaternainPat99], we have

(3.1) \begin{equation}h_{\text{top}}(N)\geq\limsup_{T\to\infty}\frac{1}{T}\log\int_Nn_T(\pi_N^{-1}(x),\bar{y})d\bar{y}=\limsup_{T\to\infty}\frac{1}{T}\log\int_{N^{\text{reg}}}n_T(\pi_N^{-1}(x),\bar{y})d\bar{y},\end{equation}

where $n_T(\pi_N^{-1}(x),\bar{y})$ denotes the number of horizontal geodesics of length at most T between $\pi_N^{-1}(x)$ and $\bar{y}$ . Since $\pi_N|_{N^{\text{reg}}}:N^{\text{reg}}\to X^{\text{reg}}$ is a Riemannian submersion, we can apply Fubini’s theorem to (3.1):

\begin{align*}h_{\text{top}}(N) & \geq\limsup_{T\to\infty}\frac{1}{T}\log\int_{N^{\text{reg}}}n_T(\pi_N^{-1}(x),\bar{y})d\bar{y}\\&=\limsup_{T\to\infty}\frac{1}{T}\log\int_{X^{\text{reg}}}\bigg(\int_{\pi_N^{-1}(y)}n_T(\pi_N^{-1}(x),\bar{y})d\bar{y}|_{\pi_N^{-1}(y)}\bigg){{d} y}.\end{align*}

Now choose $y\in X^{\text{reg}}$ and $\bar{y}\in\pi_N^{-1}(y)$ . As discussed in § 2.2, every quotient geodesic in X is the image under $\pi_N$ of a horizontal geodesic in N. Since y is a manifold point of X, every quotient geodesic from x to y is the image of a unique horizontal geodesic between $\pi_N^{-1}(x)$ and $\bar{y}$ . Therefore, letting $n_T(x,y)$ denote the number of quotient geodesics between x and y, we get $n_T(\pi_N^{-1}(x),\bar{y})=n_T(x,y)$ and hence

\begin{align*}h_{\text{top}}(N) &\geq\limsup_{T\to\infty}\frac{1}{T}\log\int_{X^{\text{reg}}}\bigg(\int_{\pi_N^{-1}(y)}n_T(x,y)d\bar{y}|_{\pi_N^{-1}(y)}\bigg){d} y\\&=\limsup_{T\to\infty}\frac{1}{T}\log\int_{X^{\text{reg}}}n_T(x,y){\text{Vol}}(\pi_N^{-1}(y)){{d} y}.\end{align*}

By a similar argument, for $y'\in\pi_M^{-1}(y)$ , one has $n_T(x,y)=n_T(\pi_M^{-1}(x),y')$ . Altogether, we get that

(3.2) \begin{equation}h_{\text{top}}(N)\geq\limsup_{T\to\infty}\frac{1}{T}\log\int_{X^{\text{reg}}}n_T(\pi_M^{-1}(x),y'){\text{Vol}}(\pi_N^{-1}(y)){{d} y}.\end{equation}

Note that the horizontal geodesics of length at most T between $\pi_M^{-1}(x)$ and y’ correspond to the critical points of the energy functional $E:\Omega^T(\pi_M^{-1}(x),y')\to{\mathbb{R}}$ . Since a generic y’ is not a focal point of the fiber $\pi_M^{-1}(x)$ , the energy functional is a Morse function and, by the Morse inequality,

\[n_T(\pi_M^{-1}(x),y')\geq\sum_j b_j\big(\Omega^T(\pi_M^{-1}(x),y')\big).\]

But for any fiber L of $\pi_M$ , the spaces $\Omega(\pi_M^{-1}(x),y')$ and $\Omega(L,y')$ are homotopy equivalent: any choice of a path $\gamma$ in $X^{\text{reg}}$ between x and $\pi(L)$ induces a map $\Omega(\pi_M^{-1}(x),y')\to\Omega(L,y')$ sending a curve c to $c\star \bar{\gamma}_c$ , where $\bar{\gamma}_{c}$ is the unique horizontal lift of $\gamma$ starting at c(1). It is easy to check that this map is a homotopy equivalence. Hence,

(3.3) \begin{equation}n_T(\pi_M^{-1}(x),y')\geq\sum_j b_j\big(\Omega^T(L,y')\big),\end{equation}

where L is a fixed fiber of $\pi_M$ . Putting together Equations (3.2) and (3.3), we get

\begin{align*}h_{\text{top}}(N) & \geq\limsup_{T\to\infty}\frac{1}{T}\log\int_{X^{\text{reg}}}\bigg(\sum_j b_j(\Omega^T(L,y'))\bigg){\text{Vol}}(\pi_N^{-1}(y)){{d} y}\\& \geq \limsup_{i\to\infty}\frac{1}{Ci}\log\int_{X^{\text{reg}}}\bigg(\sum_j b_j(\Omega^{Ci}(L,y'))\bigg){\text{Vol}}(\pi_N^{-1}(y)){{d} y},\end{align*}

where C is the constant discussed in Proposition 3.3 satisfying $H_j(\Omega(L,y'))=\iota_*H_j(\Omega^{Ck}(L,y'))$ for every k and every $j\leq k$ . We thus have $b_j(\Omega^{Ci}(L,y'))\geq b_j(\Omega(L,y'))$ for all $j\leq i$ , and hence

\[\sum_jb_j\big(\Omega^{Ci}(L,y')\big)\geq\sum_{j\leq i}b_j\big(\Omega(L,y')\big).\]

Therefore, taking $i_0$ to be large enough such that $B_{Ci_0}(x)\supseteq X^{\text{reg}}$ , it follows that

\begin{align*}h_{\text{top}}(N) & \geq\limsup_{i\to\infty}\frac{1}{Ci}\log\int_{B_{Ci}(x)\cap X^{\text{reg}}}\bigg(\sum_{j\leq i}b_j\big(\Omega(L,y')\big)\bigg){\text{Vol}}(\pi_N^{-1}(y)){{d} y}\\& =\limsup_{\substack{i\to\infty\\ i\geq i_0}}\frac{1}{Ci}\log\bigg(\sum_{j\leq i}b_j\big(\Omega(L,y')\big)\int_{B_{Ci}(x)\cap X^{\text{reg}}}{\text{Vol}}(\pi_N^{-1}(y)){{d} y}\bigg)\\& =\limsup_{\substack{i\to\infty\\ i\geq i_0}}\frac{1}{Ci}\log\bigg(\sum_{j\leq i}b_j\big(\Omega(L,y')\big)\int_{ X^{\text{reg}}}{\text{Vol}}(\pi_N^{-1}(y)){{d} y}\bigg)\\& =\limsup_{\substack{i\to\infty\\ i\geq i_0}}\frac{1}{Ci}\log\bigg(\sum_{j\leq i}b_j\big(\Omega(L,y')\big){\text{Vol}}(N)\bigg)\\& =\limsup_{\substack{i\to\infty\\ i\geq i_0}}\frac{1}{Ci}\bigg(\log\sum_{j\leq i}b_j\big(\Omega(L,y')\big)+\log{\text{Vol}}(N)\bigg)\\& =\limsup_{\substack{i\to\infty}}\frac{1}{Ci}\log\sum_{j\leq i}b_j\big(\Omega(L,y')\big).\end{align*}

The above equation, together with the assumption that the topological entropy of N is zero, implies that the sequence of the Betti numbers of $\Omega(L,y')$ has sub-exponential growth.

Now, consider the fibration $\Omega(L,y')\to L\to M$ . By assumption, L is nilpotent and satisfies $\sum_{i\geq 2}\dim\pi_i(L){\otimes}{\mathbb{Q}}<\infty$ . Therefore, M is rationally elliptic by Theorem A.1 in [Reference Grove, Wilking and YeagerGWY19].

We end this section with the proof of the main result of the paper, which is a special case of the following proposition.

4. Non-negatively curved, cohomogeneity 3 manifolds

In this section, we apply Theorem A to prove rational ellipticity of simply connected, non-negatively curved Riemannian manifolds which admit cohomogeneity 3, infinitesimally polar actions. Before proceeding, we collect some lemmas required for the proof.

Notice the following things about ${\mathcal{V}}$ and ${\mathcal{K}}$ .

1. (1) They are distributions. On the one hand, it is clear that $\dim{\mathcal{K}}_q=\operatorname{codim}(L_p)$ is independent of q and that ${\mathcal{K}}$ is a distribution. As for ${\mathcal{V}}$ , notice that, since the isotropy group $G_q$ is contained in $G_{F(q)}$ for every $q\in B$ , one has $\dim{\mathcal{V}}_q=\dim{\mathcal{V}}_{F(q)}=\dim G-\dim G_{F(q)}=\dim L_p$ for every $q\in B$ . Hence, the dimension is constant, and it is easy to see that ${\mathcal{V}}$ is then a distribution as well.

2. (2) They are G-invariant. On the one hand, since F is G-equivariant, we have $F(h\cdot q)=h\cdot F(q)$ for any $h\in G$ , and thus $(d_{h\cdot q}F)\circ h_*=h_*\circ d_qF$ , which implies that $h_*({\mathcal{K}}_q)\subseteq{\mathcal{K}}_{h\cdot q}$ and thus ${\mathcal{K}}$ is G-invariant. Furthermore, given any $h\in G$ and letting $h:M\to M$ denote the corresponding smooth map, we have:

   \begin{align*}{\mathcal{V}}_{h\cdot q}&= \{X^*_{h\cdot q}\mid X\in{\mathfrak{g}},\, X\perp{\mathfrak{g}}_{h\cdot F(q)}\}\\&= \{X^*_{h\cdot q}\mid X\in{\mathfrak{g}},\, X\perp Ad(h){\mathfrak{g}}_{F(q)}\}&& \textrm{(since ${\mathfrak{g}}_{h\cdot F(q)}=Ad(h){\mathfrak{g}}_{F(q)}$)}\\&= \{(Ad(h)Y)^*_{h\cdot q}\mid Y\in{\mathfrak{g}},\, Y\perp {\mathfrak{g}}_{F(q)}\}&& \textrm{(since $Ad(h):{\mathfrak{g}}\to {\mathfrak{g}}$ is an isometry)}\\&= \{h_*(Y^*_q)\mid Y\in{\mathfrak{g}},\, Y\perp {\mathfrak{g}}_{F(q)}\}&& \textrm{(since $h_*(Y^*_q)=(Ad(h)Y)^*_{h\cdot q}$)}\\&= h_*{\mathcal{V}}_q.\end{align*}
   Therefore, ${\mathcal{V}}$ is G-invariant as well.

3. (3) They are complementary.

   • – As observed above, $\dim {\mathcal{V}}+\dim {\mathcal{K}}=\dim B$ .

   • – For all $q\in B$ , ${\mathcal{K}}=\ker d_qF$ while $d_qF|_{{\mathcal{V}}_q}:{\mathcal{V}}_q\to {\mathcal{V}}_{F(q)}=T_{F(q)}L_p$ is injective, therefore ${\mathcal{K}}\cap {\mathcal{V}}=0$ .

By the discussion above, every vector in TB can be written uniquely as $x+v$ , where $x\in {\mathcal{K}}$ and $v\in {\mathcal{V}}$ . Now, define the metric g on B as follows: given $q=f([h,w])=\exp_{h\cdotp}(h_*w)$ , let

\[g_q(x_1+v_1,x_2+v_2)=\hat{g}_q(v_1,v_2)+\tilde{g}_{\exp_p^{-1}(h^{-1}\cdot q)}((\exp_p^{-1})_*\circ(h^{-1})_*x_1,(\exp_p^{-1})_*\circ(h^{-1})_*x_2).\]

This metric has the following properties.

1. (1) g is well defined and a smooth metric. Given a different representation $q=f([hm^{-1}, m_*w])$ for some $m\in G_p$ , we have:

   
   where the second equality follows from the $G_p$ -equivariance of the exponential map and the third equality holds because $\tilde{g}$ is $G_p$ -invariant. This proves well-definedness. Since ${\mathcal{V}}$ and ${\mathcal{K}}$ and complementary, it follows that g is a metric. Finally, since ${\mathcal{V}}$ and ${\mathcal{K}}$ are smooth, the projections onto ${\mathcal{V}}$ and ${\mathcal{K}}$ are smooth, which implies that both terms in the definition of g are smooth.

2. (2) g is G-invariant. Given $z_i=x_i+v_i\in T_qB$ , $i=1,2$ , with $q=f([h,w])$ , and given $m\in G$ , it follows from G-invariance of ${\mathcal{V}}$ and ${\mathcal{K}}$ that $m_*z_i=m_*x_i+m_*v_i$ is the decomposition of $m_*z_i$ into ${\mathcal{V}}$ and ${\mathcal{K}}$ -components. Furthermore, since $m\cdot q=f([mh,w])$ , the second term in the expression for $g_{m\cdot q}(m_*x_1+m_*v_1,m_*x_2+m_*v_2)$ equals

   
   Altogether, we get that
   

All that is left to prove is that the exponential map $(N,\tilde g) \to (B, g)$ induces an isometry between the orbit spaces $N/{G_p}$ and $B/G$ . It is enough to check this on the regular part, so let $y_*\in T_{w_*}(N/G_p)^{\textrm{reg}}$ , and let $y\in T_wN$ denote a $\tilde{g}$ -horizontal lift of $y_*$ via the canonical projection $N\to N/{G_p}$ . Define $q:=\exp_p w$ , and notice $x:=(\exp_p)_*(y)\in {\mathcal{K}}_{q}$ . In particular, x is g-perpendicular to ${\mathcal{V}}$ (since ${\mathcal{V}}$ and ${\mathcal{K}}$ are g-orthogonal by the definition of g) and x is g-perpendicular to all action fields $X^*$ with $X\in {\mathfrak{g}}_{F(q)}$ (because the restriction of g to ${\mathcal{K}}$ is equal to $\tilde{g}$ and y is $\tilde{g}$ -horizontal). By the definition of ${\mathcal{V}}$ , it then turns out that x is g-perpendicular to all action fields $X^*$ with $X\in {\mathfrak{g}}$ , or equivalently, x is g-horizontal. Let $q_*$ denote the image of q in $B/G$ , and let $x_*\in T_{q_*}(B/G)$ denote the projection of x. We conclude that

\[\|x_*\|_{B/G}\stackrel{\rm(I)}{=}\|x\|_g\stackrel{\rm(II)}{=}\|y\|_{\tilde{g}}=\|y_*\|_{N/{G_p}},\]

where equality (I) follows from the fact that $(B,g)\to B/G$ is a Riemannian submersion on the regular part and x is a g-horizontal lift of $x_*$ , and (II) follows from the definition of g. This proves the lemma.

The following result is briefly discussed in a Remark in [Reference Grove, Wilking and YeagerGWY19]. We provide a more elaborate proof here.

By Theorem 1.2 in [Reference MendesMen16], there exists a $G_{p}$ -invariant metric $\tilde{g}$ on $N:=\cup_{h\in G_{p}}h\cdot\bar{U}$ such that the maps $(N,\tilde{g})/G_p\to(\bar{U},\bar{g})/W_p\to (U,g_0)$ are all isometries. By Lemma 4.1, there exists a G-invariant metric g on $\pi^{-1}(U)$ such that the map $(N,\tilde{g})/G_p\to (\pi^{-1}(U),g)/G$ is an isometry. In particular, the projection $(\pi^{-1}(U),g)\to (\pi^{-1}(U),g)/G\simeq (U,g_0)$ is a manifold submetry.

For the next result, recall that given a Riemannian metric g on a manifold M, its cometric $g^*$ is the Euclidean structure on $T^*M$ obtained by declaring an orthonormal basis of $T^*_pM$ as the dual basis of an orthonormal basis of $T_pM$ .

Proof. Let $X^{pr}$ be the manifold part (or principal part) of X, with Riemannian metric $g_0$ , and let $M^{pr}:=\pi^{-1}(X^{pr})$ . Around $p_*\in X^{pr}\cap X_i$ and $p\in \pi^{-1}(p_*)\in M_i$ , the map $\pi$ is a Riemannian submersion and, by definition, the adjoint map $(d_p\pi)^*:(T_{p_*}^*X,g_0)\to (T_p^*M_i,g_i^*)$ is an isometric immersion. In fact, the image of $(d_p\pi)^*$ is $\operatorname{Ann}(T_pL_p):=\{\alpha\in T_p^*M\mid \alpha(T_pL_p)=0\}$ independently of the metric. In particular, if $p\in M^{pr}\cap M_1\cap M_2$ , then

\[(d_p\pi)^*:(T_{p_*}^*X,g_0)\to (\operatorname{Ann}(T_pL_p),g_i^*)\]

is an isometry with respect to both cometrics. In particular, if $g^*=f_1(p)g_1^*+f_2(p)g_2^*$ , then $(d_p\pi)^*:(T_{p_*}^*X,g_0)\to(\operatorname{Ann}(T_pL_p),g^*)$ is also an isometry, that is, $\pi:(M^{pr},g)\to(X^{pr},g_0)$ is a Riemannian submersion. Since all fibers of $\pi$ are already assumed to be smooth, and general fibers of $\pi$ are Hausdorff limits of principal fibers, we obtain that $\pi:(M,g)\to X$ is a manifold submetry.

With Proposition 4.2 in hand, we proceed to the proof of Theorem C.

In the first two cases, it is known that the metrics for N’ above have $h_{\mathrm{top}}(N')=0$ . In these cases Proposition 4.2 gives a G-invariant metric on M such that $M/G$ is isometric to $N'/\Gamma$ , therefore M is rationally elliptic by Theorem A.

In the last case of $N'=S^1\times S^2_{p,q}$ , this orbifold is diffeomorphic to $S^1\times(S^3/S^1)$ , where $S^1\subset {\mathbb{C}}$ acts on $S^3\subset {\mathbb{C}}^2$ via the weighted Hopf action $w\cdot (z_1, z_2)=(w^pz_1,w^qz_2)$ . As such, there is a rotationally symmetric metric $(S^2_{p,q},g_{\mathrm{quot}})$ such that $(S^3,g_{\mathrm{round}})\to (S^2_{p,q},g_{\mathrm{quot}})$ is a submetry. Furthermore, the Ricci soliton metric $(S^2_{p,q},g_{\mathrm{sol}})$ is rotationally symmetric as well, and with the same two fixed points. In particular, we can identify the isometry groups of $(S^2_{p,q},g_{\mathrm{sol}})$ and $(S^2_{p,q},g_{\mathrm{quot}})$ (thus $\Gamma\subset\operatorname{Isom}(S^2_{p,q}, g_{\mathrm{quot}})$ ). Letting $N=S^1\times S^3$ with the product of round metrics, we can consider $X:= S^1\times(S^2_{p,q}/\Gamma)$ , with the metric induced by $g_{\mathrm{quot}}$ , as the base of the manifold submetry $N\to N'\to X$ with $h_{\mathrm{top}}(N)=0$ . Proposition 4.2 gives a G-invariant metric on M such that $M/G$ is isometric to X, and from Proposition 3.4 we have that M is rationally elliptic.