The Ricci curvature and the normalized Ricci flow on the Stiefel manifolds $\operatorname{SO}(n)/\operatorname{SO}(n-2)$

1.   Introduction

One of powerful tools to study the evolution of Riemannian metrics on a given Riemannian manifold $\mathscr{M}^d$ is the normalized Ricci flow equation

introduced by Hamilton ^[1] for a family of a Riemannian metrics ${\bf{g}}(t)$ on $\mathscr{M}$, where $\operatorname{Ric}_{{\bf{g}}}$ and $S_{{\bf{g}}}$ are the Ricci tensor and the scalar curvature of a metric ${\bf{g}}$, respectively. Many papers are devoted to a class of Riemannian manifolds $\mathscr{M}$ called homogeneous, on which the isometry group $\operatorname{Isom}(\mathscr{M})$ acts transitively, see ^[2,3,4,5]. Any Riemannian homogeneous manifold $\mathscr{M}$ can be identified (is diffeomorphic) to some homogeneous space $G/H$ with $G = \operatorname{Isom}(\mathscr{M})$ being a Lie group according to the Myers and Steenrod theorem and $H = G_m$ the isotropy subgroup at a given point $m\in \mathscr{M}$. A large class consists of reductive homogeneous spaces $G/H$, where $G$ is a compact and semisimple Lie group and $H$ is a connected closed subgroup of $G$. Let $\mathfrak{g}$ and $\mathfrak{h}$ be the corresponding Lie algebras of $G$ and $H$, and $\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{p}$ be a reductive decomposition of $\mathfrak{g}$ orthogonal with respect to the Killing form $B$ defined on $\mathfrak{g}$ so that $\operatorname{Ad}(H)\mathfrak{p}\subset \mathfrak{p}$. Since $B$ is negative definite, $-B$ defines an $\operatorname{Ad}(G)$-invariant inner product $\langle\cdot, \cdot\rangle = -B(\cdot, \cdot)$ on $\mathfrak{g}$. Assume that $\mathfrak{p}$ admits a decomposition $\mathfrak{p}_1\oplus \cdots \oplus \mathfrak{p}_k$ into pairwise inequivalent irreducible $\operatorname{Ad}(H)$-modules of dimensions $d_i = \operatorname{dim}\mathfrak{p}_i$ such that $d_1+\cdots + d_k = d = \operatorname{dim} (G/H)$. Then, any $G$-invariant symmetric covariant 2-tensor on $G/H$ can be written as $\left.c_1\langle\cdot, \cdot\rangle\right|_{\mathfrak{p}_1}+\cdots + \left.c_k \langle\cdot, \cdot\rangle\right|_{\mathfrak{p}_k}$, where $c_1, \dots, c_k$ are some real numbers. In particular, so is any $G$-invariant metric ${\bf g}(\cdot, \cdot) = \left.x_1\langle\cdot, \cdot\rangle\right|_{\mathfrak{p}_1}+\cdots + \left.x_k \langle\cdot, \cdot\rangle\right|_{\mathfrak{p}_k}$ on $G/H$, where $x_i\in \mathbb{R}_{+}$, $i = 1, \dots, k$. The Ricci tensor of this metric has the form $\operatorname{Ric}_{\bf g}(\cdot, \cdot) = \left.x_1 {\bf r}_1\langle\cdot, \cdot\rangle\right|_{\mathfrak{p}_1}+\cdots + \left.x_k {\bf r}_k\langle\cdot, \cdot\rangle\right|_{\mathfrak{p}_k}$, where ${\bf r}_i = {\bf r}_i(x_1, \dots, x_k)$ are components of the Ricci tensor ^[6]. The decomposition $\mathfrak{p} = \mathfrak{p}_1\oplus \cdots \oplus \mathfrak{p}_k$ allows (1.1) to split into a system of $k$ nonlinear autonomous differential equations

In particular, an interesting question is finding if the normalized Ricci flow preserves the positivity of the curvature of Riemannian metrics on a given manifold. In ^[7,8] some results were obtained concerning this question on the Wallach spaces $\operatorname{SU}(3)/T^2$, $\operatorname{Sp}(3)/\operatorname{Sp}(1)\times \operatorname{Sp}(1)\times \operatorname{Sp}(1)$, $F_4/\operatorname{Spin}(8)$. The case of generalized Wallach spaces with coinciding parameters $a_1 = a_2 = a_3\in (0, 1/2)$ was considered in ^[9]. Some extensions of the results of ^[9] can be found in ^[10,11]. Recently, in ^[12] the results of ^[9] were generalized to the general case $a_1, a_2, a_3\in (0, 1/2)$ according to the classification of ^[13]:

Theorem 1 (Theorem 5 in ^[12]). The normalized Ricci flow (1.1) evolves certain invariant Riemannian metrics with positive Ricci curvature to metrics with mixed Ricci curvature on every generalized Wallach space with $a_1+a_2+a_3\le 1/2$.

Theorem 2 (Theorem 6 in ^[12]). The following assertions hold for generalized Wallach spaces with $a_1+a_2+a_3 > 1/2$:

1) The normalized Ricci flow (1.1) evolves all invariant Riemannian metrics with positive Ricci curvature to metrics with positive Ricci curvature if $\theta \ge \max\left\{\theta_1, \theta_2, \theta_3\right\}$;

2) The normalized Ricci flow (1.1) evolves certain metrics with positive Ricci curvature to metrics with positive Ricci curvature if $\theta \ge \max\left\{\theta_1, \theta_2, \theta_3\right\}$ fails,

where $\theta: = a_1+a_2+a_3-1/2$ and $\theta_i: = a_i-\dfrac{1}{2}+\dfrac{1}{2}\sqrt{\dfrac{1-2a_i}{1+2a_i}}$, $i = 1, 2, 3$.

Theorem 3 (Theorem 7 in ^[12]). The following assertions hold for $\operatorname{SO}(k+l+m)/\operatorname{SO}(k)\times \operatorname{SO}(l)\times \operatorname{SO}(m)$, $k\ge l\ge m > 1$ (denoted by $\operatorname{GWS}$ 1 according to ^[13]). Under the normalized Ricci flow (1.1)

1) All invariant Riemannian metrics with positive Ricci curvature can be evolved into metrics with positive Ricci curvature if either $k\le 11$ or one of the conditions $2 < l+m\le X(k)$ or $l+m\ge Y(k)$ is satisfied at each fixed $k\in \{12, 13, 14, 15, 16\}$;

2) At least some invariant Riemannian metrics with positive Ricci curvature can be evolved into metrics with positive Ricci curvature if $k\ge 17$ or if $X(k) < l+m < Y(k)$ for $k\in \{12, 13, 14, 15, 16\}$;

3) The number of $\operatorname{GWS}$ 1 spaces on which any original metric with $\operatorname{Ric} > 0$ maintains $\operatorname{Ric} > 0$ is finite, whereas there are infinitely (countably) many $\operatorname{GWS}$ 1 on which $\operatorname{Ric} > 0$ can be preserved at least for some original metrics with $\operatorname{Ric} > 0$,

where $X(k): = \dfrac{2k(k-2)}{k+2+\sqrt{k^2-12k+4}}-k+2$ and $Y(k): = \dfrac{2k(k-2)}{k+2-\sqrt{k^2-12k+4}}-k+2$.

Since $(l, m)\ne (1, 1)$, the case of the space $\operatorname{SO}(n)/\operatorname{SO}(n-2)\times \operatorname{SO}(1)\times \operatorname{SO}(1)$ cannot be covered by Theorem 3. In the present paper we study the above question on the preservation of positivity of the Ricci curvature for the homogeneous space $\operatorname{SO}(n)/\operatorname{SO}(n-2)$ diffeomorphic to the Stiefel manifold $V_2\mathbb{R}^n$. In general $V_k\mathbb{R}^n$, $k\le n$, can be defined as the set of $n\times k$ matrices $A$ with real entries such that $A^tA = I_k$, where $A^t$ means the transpose of $A$ and $I_k$ is the $k\times k$ identity matrix. $V_k\mathbb{R}^n$ becomes a compact smooth manifold in the subspace topology inherited from the topology of $\mathbb{R}^{n\times k}$. It is known that for $k < n$, the group $\operatorname{SO}(n)$ acts on $V_k\mathbb{R}^n$ transitively and $V_k\mathbb{R}^n$ is diffeomorphic to the reductive homogeneous space $\operatorname{SO}(n)/\operatorname{SO}(n-k)$. In the case of these spaces there are equivalent submodules which may cause a complicated decomposition of the Ricci tensor. Although the decomposition of $\mathfrak{p}$ admits two equivalent submodules $\mathfrak{p}_1$ and $\mathfrak{p}_2$ causing the dependence of an $\operatorname{SO}(n)$-invariant metric of $\operatorname{SO}(n)/\operatorname{SO}(n-2)$ on four parameters, it was proved in ^[14] that $\operatorname{SO}(n)$-invariant metrics on $\operatorname{SO}(n)/\operatorname{SO}(n-2)$ can be reduced to a diagonal form and described by an $\operatorname{Ad}(\operatorname{SO}(n-2))$-invariant inner product

on $\mathfrak{p} = \mathfrak{p}_1\oplus \mathfrak{p}_2\oplus \mathfrak{p}_3$ (see also ^[15]). The corresponding Ricci tensor is $\operatorname{Ric}_{\bf g}(\cdot, \cdot) = \left.x_1 {\bf r}_1\langle\cdot, \cdot\rangle\right|_{\mathfrak{p}_1}+ \left.x_2 {\bf r}_2\langle\cdot, \cdot\rangle\right|_{\mathfrak{p}_2}+ \left.x_3 {\bf r}_3\langle\cdot, \cdot\rangle\right|_{\mathfrak{p}_3}$. Clearly, $d_1 = \operatorname{dim}\mathfrak{p}_1 = n-2 = \operatorname{dim}\mathfrak{p}_2 = d_2$ and $d_3 = \operatorname{dim}\mathfrak{p}_3 = 1$ with $d = 2n-3$.

The main result of this paper is contained in the following theorem.

Theorem 4. On every Stiefel manifold $\operatorname{SO}(n)/\operatorname{SO}(n-2)$ with $n\ge 3$, the normalized Ricci flow (1.1) evolves any metric (1.3) to a metric with positive Ricci curvature.

2.   Results

The following expressions were found in ^[16] for the components of the Ricci tensor of the metric (1.3):

Then, the scalar curvature $S_{\bf g} = d_1{\bf r}_1+d_2 {\bf r}_2+d_3{\bf r}_3$ takes the form

Substituting (2.1) and (2.2) into (1.2) the following system of ordinary differential equations can be obtained on $\operatorname{SO}(n)/\operatorname{SO}(n-2)$, $n\ge 3$:

where

2.1.   The set $\mathscr{R}_{+}$ and its structural properties

Denote by $\mathscr{R}_{+}$ the set of metrics (1.3) which admit positive Ricci curvature

Observe that ${\bf r}_3 = 0$ is equivalent to $(x_1-x_2+x_3)(x_2-x_1+x_3) = 0$ for every $n\ge 3$. Therefore, $\mathscr{R}_{+}$ is bounded by two planes $-x_1+x_2+x_3 = 0$ and $x_1-x_2+x_3 = 0$ in any case, where $x_1, x_2, x_3 > 0$. Denote them by $\Pi_1$ and $\Pi_2$, respectively. Let $\Gamma_1$ and $\Gamma_2$ be those components of the cones ${\bf r}_1 = 0$ and ${\bf r}_2 = 0$ defined by the equations (see Figure 1)

respectively. We denote the other components

by $\Gamma_1^{-}$ and $\Gamma_2^{-}$, respectively (the lower surfaces in Figure 1 in magenta and aquamarine). Let $\Pi_3$ be the plane $x_1+x_2-x_3 = 0$ and $L$ be the ray $\big(p, p, (2n-4)p\big)$, $p > 0$ (depicted in blue color in Figure 1).

Lemma 1. For all $n\ge 3$, the set $\mathscr{R}_{+}$ has the boundary

with $\widetilde{\Gamma}_1 = \left\{(\mu t, \nu t, t) \; \big| \; t > 0, \, 0 < \nu\le \widetilde{\nu}\right\}\subset \Gamma_1$ and $\widetilde{\Gamma}_2 = \left\{(\nu t, \mu t, t) \; \big| \; t > 0, \, 0 < \nu \le \widetilde{\nu}\right\}\subset \Gamma_2$, where $\mu = \sqrt{\nu^2-2(n-2)\nu+1}$ and $\widetilde{\nu} = (2n-4)^{-1}$. Moreover, $\Pi_1 \cap \Pi_2 = \Pi_1 \cap \widetilde{\Gamma}_1 = \Pi_2 \cap \widetilde{\Gamma}_2 = \emptyset$ and $\widetilde{\Gamma}_1\cap \widetilde{\Gamma}_2 = L$ for all $n\ge 3$.

Proof. Let us consider $\Gamma_1\cap \Gamma_2$. The system of equations

is equivalent to the system of ${\bf r}_1 = 0$ and ${\bf r}_2 = 0$. It is easy to see that $x_3 = (n-2)(x_1+x_2)$ is a consequence of the system corresponding to $\Gamma_1\cap \Gamma_2$. Substituting this expression for $x_3$ into the second equation in the system, we obtain

If $n\ge 4$, then $x_1 = x_2$. Hence, the straight line $x_1 = x_2 = p > 0$, $x_3 = 2(n-2)p$ denoted by $L$ is a common part of the cones $\Gamma_1$ and $\Gamma_2$. If $n = 3$, then (2.4) implies that $\Gamma_1 = \Gamma_2 = \Pi_3$.

The system above also admits the extra solution $x_1 = x_2$, $x_3 = 0$. Obviously, it corresponds to $\Gamma_1^{-}\cap \Gamma_2^{-}$.

The component $\Gamma_1$ of the cone ${\bf r}_1 = 0$ can be parameterized as

where $\mu = \sqrt{\nu^2-2(n-2)\nu+1}$ and

is the smallest root of the quadratic equation $\nu^2-2(n-2)\nu+1 = 0$ (by symmetry, for the component $\Gamma_2$ of ${\bf r}_2 = 0$, we have $x_2 = \mu t$, $x_1 = \nu t$, $x_3 = t$).

For all $n\ge 4$ the conic components $\Gamma_1$ and $\Gamma_2$ have extra pieces not relating to the set $\mathscr{R}_{+}$. Since $\Gamma_1 \cap \Gamma_2 = L$ then necessarily $\mu = \nu$, implying the critical value $\widetilde{\nu} = (2n-4)^{-1}$. It is easy to see that the useful part $\widetilde{\Gamma}_1$ of $\Gamma_1$ can be parameterized by values $t > 0$ and $0 < \nu\le \widetilde{\nu}$. The extra part of $\Gamma_1$ corresponds to the values $t > 0$ and $\tilde{\nu} < \nu < l$. The description of $\widetilde{\Gamma}_2$ is obvious from symmetry. The equality $\widetilde{\Gamma}_1 \cap \widetilde{\Gamma}_2 = L$ is obvious too. The system of the equations $-x_1+x_2+x_3 = 0$ and $x_1-x_2+x_3 = 0$ implies that $x_3 = 0$. Analogously, it follows from $\Pi_1 \cap \Gamma_1$ and $\Pi_2 \cap \Gamma_2$ that $x_2 = 0$ and $x_1 = 0$. Therefore, $\Pi_1 \cap \Pi_2 = \Pi_1 \cap \Gamma_1 = \Pi_2 \cap \Gamma_2 = \emptyset$ for $x_i > 0$ implying $\Pi_1 \cap \Pi_2 = \Pi_1 \cap \widetilde{\Gamma}_1 = \Pi_2 \cap \widetilde{\Gamma}_2 = \emptyset$. □

2.2.   The vector field ${\bf V}$ on the boundary $\partial(\mathscr{R}_{+})$

Introduce the vector field

associated with the differential system (2.3).

Lemma 2. For every $n\ge 3$, the vector field ${\bf V}$ associated to the system (2.3), when restricted to its boundary $\partial(\mathscr{R}_{+})$ takes values in the domain $\mathscr{R}_{+}$.

Proof. By Lemma 1 the planes $\Pi_1$ and $\Pi_2$ bound the domain $\mathscr{R}_{+}$ for every $n\ge 3$. Denote by ${\bf n}_1$ and ${\bf n}_2$ their normals $(-1, 1, 1)$ and $(1, -1, 1)$. We claim that the inner product $({\bf V}, {\bf n}_1)$ is positive at every point of $\Pi_1$ for all $n\ge 3$. Indeed, $\Pi_1$ can be parameterized as $x_2 = u$, $x_3 = v$, $x_1 = u+v$ with $u, v > 0$. Then,

on $\Pi_1$ for $n\ge 3$, where

The same inequality $({\bf V}, {\bf n}_2) = \frac{2}{n-2} > 0$ holds on the plane $\Pi_2$ for all $n\ge 3$. Indeed, $\Pi_2$ can be parameterized as $x_1 = u$, $x_3 = v$, $x_2 = u+v$ and $f_1\big|_{\Pi_2} = f_2\big|_{\Pi_1}$, $f_2\big|_{\Pi_2} = f_1\big|_{\Pi_1}$, $f_3\big|_{\Pi_2} = f_3\big|_{\Pi_1}$.

The case $n = 3$. According to Lemma 1, we have one more plane $\Pi_3$ that bounds the domain $\mathscr{R}_{+}$. Note that $f_1, f_2$, and $f_3$ take the following forms on $\Pi_3$:

Then, we obtain $({\bf V}, {\bf n}_3) = f_1\big|_{\Pi_3}+f_2\big|_{\Pi_3} -f_3\big|_{\Pi_3} = 2 > 0$ on $\Pi_3$, where ${\bf n}_3 = (1, 1, -1)$ is the normal of $\Pi_3$.

Since the normals ${\bf n}_1$, ${\bf n}_2$ and ${\bf n}_3$ are directed inside the domain $\mathscr{R}_{+}$ the inequalities $({\bf V}, {\bf n}_i) > 0$ imply that the vector field ${\bf V}$ is directed towards $\mathscr{R}_{+}$ on every point of the boundary $\partial(\mathscr{R}_{+}) = \Pi_1\cup\Pi_2\cup \Pi_3$.

The case $n\ge 4$. Then, $\partial (\mathscr{R}_{+}) = \Pi_1 \cup\Pi_2 \cup \widetilde{\Gamma}_1 \cup \widetilde{\Gamma}_2$ by Lemma 1. It suffices to consider the surface $\widetilde{\Gamma}_1 = \left\{(\mu t, \nu t, t)\; | \; t > 0, \, \nu\in \big(0, \widetilde{\nu}\big]\right\}$, where $\widetilde{\nu} = (2n-4)^{-1}$.

The components of the normal ${\bf m} = \nabla {\bf r}_1 = \left(\frac{\partial{\bf r}_1}{\partial x_1}, \frac{\partial{\bf r}_1}{\partial x_2}, \frac{\partial{\bf r}_1}{\partial x_3}\right)$ to the conic component $\Gamma_1$ have the forms

It is easy to check that $\frac{1}{2(n-2)} < n-2-\sqrt{(n-1)(n-3)}\le \frac{1}{n-2}$ for $n\ge 3$. Then, the inequalities $0 < \nu\le \widetilde{\nu} < l < \frac{1}{n-2} < n-2$ yield that

on $\Gamma_1$ and, in particular, on $\widetilde{\Gamma}_1$ implying that ${\bf m}$ is directed inside the domain $\mathscr{R}_{+}$ for all $n\ge 4$.

We claim that the inner product $({\bf V}, {\bf m})$ is positive on every point of $\widetilde{\Gamma}_1$. Indeed, by calculations in Maple, we obtain that $\operatorname{sign}({\bf V}, {\bf m}) = \operatorname{sign} (F-G)$, where

Since $F > 0$ and $G > 0$ for all $n\ge 4$ and all $\nu\in \big(0, \widetilde{\nu}\big]$ the sign of $F-G$ coincides with the sign of the polynomial

We claim that $p(\nu) > 0$ for all $\nu\in \big(0, \widetilde{\nu}\big]$ and all $n\ge 4$. It is more convenient to consider the interval $[0, 1]\supset \big(0, \widetilde{\nu}\big]$ instead of $\big(0, \widetilde{\nu}\big]$. Since the discriminant

of the polynomial $p(\nu)$ is positive for every $n\ge 4$ then $p(\nu)$ has no multiple roots in $[0, 1]$. Therefore, we can use Sturm's method for our goal. Construct the sequence of Sturm polynomials $p_0 = p$, $p_1 = p_0'$, $p_{i+1} = -\operatorname{rem}(p_{i-1}, p_i)$, where $\operatorname{rem}(p_{i-1}, p_i)$ is the remainder of the division of $p_{i-1}$ by $p_i$, $i = 1, 2, 3$. Then, for $\nu = 0$ we obtain using Maple that

where

for $n\ge 4$, where $\alpha_4\ne 0$ for $n\ge 4$ since the unique acceptable divisor $n = 467$ of $1401$ does not satisfy $\alpha_4 = 0$.

Evaluate now $p_i$ at $\nu = 1$:

where $\alpha_6 = 24+60(n-3)+63(n-3)^2+35(n-3)^3+10(n-3)^4+(n-3)^5 > 0$.

It follows that the Sturm sequence has the same number of sign changes, namely $1$, for both $\nu = 0$ and $\nu = 1$ under the condition $n\ge 4$. Therefore, $p(\nu)$ has no real roots in the interval $[0, 1]$ for all $n\ge 4$. Since $p(0) > 0$ we have $p(\nu) > 0$ for all $\nu\in [0, 1]$, in particular, this is true for all $\nu\in \big(0, \widetilde{\nu}\big]$. Therefore, $({\bf V}, {\bf m}) > 0$ on $\widetilde{\Gamma}_1$. Since the normal ${\bf m}$ is directed into the domain $\mathscr{R}_{+}$, as shown above the vector field ${\bf V}$ is directed towards $\mathscr{R}_{+}$ on the surface $\widetilde{\Gamma}_1$. By symmetry, the analogous assertion holds for ${\bf V}$ on the surface $\widetilde{\Gamma}_2\subset \Gamma_2$. □

2.3.   Invariant sets and singular points of the system (2.2)

Let $\Sigma$ be the surface defined by the equation $\operatorname{Vol}(x_1, x_2, x_3) = c$, $c > 0$, where

is the volume function for the metric (1.3). Introduce the following subsets of $\Sigma$ (curves on $\Sigma$) as well

In Figure 1 the curves $I_1, I_2, I_3$ are depicted in gold for $c = 1$ and $n = 4$.

Lemma 3. The following assertions hold for the system (2.3):

1) The algebraic surface $\Sigma$ is invariant for every $n\ge 3$ and every $c > 0$;

2) The curve $I_3$ is invariant for every $n\ge 3$ and every $c > 0$;

3) The curves $I_1$ and $I_2$ are also invariant for every $c > 0$ if $n = 3$.

4) For all $n\ge 3$ the following inclusions are true:

Proof. 1) Actually the first assertion can easily be proved for system (1.2). Put $\operatorname{Vol} = x_1^{d_1}\cdots x_k^{d_k}$ and $U: = \operatorname{Vol}-c$. Then, the invariance of $\Sigma$ is equivalent to the inner product $(\nabla U, {\bf V})$ vanishing on it. Indeed

due to $S_{\bf g} = \sum_{i = 1}^kd_i{\bf r}_i$ and $d = \sum_{i = 1}^kd_i$ (see also Lemma 2 in ^[17]).

2) The invariance of $I_3$ is equivalent to the vector field ${\bf V}$ being parallel to the tangent of $I_3$ at its every point. Direct calculations confirm this property:

3) It suffices to observe that $I_1, I_2$, and $I_3$ can be obtained by cyclic permutations in $(\tau, \tau, c\tau^{-2})$ in the case $n = 3$. Moreover, ${\bf x}^0 = \big(\sqrt[3]{c}, \sqrt[3]{c}, \sqrt[3]{c}\big)$ by Lemma 4. However, for $n\ge 4$, the curves $I_1$ and $I_2$ cannot pass through the point ${\bf x}^0$ and hence cannot be invariant.

4) Choose $I_1$. Substituting $x_1 = c^{\tfrac{1}{n-2}}\tau^{\tfrac{1-n}{n-2}}$, $x_2 = \tau$ and $x_3 = \tau$ into $\Pi_1$ and $\Pi_2$ we obtain

Analogously ${\bf r}_1$ and ${\bf r}_2$ take the following forms on $I_1$:

Then, $\Pi_2\big|_{I_1} > 0$ and ${\bf r}_1\big|_{I_1} > 0$ for $\tau > 0$ and $n\ge 3$. But $\Pi_1\big|_{I_1} > 0$ only for $\tau > \tau_1$, where $\tau_1: = 2^{2-n}c$ is the single root of the equation $2-c^{\tfrac{1}{n-2}} \tau^{-\tfrac{1}{n-2}} = 0$.

Note that ${\bf r}_2\big|_{I_1} > 0$ for all $n\ge 3$ if $\tau > \tau_1$:

By symmetry, for $I_2$ we have

Now consider $I_3$. Then,

Therefore, ${\bf r}_1\big|_{I_3} = {\bf r}_2\big|_{I_3} > 0$ for $n\ge 3$ and $\tau > \tau_2: = \left(\dfrac{2n-4}{c}\right)^{\tfrac{1}{3-2n}}$. □

Remark 1. As follows from (2.7), $I_1$ intersects the cone ${\bf r}_2 = 0$ at the point $\left(c^{\tfrac{1}{n-2}}\, \tau_0^{\tfrac{1-n}{n-2}}, \tau_0, \tau_0\right)$, where $\tau_0$ is defined from the equation $c^{\tfrac{1}{n-2}}\, \tau_0^{-\tfrac{1}{n-2}} = 2(n-2)$ for all $n\ge 3$. However, this point does not belong to the set $\mathscr{R}_{+}$ since $\Pi_1\big|_{I_1} = \tau\left(2-c^{\tfrac{1}{n-2}}\, \tau^{-\tfrac{1}{n-2}}\right)\le 0$ at $\tau = \tau_0$. Indeed, $2-2(n-2) = 6-3n\le 0$ for $n\ge 3$. Actually, here we deal with the intersection of $I_1$ with another component $\Gamma_2^{-}$ of the cone ${\bf r}_2 = 0$, establishing that $\Gamma_2^{-}$ is the extra component which has no relation to the set $\mathscr{R}_{+}$ (the lower surface in aquamarine in Figure 1). Thus, $I_1$ does not intersect the useful component $\Gamma_2$ of ${\bf r}_2 = 0$ (see (2.4)). A similar conclusion can be obtained for $I_2$, by interchanging the indices by symmetry.

Lemma 4. The following assertions hold for the system (2.3) at a fixed $c > 0$:

1) For every $n\ge 3$ the system (2.3) has the unique family of one-parameter singular (equilibrium) points given by the formula

The actual value $q = q_0$ of the parameter $q$ at which ${\bf x}^0$ belongs to $\Sigma$ is

2) If $n = 3$, then $\Sigma$ is the stable manifold of ${\bf x}^0$ with the tangent space $E^s = \operatorname{Span}\{(-1, 1, 0), (-1, 0, 1)\}$ and, hence, ${\bf x}^0\in \Sigma$ is a stable node. Moreover, $I_1, I_2$ and $I_3$ are one-dimensional stable submanifolds for ${\bf x}^0$.

3) If $n\ge 4$, then ${\bf x}^0$ is a saddle with the tangent spaces $E^s = \operatorname{Span}\left\{\big(1, 1, -4(n-2)^2(n-1)^{-1}\big)\right\}$ and $E^u = \operatorname{Span}\{(-1, 1, 0)\}$ of corresponding manifolds (separatrices) on $\Sigma$. Moreover, the stable separatrix is exactly $I_3$ for all $n\ge 4$.

4) For all $n\ge 3$, the singular point ${\bf x}^0$ also admits the center manifold with the tangent $E^c = \operatorname{Span}\big\{(1, 1, \kappa)\big\}$.

Proof. Fix $c > 0$ and use ideas of ^[17].

1) Every singular (equilibrium) point of the system (1.2) is also an invariant Einstein metric of a considered space $G/H$. Conversely, every Einstein metric of $G/H$ is a singular point of (1.2). Such a conclusion easily follows from the observation that the equalities $f_1 = \cdots = f_k = 0$ and ${\bf r}_1 = \cdots = {\bf r}_k = d^{-1}S_{\bf g}$ are equivalent. In particular, according to ^[14,18], the Stiefel manifold $\operatorname{SO}(n)/\operatorname{SO}(n-2)$ admits the unique $\operatorname{SO}(n)$-invariant Einstein metric $\left(1, 1, \kappa\right)$ up to scale for every $n\ge 3$ which is also the unique singular point of the system (2.3). The actual value $q_0$ providing ${\bf x}^0\in \Sigma$ can easily be found from the condition

To establish that ${\bf x}^0\in \mathscr{R}_{+}$ for all $n\ge 3$, it suffices to check the definition of the set $\mathscr{R}_{+}$:

Denote by $\chi(\lambda, {\bf x})$ the characteristic polynomial of the Jacobian matrix ${\bf J}({\bf x})$ of the vector field ${\bf V}({\bf x})$ associated with the differential system (2.3), ${\bf x} = (x_1, x_2, x_3)$. Then, the cubic equation

admits three real roots

for ${\bf x} = {\bf x}^0 = \left(q, q, \kappa q\right)$.

2) If $n = 3$, then $\lambda_1 = \lambda_2 = -q^{-1} < 0$. Therefore ${\bf x}^0$ is a stable (attracting) node on every $\Sigma$ with the stable manifold $W^s = \Sigma$ admitting the tangent space spanned by the eigenvectors $(-1, 1, 0)$ and $(-1, 0, 1)$ corresponding to the eigenvalue $\lambda_1$ of multiplicity $2$.

In addition, $I_1, I_2$ and $I_3$ are invariant for $n = 3$ by Lemma 3. Moreover, $I_1, I_2$ and $I_3$ are also subsets of the stable manifold $\Sigma$ and hence each of them is stable.

3) For $n\ge 4$, we have $\lambda_1\lambda_2 < 0$. Therefore, ${\bf x}^0$ is a saddle on $\Sigma$ with one dimensional stable and unstable manifolds (separatrices) both contained in $\Sigma$. The tangent space $E^s$ of the stable separatrix is spanned by the eigenvector $\big(1, 1, -4(n-2)^2(n-1)^{-1}\big)$ corresponding to $\lambda_2 = -q^{-1} < 0$. The eigenvector $(-1, 1, 0)$ corresponds to $\lambda_1 > 0$.

In addition, $I_3$ maintains the property of invariance for all $n\ge 4$ by Lemma 3 and in fact it is the stable manifold of the saddle ${\bf x}^0$. Indeed, since $\dot{x}_1(\tau)\equiv 1$, $\dot{x}_2(\tau)\equiv 1$, $\dot{x}_3(\tau) = (4-2n)c\tau^{3-2n}$ we have

for $\tau = q_0$ corresponding to the saddle point ${\bf x}^0 = (q_0, q_0, \kappa q_0)\in I_3\subset \Sigma$. Therefore, the vector $\big(1, 1, -4(n-2)^2(n-1)^{-1}\big)$ is tangent to $I_3$ at ${\bf x}^0$. On the other hand, this vector spans the stable eigenspace $E^s$ of ${\bf x}^0$ as shown above. Therefore, the stable manifold (separatrix) $W^s$ of ${\bf x}^0$ coincides with $I_3$ for every $n\ge 4$.

4) Finally, $\lambda_3 = 0$ is responsible for the center (slow) manifold with the tangent space spanned by the eigenvector $(1, 1, \kappa)$ for all $n\ge 3$. □

Remark 2. Actually, we are interested in the special case $c = 1$ which corresponds to the metrics (1.3) of unit volume and, therefore, has a purely geometric meaning. Studies of (2.3) on $\Sigma$ with an arbitrary $c > 0$ can easily be reduced to the "geometric" case, $c = 1$, by the change of variables $x_i = X_i \sqrt[3]{c}$ and $t = \tau \sqrt[3]{c}$ in (2.3) based on homogeneity of the functions $f_i$ and autonomy of the system (2.3) (see also ^[17]). Thus, in the sequel we assume that $c = 1$ without loss of generality.

2.4.   Structural properties of the set $\Sigma \cap\mathscr{R}_{+}$

We need the sets $\gamma_1 = \Sigma \cap \widetilde{\Gamma}_1$, $\gamma_2 = \Sigma \cap \widetilde{\Gamma}_2$ (the curves in red and teal respectively in Figure 1) and $\pi_1 = \Sigma \cap \Pi_1$, $\pi_2 = \Sigma \cap \Pi_2$ (the curves in blue color there).

Lemma 5. For all $n\ge 3$, the set $\Sigma\cap \mathscr{R}_{+}$ is bounded by the smooth and connected curves $\pi_1, \pi_2, \gamma_1$ and $\gamma_2$ such that:

1) $\pi_1 \cap \pi_2 = \pi_1 \cap \gamma_1 = \pi_2 \cap \gamma_2 = \emptyset$ and the components in the pairs $(\pi_1, \pi_2)$, $(\pi_1, \gamma_1)$ and $(\pi_2, \gamma_2)$ approach each other arbitrarily closely at infinity (which we denote by $\pi_1 \rightarrow \pi_2$, $\pi_1 \rightarrow \gamma_1$ and $\pi_2 \rightarrow \gamma_2$);

2) The curves $\gamma_1$ and $\gamma_2$ have a single common point $P_{12}\big(\overline{p}, \overline{p}, (2n-4)\overline{p}\big)$, where $\overline{p} = \left(\dfrac{c}{2n-4}\right)^{\tfrac{1}{2n-3}}$.

Proof. Intersections and long time behaviors of the curves $\pi_i, \gamma_i$, $i = 1, 2$. Since $\gamma_1\cap \gamma_2 = (\widetilde{\Gamma}_1\cap \widetilde{\Gamma}_2)\cap \Sigma$ and $\widetilde{\Gamma}_1\cap \widetilde{\Gamma}_2 = L$ by Lemma 1, the set $\gamma_1\cap \gamma_2$ consists of the single point $P_{12}$, where $\overline{p}$ is defined as the unique root of the equation $p^{n-2}p^{n-2}(2n-4)p = c$, for a given $n\ge 3$.

By analogy, $\pi_1 \cap \pi_2 = \pi_1 \cap \gamma_1 = \pi_2 \cap \gamma_2 = \emptyset$ easily follows from $\Pi_1 \cap \Pi_2 = \Pi_1 \cap \widetilde{\Gamma}_1 = \Pi_2 \cap \widetilde{\Gamma}_2 = \emptyset$, also known from Lemma 1, for all $n\ge 3$. To prove that $\pi_1$ and $\gamma_1$ approximate each other at infinity we use the approach developed in ^[12]: we will show $\pi_1\rightarrow I_2$ and $\gamma_1\rightarrow I_2$ instead of directly showing that $\pi_1 \rightarrow \gamma_1$ (note that it is quite difficult to derive it directly from the corresponding systems which define $\pi_1$ and $\gamma_1$). According to Lemma 3, we know that $I_1, I_2, I_3\subset \mathscr{R}_{+}$ for sufficiently large $\tau$ (more precisely, for all $\tau > \max\{\tau_1, \tau_2\}$).

Moreover, Lemma 3 also implies the following limits for every fixed $n\ge 3$ (see formulas (2.6)–(2.8)):

Therefore, $\pi_1 \rightarrow \gamma_1$ at infinity as close as we want. Then, $\pi_2 \rightarrow \gamma_2$ is clear from symmetry. Finally, $\Pi_1\big|_{I_3} = \Pi_2\big|_{I_3} = \tau^{-2(n-2)}\rightarrow 0$ as $\tau\to +\infty$, implying that $\pi_1 \rightarrow \pi_2$.

Smoothness and connectedness of the curves $\pi_i, \gamma_i$, $i = 1, 2$. Substituting $x_2 = tx_1$, where $t > 0$, into ${\bf r}_3 = 0$, we easily get that $(t-1)^2x_1^2-x_3^2 = 0$. Taking into account the condition $x_1^{n-2}x_2^{n-2}x_3 = c$ (the equation of the surface $\Sigma$), we obtain the parametric equations

for $\pi_1$ and $\pi_2$ with $t\in (0, +\infty)\setminus \{1\}$ so that the interval $(0, 1)$ corresponds to $\pi_1$ and $(1, +\infty)$ corresponds to $\pi_2$. By analogy, putting $x_2 = tx_3$ in ${\bf r}_1 = 0$, the equations

can be obtained for the curve $\Sigma\cap \Gamma_1$ with $\Psi(t) = \sqrt{t^2-2(n-2)t+1}$ and $l$ is given in (2.5). The representation $x_1 = \psi_2(t)$, $x_2 = \psi_1(t)$, $x_3 = \psi_3(t)$ is clear for the curve $\Sigma\cap \Gamma_2$ due to symmetry, $t\in (0, l)$. It is clear that $\gamma_1\subset \Sigma\cap \Gamma_1$ and $\gamma_2 \subset \Sigma\cap \Gamma_2$, and hence $\gamma_1$ and $\gamma_2$ can be parameterized by the same functions $\psi_1, \psi_2, \psi_3$, but with $t\in \Delta$, where $\Delta$ is some interval such that $\Delta\subset (0, l)$. To clarify $\Delta$, recall Lemma 1, according to which the curves $\Sigma\cap \Gamma_1$ and $\Sigma\cap \Gamma_2$ leave tails (extra pieces) not related to the set $\Sigma \cap \mathscr{R}_{+}$ after their intersection in the case $n\ge 4$, and coincide if $n = 3$. So to eliminate those extra pieces, take into account $x_1 = x_2$ at the unique point $P_{12}\in \gamma_1 \cap \gamma_2$, which implies the equation $\Psi(t) = t$ with the unique root $\widetilde{t} = (2n-4)^{-1}$. Thus, in what follows we assume that $\Delta = \big(0, \widetilde{t}\, \big]$.

Now it is easy to see that the functions $\phi_1, \phi_2$, and $\phi_3$ are differentiable on the sets $(0, 1)$ and $(1, +\infty)$. Therefore, the curves $\pi_1$ and $\pi_2$ are smooth. The curve $\pi_1$ must be connected, as it is the image of the connected set $(0, 1)$ under the continuous function $t \mapsto \big(\phi_1(t), \phi_2(t), \phi_3(t)\big)$, $t\in (0, 1)$. So is $\pi_2$, being the continuous image of the connected set $(1, +\infty)$. Smoothness and connectedness of $\gamma_1$ and $\gamma_2$ follow analogously from the differentiability of the functions $t \mapsto \big(\psi_1(t), \psi_2(t), \psi_3(t)\big)$ and $t \mapsto \big(\psi_2(t), \psi_1(t), \psi_3(t)\big)$, where $t\in \Delta$. □

Proof of Theorem 4. Step 1. Any trajectory of (2.3) originating in $\mathscr{R}_{+}$ remains there forever. Indeed, by Lemma 2, the vector field ${\bf V}$ associated with the system (2.3) is directed into $\mathscr{R}_{+}$ on every point of $\partial(\mathscr{R}_{+})$, for all $n\ge 3$. Therefore, no trajectory of (2.3) can leave $\mathscr{R}_{+}$: if ${\bf x}(0)\in \mathscr{R}_{+}$ then ${\bf x}(t)\in \mathscr{R}_{+}$ for all $t > 0$. According to the definition of the set $\mathscr{R}_{+}$, this means that (1.1) preserves the positivity of the Ricci curvature of metrics (1.3) on $\operatorname{SO}(n)/\operatorname{SO}(n-2)$: metrics with positive Ricci curvature can be evolved only into metrics with positive Ricci curvature.

Step 2. Any trajectory originating in the exterior of $\mathscr{R}_{+}$ enters $\mathscr{R}_{+}$ in finite time. To prove this assertion we use Lemma 4. It suffices to study (2.3) on $\Sigma \cap \mathscr{R}_{+}$, where $\Sigma$ are the invariant surfaces of (2.3) responsible for dominant motions of its trajectories corresponding to nonzero eigenvalues $\lambda_1$ and $\lambda_2$, according to Lemmas 3 and 4. Any movement caused by $\lambda_3 = 0$, for all $n\ge 3$, can be neglected because it can only occur within the domain $\mathscr{R}_{+}$ itself, along the eigenvector $(1, 1, \kappa)$ (as slow transitions between different invariant surfaces):

By Lemma 4, for every $n\ge 3$, there exists $q = q_0$ such that ${\bf x}^0\in \Sigma\cap \mathscr{R}_{+}$. Therefore, every trajectory of (2.3) originated in $\Sigma\cap \operatorname{ext}\mathscr{R}_{+}$ must intersect the boundary $\Sigma\cap \partial(\mathscr{R}_{+})$ in finite time and enter the set $\Sigma\cap \mathscr{R}_{+}$ governed by the stable manifold $W^s = \Sigma$ (and by the stable submanifolds $I_1, I_2$, and $I_3$ as well) of the unique stable node ${\bf x}^0$ in the case $n = 3$, and by the separatrices $W^s = I_3\subset \Sigma$ and $W^u\subset \Sigma$ of ${\bf x}^0$ being the unique saddle for every $n\ge 4$ (see Figure 2). The domain $\Sigma \cap \mathscr{R}_{+}$ is able to receive all trajectories due to the fact that it is unbounded, with the boundary $\Sigma \cap \partial(\mathscr{R}_{+})$ consisting of the curves $\pi_1, \pi_2, \gamma_1$, and $\gamma_2$ in $\mathbb{R}^3$ such that $\pi_1\cap \pi_2 = \pi_1\cap \gamma_1 = \pi_2\cap \gamma_2 = \emptyset$, according to Lemma 5. This is also quite consistent with the facts established above. Indeed, if $\Sigma \cap \partial(\mathscr{R}_{+})$ was a bounded set, then in order to satisfy Lemma 2, the dynamics of the system (2.3) would be different from those described in Lemma 4 contradicting it. Another significant circumstance that causes all trajectories to enter $\mathscr{R}_{+}$ and remain there forever for all $n\ge 4$ is that the separatrices $W^s$ and $W^u$ are subsets of $\mathscr{R}_{+}$ at infinity since none of them can reach (intersect or touch) the boundary $\Sigma\cap \partial(\mathscr{R}_{+})$, according to Lemma 2.

Thus, we proved that on the Stiefel manifold $\operatorname{SO}(n)/\operatorname{SO}(n-2)$, $n\ge 3$, the normalized Ricci flow (1.1) evolves all metrics (1.3) (of mixed or positive Ricci curvature) into metrics with positive Ricci curvature. Theorem 4 is proved. □

3.   Planar illustrations

Clearly, the volume function $\operatorname{Vol} = x_1^{n-2}x_2^{n-2}x_3$ is a first integral of (2.3). Based on this fact, the system (2.3) could equivalently be reduced to the following planar system:

where $\widetilde{f}_i(x_1, x_2): = f_i\left(x_1, x_2, \varphi(x_1, x_2)\right)$ and $\varphi(x_1, x_2): = (x_1x_2)^{2-n}$. The corresponding singular point of (3.1) is $\widetilde{{\bf x}}^0 = (q_0, q_0)$, where $q_0 = \sqrt[2n-3]{(n-1)(2n-4)^{-1}}$.

On the other hand, it is well known that $\chi(\lambda, {\bf x}^0) = \lambda^3-\rho\lambda^2+\delta\lambda$, where $\rho = \operatorname{trace}\widetilde{{\bf J}}(\widetilde{{\bf x}}^0)$ and $\delta = \operatorname{det}\widetilde{{\bf J}}(\widetilde{{\bf x}}^0)$ are the trace and the determinant, respectively, of the Jacobian matrix $\widetilde{{\bf J}}(\widetilde{{\bf x}})$ of the system (3.1), evaluated at its singular point $\widetilde{{\bf x}} = \widetilde{{\bf x}}^0$. Consequently,

According to the theory of planar dynamical systems, $\widetilde{{\bf x}}^0$ is a hyperbolic stable node of (3.1) for $n = 3$ since $\delta > 0$, $\rho < 0$, and $\sigma = 0$. Clearly, $\widetilde{{\bf x}}^0$ is a hyperbolic saddle of (3.1) for all $n\ge 4$ since $\delta < 0$. These results are illustrated in Figure 3.

Using the idea in ^[9], the dynamics of (2.3) can be illustrated on the plane $x_1+x_2+x_3 = 1$ (denoted by $\Pi_0$) preserving the dihedral symmetry of the problem. In Figure 4, the relevant results are depicted, where the edges of the big triangle correspond to $x_i = 0$ on $\Pi_0$; the curves $\Pi_0\cap \widetilde{\Gamma}_1$, $\Pi_0\cap \widetilde{\Gamma}_2$ and $\Pi_0\cap \Pi_1$, $\Pi_0 \cap \Pi_2$ are depicted in red, teal, and blue, respectively; the yellow circle (or ellipse) represents the curve defined by the systems of the equations $x_0^2-(2n-4)(x_1+x_2)x_0+(x_1-x_2)^2 = 0$ and $x_1+x_2+x_3 = 1$, where the first one is equivalent to $S_{{\bf g}} = 0$. These illustrations confirm the well known general fact that the normalized Ricci flow preserves the positivity of the scalar curvature of invariant metrics on every compact homogeneous space (see ^[19,20]).

4.   Conclusions

The geometric problem considered in the article can be of some interest from the point of view of the dynamical systems. It is not easy to find the smallest invariant set (or minimal invariant sets) of the flow. In Theorem 4 we proved that, for every invariant set $\Sigma$ of the system (2.3) defined as $x_1^{n-2}x_2^{n-2}x_3 = c$, there exists its invariant subset $\Sigma\cap \mathscr{R}_{+}$. Similar results were obtained in ^[9,12,17] for (1.1) reduced to a dynamical system on generalized Wallach spaces; see for instance the case that $a\in (1/6, 1/4)\cup (1/4, 1/2)$ in [9, Theorem 3], the case that $a = 1/6$ in [9, Theorem 4] and the case that $a\in [1/4, 1/2)$ in [17, Theorem 2] (see cases 4 and 5), which concern invariance of certain sets related to generalized Wallach spaces with $a_1 = a_2 = a_3 = a$. The case that $a_1+ a_2+ a_3 > 1/2$ was studied in [12, Theorem 6], where some additional conditions were found on the parameters $a_1, a_2, a_3$, which provide the invariance of a set analogous to $\mathscr{R}_{+}$.

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