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Some classes of almost Hermitian structures that can be realized on N. A. Daurtseva Novosibirsk State University, Novosibirsk, Russia
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     § 1. IntroductionIt is well known that there exists an almost complex structure on a 6-sphere, which is induced by the embedding of the sphere in $\mathbb{R}^7$ and multiplication in the algebra of octonions. It is often called the Cayley structure; it is invariant under the action of the group $G_2$ on $S^6$ and, in combination with a metric of constant curvature (the round metric on the sphere) defines a nearly Kähler structure on $S^6$. Moreover, for each orthogonal transformation in $O(7)$ there exists an embedding of $G_2$ in $O(7)$ which induces a Cayley structure of its own. The set of Cayley structures inducing the same orientation on $S^6$ forms the seven-dimensional space $\mathbb{R}P^7$ (see [1]). All these structures are isometric and define nearly Kähler structures on the round sphere $S^6$ (see [2]). As concerns other almost Hermitian structures, it is known that they are not Hermitian (see [3]), that is, almost complex structures compatible with the round metric are never integrable. Moreover, even structures such that the associated metric is ‘sufficiently close’ to the round one cannot be Hermitian (see [4]). The most recent result on the properties of almost Hermitian structures on $S^6$ is due to Foscolo and Haskins [5] (2017). It claims that there exists an inhomogeneous nearly Kähler structure on $S^6$ that is distinct from the Cayley structure. It is constructed for the set of almost Hermitian structures which are invariant under the action of cohomogeneity one of the group $\mathrm{SU}(2)\times \mathrm{SU}(2)$ on the 6-sphere. A natural question here is as follows: which classes of almost Hermitian structures can be realized on the round sphere if one limits oneself to structures of cohomogeneity one? The answer can be useful, in particular, for understanding the geometry of the nearly Kähler structure whose existence was established in [5]. In this paper we investigate almost Hermitian structures of cohomogeneity one on $S^6$. We prove the existence of a whole family of semi-Kähler structures on the round sphere. All of them are obtained by a deformation of the Cayley structure. In addition, we show that some of these structures can be deformed to quasi-Kähler structures with the same 2-form, but with metric distinct from a constant curvature metric. We prove that all structures of cohomogeneity one in the class $\mathcal{W}_1\oplus\mathcal{W}_3\oplus\mathcal{W}_4$ on the round sphere reduce to the Cayley structure, while structures in the class $\mathcal{W}_2\oplus\mathcal{W}_4$ are impossible on it. § 2. Classes of almost Hermitian manifolds2.1. The Gray-Hervella classificationLet $(M,g_0,J)$ be an almost Hermitian manifold; the Riemannian metric $g_0$ and the almost complex structure $J$ define uniquely the antisymmetric 2-form $\omega_J(\cdot,\cdot)=g_0(J\cdot,\cdot)$. Consider the 3-form $\nabla\omega_J$, where $\nabla$ is the Levi-Civita connection. Then
$$
\begin{equation*}
\nabla\omega_J(X,Y,Z)=-\nabla\omega_J(X,Z,Y)=-\nabla\omega_J(X,JY,JZ).
\end{equation*}
\notag
$$
Let $W$ be the space of tensors $\alpha$ on the vector space $\mathbb{R}^{2n}$ with the same symmetries as $\nabla\omega_J$. The Gray-Hervella classification [6] of almost Hermitian manifolds is based on the decomposition of $W$ into a direct sum of four $U(n)$-irreducible components: $W= W_1\oplus W_2\oplus W_3\oplus W_4$. Using this decomposition we can construct 16 $U(n)$-invariant subspaces: $\varnothing$, $W_i$, $W_i\oplus W_j$, $W_i\oplus W_j\oplus W_k$, $W$. Each subspace satisfies certain restrictions on the form of $\alpha$. We say that an almost Hermitian manifold is in the class $\mathcal{W}_i\oplus\mathcal{W}_j$ if $(\nabla\omega_J)_x\in W_i\oplus W_j$ for all $x\in M$. In this way we can distinguish 16 classes of almost Hermitian manifolds. Here are some of them:
2.2. Equivalent definitionsAn almost complex structure on $M$ induces a decomposition
$$
\begin{equation*}
\Lambda^1_{\mathbb C}(M)=\Lambda^{1,0}(M)\oplus\Lambda^{0,1}(M)
\end{equation*}
\notag
$$
of the space of complex 1-forms on $M$, where $\Lambda^{1,0}$ consists of the 1-forms $\alpha$ such that $\alpha(J\cdot\,)=i\alpha(\,\cdot\,)$. In a similar way, for $m\geqslant 1$
$$
\begin{equation*}
\Lambda^m_{\mathbb{C}}(M)=\bigoplus_{p+q=m}\Lambda^{p,q}(M),\quad\text{where} \quad \Lambda^{p,q}=\wedge^p(\Lambda^{1,0})\otimes\wedge^q(\Lambda^{0,1}).
\end{equation*}
\notag
$$
The exterior derivative splits into a sum: $d=d^{2,-1}+d^{1,0}+d^{0,1}+d^{-1,2}$. By Leibniz’s rule its component $d^{-1,2}\colon \Lambda^{1,0}\to\Lambda^{0,2}$ is linear with respect to smooth functions $f$ on $M$: $d^{-1,2}f\alpha=fd^{-1,2}\alpha$; it is precisely the Nijenhuis tensor. In the six-dimensional case we can conveniently describe this tensor as follows. Let $\alpha^1, \alpha^2$ and $ \alpha^3$ be linearly independent 1-forms in $\Lambda^{1,0}$ defined on an open set $U\subset M$. Then
$$
\begin{equation*}
d^{-1,2}\colon \begin{pmatrix} \alpha^1\\ \alpha^2\\ \alpha^3 \end{pmatrix}=N(\alpha) \begin{pmatrix} \overline{\alpha^2\wedge\alpha^3} \\ \overline{\alpha^3\wedge\alpha^1} \\ \overline{\alpha^1\wedge\alpha^2} \end{pmatrix},
\end{equation*}
\notag
$$
where $N(\alpha)\colon U\to M(\mathbb{C},3,3)$. Using our notation we can give equivalent definitions for some classes of almost Hermitian manifolds. For instance, $(M,g_0,J)$ is nearly Kähler if and only if $d\omega$ has type $(3,0)+(0,3)$ and the Nijenhuis tensor is totally antisymmetric. The fact that $(M,g_0,J)$ is quasi-Kähler is equivalent to $d\omega$ having type $(3,0)+(0,3)$. In the case of a 6-sphere embedded in $\mathbb{R}^7$ in the standard way, it is natural to consider the metric $g_0$ induced on it. It has a constant curvature, and we call it round in what follows; we also call the manifold $(S^6,g_0)$ the round sphere. Also, regarding elements of $\mathbb{R}^7=\operatorname{Im}\mathbb{O}$ as imaginary octonions, we can define on $S^6$ the almost complex structure $J^C_x(y)=y\cdot x$, where $x\in S^6$, $y\in T_xS^6$, and $\cdot$ denotes multiplication in the Cayley algebra. It is known that $(S^6,g_0, J^C)$ is strictly nearly Kähler. As regards the other almost complex structures preserving the round metric, it is known (see [3]) that they cannot be integrable, that is, $(S^6,g_0,J)\notin\mathcal{W}_3\oplus\mathcal{W}_4$. The following question arises: which classes of almost Kähler structures can be realized on $(S^6, g_0)$? § 3. Manifolds of cohomogeneity one3.1. The main definitionsLet $G$ denote a compact connected Lie group acting on a connected closed manifold $M$. We call the action of the compact Lie group $G$ on $M$ an action of cohomogeneity one if the orbit space $M/G$ is one-dimensional. For a compact manifold $M$ with finite fundamental group the orbit space $M/G$ is a one-dimensional Hausdorff topological space homeomorhic to a line segment (see [7] and [8]). This is a so-called interval cohomogeneity one manifold one. Let $(M,g)$ be a Riemannian interval cohomogeneity one manifold. Then $M/G=[a_1,a_2]$; also let $\pi\colon M\to [a_1,a_2]$ be the natural projection of $M$ onto the orbit space of the action of $G$. Let $\gamma\colon [a_1,a_2]\to M$ be a normal geodesic, that is, a geodesic intersecting orbits orthogonally. Different orbits correspond to interior points and endpoints of $[a_1,a_2]$, namely, orbits corresponding to interior points $t\in(a_1,a_2)$ are diffeomorphic to the homogeneous space $G/K$, where $K$ is the stabilizer $G_{\gamma(t)}$ of the interior point $\gamma(t)$ of the geodesic. We call them principal orbits. and we call $K$ the principal isotropy subgroup. The orbits $M_1=\pi^{-1}(a_1)$ and $M_2=\pi^{-1}(a_2)$ corresponding to endpoints are diffeomorphic to the homogeneous spaces $G/H_1$ and $G/H_2$, where the $H_i=G_{\gamma(a_i)}$ are singular isotropy subgroups containing $K$. The orbits $M_1$ and $M_2$ have a lower dimension than the principal orbit $G/K$; they are said to be singular. Thus, we have $M=M_1\cup M_{\mathrm{reg}}\cup M_2$, where $M_{\mathrm{reg}}$ is the principal set corresponding to the interior of the interval $[a_1,a_2]$ and diffeomorphic to $(a_1,a_2)\times G/K$ (see [9]). Let $D_i$ is a disc of radius 1 orthogonal to $M_i$ at the point $\gamma(a_i)$. The group $H_i$ acts on $D_i$, and this action is transitive on the boundary $\partial D_i=S^{l_i}=H_i\cdot \gamma(t_0)=H_i/K$, where $t_0$ is an interior point of the interval. The slice theorem claims that tubular neighbourhoods of singular orbits have the form (see [8])
$$
\begin{equation*}
\pi^{-1}[a_1,t_0]\approx G\times_{H_1}D_1\quad\text{and} \quad \pi^{-1}[t_0,a_2]\approx G\times_{H_2}D_2.
\end{equation*}
\notag
$$
Thus, $M$ decomposes into a union of two disc bundles which are glued along their common boundary $\pi^{-1}(t_0)\approx G/K$:
$$
\begin{equation*}
M\approx G\times_{H_1}D_1\bigcup_{G/K}G\times_{H_2}D_2\quad\text{and} \quad S^{l_i}=\partial D_i\approx H_i/K.
\end{equation*}
\notag
$$
This enables one to describe $M$ in terms of the groups $K\subset H_1, H_2\subset G$. 3.2. Nearly Kähler manifolds of cohomogeneity oneIn [10] and [11] the authors classified all six-dimensional compact manifolds of cohomogeneity one on which a strictly nearly Kähler structure can be introduced. Subsequently, this list was corrected in [5] (Table 1). Table 1.A classification of the group diagrams of nearly Kähler manifolds. Here $\Delta T=\{(t,t)\colon t\in T\}\subset T\times T$ for the subgroup $T\subset G$
The last line in Table 1 is the sine-cone, which carries a unique singular nearly Kähler metric; it is not interesting from the standpoint of constructing a complete nearly Kähler structure on $S^6$. To comprehend the geometry of the manifolds in the first four lines of Table 1, consider the conifold $\mathcal{C}$ in $\mathbb{C}^4$ defined by the equation $w_1^2+w_2^2+w_3^2+w_4^2=0$. We find its base $\mathcal{N}$ by taking its intersection with a sphere centred at the singular point:
$$
\begin{equation*}
\begin{cases} w_1^2+w_2^2+w_3^2+w_4^2=0, \\ |w_1|^2+|w_2|^2+|w_3|^2+|w_4|^2=r^2. \end{cases}
\end{equation*}
\notag
$$
After going over to the real and imaginary parts,
$$
\begin{equation*}
x=(x_1, x_2, x_3, x_4),\quad y=(y_1, y_2, y_3, y_4),\qquad x_i=\operatorname{Re}(w_i),\quad y_i=\operatorname{Im}(w_i);
\end{equation*}
\notag
$$
we can define $\mathcal{N}$ by the equations
$$
\begin{equation*}
\begin{cases} (x,x)=\dfrac12r^2, \\ (y,y)=\dfrac12r^2, \\ (x,y)=0. \end{cases}
\end{equation*}
\notag
$$
The first equation defines $S^3$, while the second and third define an $S^2$-bundle over $S^3$ (see [12]). Since all such fibrations over $S^3$ are trivial, we have topologically ${\mathcal{N}=S^2\times S^3}$, and $\mathcal{C}$ is a cone over $S^2\times S^3$. There are two scenarios of smoothing the singular point (Figure 1). The first is usually called a deformation (see [12]) or a smoothing (see [5]); its idea is in replacing the singular point by $S^3$. In taking it we go over to the condition $w_1^2+w_2^2+w_3^2+w_4^2=\epsilon^2$, where $\epsilon\neq 0$ is a constant. The new condition describes a smoothed conifold diffeomorphic to $T^*S^3$. The second way is a small resolution, when we replace the singular point by $S^2$. Let $X=w_1+iw_2$, $Y=w_1-iw_2$, $U=-w_3+iw_4$ and $V=w_3+iw_4$; then $\mathcal{C}$ is defined by $XY-UV=0$. We replace this equation in $\mathbb{C}^4$ by the equation
$$
\begin{equation*}
\begin{pmatrix} X & U\\ V & Y \end{pmatrix} \begin{pmatrix} \lambda_1\\ \lambda_2 \end{pmatrix}=0 \quad\text{in } \mathbb{C}^4\times\mathbb{C}P^1.
\end{equation*}
\notag
$$
As a result, we obtain a small resolution of the conifold, which is the total space of the vector bundle $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ over $S^2$ . Gluing two conifolds smoothed in the first or second way along $\mathcal{N}$ we can obtain any manifold in the first four lines of Table 1. Foscolo and Haskins [5] (2017) examined the possibility for gluing smoothly along $\mathcal{N}$ two nearly Kähler deformations of Calabi-Yau structures on neighbourhoods of singular orbits of manifolds of cohomogeneity one in the first four lines of Table 1. In this way they could establish the existence of nonhomogeneous nearly Kähler structures on $S^6$ and $S^3\times S^3$. 3.3. The action of $\mathrm{SU}(2)\times \mathrm{SU}(2)$ on $S^6$We define an action of the group $G=\mathrm{SU}(2)\times \mathrm{SU}(2)$ on the round sphere $S^6$ as follows:
$$
\begin{equation*}
(g_1,g_2)\in G\colon (p,q)\in S^6\subset \operatorname{Im}(\mathbb{H})\times\mathbb{H}\to (g_1pg_1^{-1},g_2qg_1^{-1})\in S^6.
\end{equation*}
\notag
$$
Let $\gamma\colon [0,{\pi}/{2}]\to S^6$, $\gamma(\varphi)=(i\sin\varphi,\cos\varphi)$, be a normal geodesic on the round sphere. The singular isotropy subgroup $H_1$ of the point $\gamma(0)=(0,1)$ is equal to $\Delta \mathrm{SU}(2)$ and the singular orbit is $M_1=G/H_1=\{0\}\times S^3$. The singular isotropy subgroup $H_2$ of $\gamma({\pi}/{2})=(i,0)$ is equal to $U(1)\times \mathrm{SU}(2)$ and $M_2=G/H_2=S^2\times\{0\}$. The principal isotropy subgroup $K$ of a point $\gamma(\varphi)=(i\sin\varphi,\cos\varphi)$, $\varphi\in(0,{\pi}/{2})$, is equal to $\Delta U(1)$, principal orbits are diffeomorphic to $ G/K=\mathcal{N}=S^2\times S^3$, and the principal part of $S^6_{\mathrm{reg}}$ is $(0,{\pi}/{2})\times\mathcal{N}$. The round metric and the Cayley structure are invariant under this action. § 4. Almost Hermitian structures of cohomogeneity one on the round sphere4.1. The 2-form $\omega$Fix the standard basis
$$
\begin{equation*}
H=\frac12\begin{pmatrix} i & 0\\ 0 & -i \end{pmatrix}, \qquad E=\frac{1}{2\sqrt{2}}\begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}, \qquad V=\frac{1}{2\sqrt{2}}\begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix}
\end{equation*}
\notag
$$
of the Lie algebra $\mathfrak{s}\mathfrak{u}(2)$. In this basis the Lie brackets are defined by the formulae The vectors
$$
\begin{equation*}
\begin{gathered} \, U_+=(H,H),\qquad U_-=(H,-H),\qquad E_1=(E,0), \qquad V_1=(V,0), \\ E_2=(0,E)\quad\text{and}\quad V_2=(0,V) \end{gathered}
\end{equation*}
\notag
$$
form a basis of the Lie algebra $\mathfrak{s}\mathfrak{u}(2)\oplus\mathfrak{s}\mathfrak{u}(2)$. The vector $U_+$ is a basis of the Lie algebra $\Delta\mathfrak{u}(1)$, and $U_-$, $E_1$, $V_1$, $E_2$ and $V_2$ define invariant vector fields tangent to the principal orbit $S^2\times S^3=\mathrm{SU}(2)\times \mathrm{SU}(2)/\Delta U(1)$. By [5] each invariant 2-form $\omega$ on $S^6_{\mathrm{reg}}$ has the form
$$
\begin{equation*}
\omega=\lambda(\varphi)u_-\wedge d\varphi+u_0(\varphi)\omega_0+u_1(\varphi)\omega_1+u_2(\varphi)\omega_2+u_3(\varphi)\omega_3,
\end{equation*}
\notag
$$
where $u_-$, $e_1$, $v_1$, $e_2$, $v_2$ is the corresponding cobasis,
$$
\begin{equation*}
\begin{gathered} \, \omega_1=\frac{1}{12}(e_1\wedge v_1-e_2\wedge v_2), \qquad \omega_2=\frac{1}{12}(e_1\wedge v_2+e_2\wedge v_1), \\ \omega_3=\frac{1}{12}(e_1\wedge e_2+v_1\wedge v_2)\quad\text{and} \quad \omega_0=\frac{1}{12}(e_1\wedge v_1+e_2\wedge v_2). \end{gathered}
\end{equation*}
\notag
$$
Let $(g_0, J)$ be an arbitrary almost Hermitian structure of cohomogeneity one on the round sphere $S^6$. Lemma 1. The 2-form $\omega_J(\cdot,\cdot)=g_0(J\cdot,\cdot)$ corresponding to the almost Hermitian structure $(g_0, J)$ on $S^6_{\mathrm{reg}}$ is defined by the conditions Proof. At points in $\gamma$ the vector fields corresponding to $U_-$, $E_1$, $V_1$, $E_2$ and $V_2$ are defined by the formulae
$$
\begin{equation*}
\begin{gathered} \, U_-^*(\gamma(\varphi))=(0,-i\cos\varphi), \\ E_1^*(\gamma(\varphi))=\biggl(-\frac{\sin\varphi}{\sqrt{2}}k,-\frac{\cos\varphi}{2\sqrt{2}}j\biggr), \qquad V_1^*(\gamma(\varphi))=\biggl(\frac{\sin\varphi}{\sqrt{2}}j,-\frac{\cos\varphi}{2\sqrt{2}}k\biggr), \\ E_2^*(\gamma(\varphi))=\biggl(0,\frac{\cos\varphi}{2\sqrt{2}}j\biggr)\quad\text{and} \quad V_2^*(\gamma(\varphi))=\biggl(0,\frac{\cos\varphi}{2\sqrt{2}}k\biggr). \end{gathered}
\end{equation*}
\notag
$$
A direct application of the condition of compatibility of $g_0$ and $J$ at interior points of $\gamma$ completes the proof of the lemma. It order that the 2-form $\omega_J$ on the regular part of the sphere define a 2-form of cohomogeneity one on the whole sphere, certain conditions for the smooth extension of this form to the singular orbits must be fulfilled. In Lemmas 4.1 and 4.2 in [5] the authors indicated conditions for a smooth extension of a 2-form $\omega$ to the singular orbits. After correcting slightly the results in [5] for the case under consideration we obtain the following result. Lemma 2. An almost Hermitian structure $(g_0, J)$ of cohomogeneity one on $S_{\mathrm{reg}}^6$ extends smoothly to singular orbits if and only if 4.2. Structures in the class $\mathcal{W}_1\oplus\mathcal{W}_2\oplus\mathcal{W}_3$Lemma 3. If an almost Hermitian manifold $(S^6,g_0,J)$ of cohomogeneity one belongs to the class $\mathcal{W}_1\oplus\mathcal{W}_2\oplus\mathcal{W}_3$, then Proof. The class of almost Hermitian manifolds $\mathcal{W}_1\oplus\mathcal{W}_2\oplus\mathcal{W}_3$ is defined by the condition $\delta\omega_J=0$.
For differential forms $u^-d\varphi e_2v_2$ and $ u^-d\varphi e_1v_1$ we have
$$
\begin{equation*}
d(u^-d\varphi e_2v_2)=-d(u^-d\varphi e_1v_1)=\frac14\, d\varphi e_1e_2v_1v_2.
\end{equation*}
\notag
$$
The exterior differentials of the other 4-forms occurring in $*\omega_J$ are zero. Taking the assumptions of Lemma 1 into account we obtain
$$
\begin{equation*}
\delta\omega_J=2\frac{\pm(\cos^2\varphi\sin\varphi-\sin^3\varphi)-(1/3) u_1}{\sin^2\varphi}u^- \quad\text{for } \lambda=\pm\cos\varphi.
\end{equation*}
\notag
$$
The proof is complete. Taking into account the conditions from Lemma 1, for $u_3=0$ we obtain two structures:
$$
\begin{equation*}
\begin{gathered} \, \lambda=\cos\varphi, \qquad u_0=\frac32\sin\varphi(5\cos^2\varphi-2), \\ u_1=3\sin\varphi(2\cos^2\varphi-1), \qquad u_2=-\frac92\sin\varphi\cos^2\varphi, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{gathered} \, \lambda=\cos\varphi, \qquad u_0=-\frac32\sin\varphi(4\sin^4\varphi-3\sin^2\varphi+1),\\ u_1=3\sin\varphi(2\cos^2\varphi-1), \qquad u_2=\frac32\sin\varphi(4\sin^2\varphi+1)\cos^2\varphi. \end{gathered}
\end{equation*}
\notag
$$
Both structures satisfy the assumptions of Lemma 2. The first solution produces precisely the Cayley structure (see [5]), while using the second we define a structure of cohomogeneity one in the class $\mathcal{W}_1\oplus\mathcal{W}_2\oplus\mathcal{W}_3$ on the round sphere, which is distinct from the Cayley structure. We denote this new structure by $J^{123}$. At points $x$ on the curve $\gamma(\varphi)=(i\sin\varphi,\cos\varphi)=i\sin\varphi+e\cos\varphi\in \operatorname{Im}\mathbb{O}$ it can be expressed as follows in terms of right multiplication in the Cayley algebra:
$$
\begin{equation*}
J^{123}_x(Y)= \begin{cases} Y\cdot x=J^C(Y)&\text{for}\ Y\in\langle U^*_-,\dot{\gamma}\rangle, \\ Y\cdot (\widetilde{x}x\widetilde{x})&\text{for}\ Y\in\langle E^*_1, E^*_2, V^*_1, V^*_2\rangle, \end{cases}
\end{equation*}
\notag
$$
where $\widetilde{x}=(i\sin\varphi,-\cos\varphi)=i\sin\varphi-e\cos\varphi\in \operatorname{Im}\mathbb{O}$. In the case when $ u_3\neq 0$ and $|u_3|\leqslant 6\cos^2\varphi\sin^2\varphi$, using Lemma 1 we obtain
$$
\begin{equation*}
u_0 =\frac32\sin\varphi(-2\cos^4\varphi+5\cos^2\varphi-2) \pm\frac12\sqrt{36\sin^2\varphi\cos^8\varphi-\frac{\cos^4\varphi}{\sin^2\varphi}u_3^2}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
u_2 =\frac32\sin\varphi\cos^2\varphi(1-2\cos^2\varphi) \mp\frac12(\sin^2\varphi+1)\sqrt{36\sin^2\varphi\cos^4\varphi-\frac{u_3^2}{\sin^2\varphi}}
\end{equation*}
\notag
$$
on $(0,{\pi}/{2})$. A direct calculation shows that if $u_3$ satisfies the assumptions of Lemma 2, then all other assumptions of Lemma 2 are also fulfilled for $\lambda=\cos\varphi$, $u_1=3\sin\varphi(2\cos^2\varphi-1)$ and the functions $u_0$ and $u_2$ as above. Theorem 1. There exist an infinite number of cohomogeneity one structures in the class $\mathcal{W}_1\oplus\mathcal{W}_2\oplus\mathcal{W}_3$ on $(S^6, g_0)$. In particular, there exists a continuous deformation of the Cayley structure $J_0=J^C$ to $J_1=J^{123}$. Proof. Let $u_3=6\sin \pi t\sin^2\varphi\cos^2\varphi$; then for each $t\in[0,1]$ the functions and together with $\lambda=\cos\varphi$, define a semi-Kähler structure of cohomogeneity one on $(S^6,g_0)$.
The proof is complete. Note that the necessary condition in Lemma 3 also corresponds to an even stronger condition but for almost Hermitian structures with metrics distinct from $g_0$. Lemma 4. If a 2-form $\omega_J$ satisfying the assumptions of Lemmas 1 and 2 corresponds to a quasi-Kähler structure, then Proof. In the six-dimensional case the class of quasi-Kähler manifolds $\mathcal{QK}=\mathcal{W}_1\oplus \mathcal{W}_2$ can be defined as consisting of the manifolds with $\mathrm{SU}(3)$-structure $(\omega,\Omega)$ such that $d\omega=3\operatorname{Re}\Omega$. Assuming that such an $\mathrm{SU}(3)$-structure exists, we can construct the corresponding almost complex structure using Hitchin’s construction [13]:
$$
\begin{equation*}
I_{\omega}=\frac{1}{\sqrt{-\frac16\operatorname{tr}(K^2)}}K, \qquad K(X)=A(i_X\, d\omega\wedge d\omega),
\end{equation*}
\notag
$$
where $\iota_{A(\psi)}\operatorname{Vol}=\psi$ for an arbitrary 5-form $\psi$. Detailed calculations for $K$ were presented in [14]. Such a structure exists if and only if the following conditions are satisfied:
$$
\begin{equation*}
\begin{cases} \operatorname{tr}(K^2)<0,\\ \omega(I_{\omega}\cdot\,, I_{\omega}\cdot\,)=\omega(\,\cdot\,{,}\,\cdot\,),\\ \omega(\,\cdot\,, I_{\omega}\cdot\,)>0. \end{cases}
\end{equation*}
\notag
$$
The consistency condition $\omega(I_{\omega}\cdot\,, I_{\omega}\cdot\,)=\omega(\,\cdot\,{,}\,\cdot\,)$ is equivalent to $(-u_0^2+u_1^2+u_2^2+ u_3^2)'=6\lambda u_1$. For $\omega=\omega_J$ considered in Lemmas 1 and 2 this condition yields $u_1=\pm 3\sin\varphi(2\cos^2\varphi-1)$ for $\lambda=\pm\cos\varphi$.
The proof is complete. For $\omega_J$ the condition $\operatorname{tr}(K^2)<0$ is equivalent to $(f')^2+324\cos^2\varphi\sin^4\varphi>\dot{u}_0^2$, where $f^2:=u_2^2+u_3^2$. Note again that in the above situation the almost complex structure $I_{\omega_J}$ is distinct from $J$ in most cases. In the case when $I_{\omega_J}=J$, we obtain a quasi-Kähler structure on the round sphere, which is only possible for the Cayley structure. In the other cases the pair $(\omega_J, I_{\omega_J})$, when it exists, corresponds to another metric, distinct from $g_0$. For example, the form $\omega_J$, defined by the functions
$$
\begin{equation*}
\begin{gathered} \, \lambda=\cos\varphi, \qquad u_0=-\frac32\sin\varphi(4\sin^4\varphi-3\sin^2\varphi+1), \\ u_1=3\sin\varphi(2\cos^2\varphi-1), \qquad u_2=\frac32\sin\varphi(4\sin^2\varphi+1)\cos^2\varphi, \qquad u_3=0, \end{gathered}
\end{equation*}
\notag
$$
satisfies all conditions listed in the proof of Lemma 4, and in combination with the corresponding almost complex structure $I_{\omega_J}$, defines a quasi-Kähler structure of cohomogeneity one on $S^6$, but $I_{\omega_J}\neq J^{123}$. Taking Lemmas 3 and 4 into account we can state the following result. Theorem 2. Let the 2-form $\omega_J$ satisfy the conditions in Lemmas 1 and 2. If a quasi-Kähler structure $(\omega_J, I_{\omega_J})$ of cohomogeneity one exists on $S^6$, then 4.3. Structures in the class $\mathcal{W}_1\oplus\mathcal{W}_3\oplus\mathcal{W}_4$Theorem 3. An almost Hermitian manifold $(S^6,g_0,J)$ of cohomogeneity one belongs to the class $\mathcal{W}_1\oplus\mathcal{W}_3\oplus\mathcal{W}_4$ if and only if $J=J^C$. Proof. The class of almost Hermitian manifolds $\mathcal{W}_1\oplus\mathcal{W}_3\oplus\mathcal{W}_4$ is defined by the condition that the Nijenhuis tensor is totally antisymmetric. To verify this condition we look at an orthonormal basis in the space of $(1,0)$-forms $\Lambda^{*(1,0)}$ at points of the curve $\gamma$:
$$
\begin{equation*}
\alpha^1=d\varphi-i\lambda u^-, \qquad \alpha^2=e_+-iJe_+, \qquad \alpha^3=v-iJv,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
e_+=\frac{2\sin\varphi-\cos\varphi}{4}e_1+\frac{\cos\varphi}{4}e_2,
\end{equation*}
\notag
$$
and $v$ is the form orthogonal to $d\varphi$, $u^-$, $e_+$ and $Je_+$. Then
$$
\begin{equation*}
d^{-1,2}\alpha^i=N^i_{\overline{jk}}\, \overline{\alpha^j}\wedge\overline{\alpha^k}, \quad\text{where } i,j,k\in\{1,2,3\}.
\end{equation*}
\notag
$$
The condition that the manifold belongs to the class $\mathcal{W}_1\oplus\mathcal{W}_3\oplus\mathcal{W}_4$ is equivalent to the equalities $N^1_{\overline{23}}=N^2_{\overline{31}}=N^3_{\overline{12}}$ and $N^i_{\overline{jk}}=0$ for all other values of the indices. A direct verification shows that if $\operatorname{Im}(N^2_{\overline{12}})=0$, then $u_3=0$. Now, the equality $N^3_{\overline{13}} - N^2_{\overline{12}} = 0$ implies that $u_0+u_2-u_1=-3\sin\varphi\cos^2\varphi$. Then we obtain either
$$
\begin{equation*}
u_1=3\sin\varphi(2\cos^2\varphi-1)\quad\text{or}\quad u_1=3\sin\varphi.
\end{equation*}
\notag
$$
The first case produces only the Cayley structure, while in the second the conditions in Lemma 2 are not satisfied.
The proof is complete. Thus, on the round sphere $S^6$ the class $\mathcal{W}_1\oplus\mathcal{W}_3\oplus\mathcal{W}_4=\mathcal{W}_1$ is just the class of structures of cohomogeneity one. 4.4. Structures in the class $\mathcal{W}_2\oplus\mathcal{W}_4$Theorem 4. There exist no almost Hermitian structures of cohomogeneity one in the class $\mathcal{W}_2\oplus\mathcal{W}_4$ on $(S^6,g_0)$. Proof. Structures in the class $\mathcal{W}_2\oplus\mathcal{W}_4$ are defined by the condition $d\omega=\omega\wedge\theta$, where $\theta(X)=\delta\omega(JX)$. We calculate $\theta$ using the results of Lemma 3:
$$
\begin{equation*}
\theta=\frac{-6\sin\varphi\cos 2\varphi\pm2u_1}{3\cos\varphi\sin^2\varphi}\, d\varphi.
\end{equation*}
\notag
$$
The condition $d\omega_J=\omega_J\wedge\theta$ is equivalent to
$$
\begin{equation*}
\begin{cases} u_2=u_3=0,\\ \dot{u}_0=\widetilde{\theta}u_0,\\ \dot{u}_1\mp 3\cos\varphi=u_1\widetilde{\theta}, \\ \widetilde{\theta}=\dfrac{-6\sin\varphi\cos 2\varphi\pm 2u_1}{3\cos\varphi\sin^2\varphi}. \end{cases}
\end{equation*}
\notag
$$
Using the assumptions of Lemma 1 and the conditions $u_2=u_3=0$ we obtain which does not agree with the condition $\dot{u}_0=\widetilde{\theta}u_0$.
The proof is complete. 
Citation in format AMSBIB
\Bibitem{Dau23} |
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