Twistorial examples of Riemannian almost product manifolds and their Gil-Medrano and Naveira types

TL;DR

This paper constructs non-trivial Riemannian almost product structures on twistor spaces of four-manifolds, classifies their types, and provides geometric interpretations of these classes.

Contribution

It introduces new examples of Riemannian almost product structures on twistor spaces and analyzes their Gil-Medrano and Naveira types with geometric insights.

Findings

Structures are constructed on product bundles of twistor spaces.

The types of these structures are classified and interpreted geometrically.

Provides explicit examples and classifications of almost product structures.

Abstract

Non-trivial examples of Riemannian almost product structures are constructed on the product bundle of the positive and negative twistor spaces of an oriented Riemannian four-manifold. The Gil-Medrano and Naveira types of these structures are determined and a geometric interpretation of the corresponding classes is given.

Equations

$$g(v_1\wedge v_2,v_3\wedge v_4)=\frac{1}{2}det[g(v_i,v_j)].$$

$$R(X\wedge Y)(Z\wedge T)=R(X,Y)Z\wedge T+Z\wedge R(X,Y)T$$

$$g(\mathcal{R}(X\wedge Y),Z\wedge T)=g(R(X,Y)Z,T)$$

$${\Lambda }^{2}TM={\Lambda }_{-}^{2}TM\oplus {\Lambda }_{+}^{2}TM$$

$$s_1^{\pm }=E_1\wedge E_2\pm E_3\wedge E_4,s_2^{\pm }=E_1\wedge E_3\pm E_4\wedge E_2,s_3^{\pm }=E_1\wedge E_4\pm E_2\wedge E_3.$$

$$g(K_{a}X,Y)=2g(a,X\wedge Y),X,Y\in T_{\pi (a)}M.$$

$$\begin{array}[]{c}K_{a}\circ K_{b}=K_{b}\circ K_{a},\quad{\rm{if}}\quad a\in\Lambda^{2}_{\pm}T_{p}M,\>b\in\Lambda^{2}_{\mp}T_{p}M;\\[6.0pt] K_{a}\circ K_{b}=-K_{b}\circ K_{a},\quad{\rm{if}}\quad a,b\in\Lambda^{2}_{\pm}T_{p}M,\>a\perp b.\end{array}$$

$$g(R(a)b,c)=\pm g(\mathcal{R}(a),b\times c))$$

$$K_b\circ K_c=-g(b,c)Id\pm K_{b\times c},b,c\in {\Lambda }_{\pm }^2{T}_{p}M.$$

$$\mathcal{B}(X\wedge Y)=\rho (X)\wedge Y+X\wedge \rho (Y)-\frac{s}{2}X\wedge Y.$$

$$\mathcal{R}=\frac{s}{6}Id+\mathcal{B}+{\mathcal{W}}_{+}+{\mathcal{W}}_{-}$$

$$\begin{array}[]{c}{\mathcal{K}}_{\nu}X^{h}_{\varkappa}=(K_{\sigma^{+}}\circ K_{\sigma^{-}}X)^{h}_{\varkappa}\quad{\rm{for}}\quad X\in T_{p}M,\\[6.0pt] {\mathcal{K}}_{1}(V^{+},V^{-})=(V^{+},V^{-}),\quad{\mathcal{K}}_{2}(V^{+},V^{-})=(V^{+},-V^{-}),\\[8.0pt] {\mathcal{K}}_{3}(V^{+},V^{-})=-(V^{+},V^{-}),\quad{\mathcal{K}}_{4}(V^{+},V^{-})=(-V^{+},V^{-})\\[8.0pt] {\rm{for}}\quad(V^{+},V^{-})\in{\mathcal{V}}_{\varkappa}.\end{array}$$

$${\mathcal{K}}_{\nu }(V)={\epsilon }_{\nu }(V^{+},(-1{)}^{\nu +1}{V}^{-}),V=(V^{+},V^{-})\in {\mathcal{V}}_{\varkappa },\nu =1,...,4.$$

$$G_{\mathbf{t}}(X^h+V,Y^h+W{)}_{\varkappa }=g(X,Y)+t_{1}g(V^{+},W^{+})+t_{2}g(V^{-},W^{-}),$$

$$X=\sum _{\alpha =1}^4{X}^{\alpha }\frac{\partial }{\partial x_{\alpha }}$$

$$\begin{array}[]{c}X^{h}=\displaystyle{\sum\limits_{\alpha=1}^{4}(X^{\alpha}\circ\pi)\frac{\partial}{\partial\widetilde{x}_{\alpha}}}\\[8.0pt] -\displaystyle{\sum\limits_{j,k=1}^{3}y^{+}_{j}(g(\nabla_{X}s^{+}_{j},s^{+}_{k})\circ\pi)\frac{\partial}{\partial y^{+}_{k}}-\sum\limits_{j,k=1}^{3}y^{-}_{j}(g(\nabla_{X}s^{-}_{j},s^{-}_{k})\circ\pi)\frac{\partial}{\partial y^{-}_{k}}}.\end{array}$$

$$\begin{array}[]{c}[X^{h},Y^{h}]=[X,Y]^{h}\\[8.0pt] +\displaystyle{}\sum\limits_{j,k=1}^{3}y_{j}^{+}(g(R(X\wedge Y)s_{j}^{+},s_{k}^{+})\circ\pi)\frac{\partial}{\partial y_{k}^{+}}+\sum\limits_{j,k=1}^{3}y_{j}^{-}(g(R(X\wedge Y)s_{j}^{-},s_{k}^{-})\circ\pi)\frac{\partial}{\partial y_{k}^{-}}.\end{array}$$

$$T_{\omega }({\Lambda }_{\pm }^2{T}_{\pi (\omega )}M)\cong {\Lambda }_{\pm }^2{T}_{\pi (\omega )}M,\omega \in {\mathcal{Z}}_{\pm },$$

$$[X^h,Y^h{]}_{\varkappa }=[X,Y{]}_{\varkappa }^h+R_p(X\wedge Y)\varkappa ,\varkappa =({\sigma }^{+},{\sigma }^{-})\in {\mathcal{Z}}_{+}\times {\mathcal{Z}}_{-},p=\pi (\varkappa ),$$

$$G_{\mathbf{t}}(R(X,Y)\varkappa ,V)=g(\mathcal{R}(t_1{\sigma }^{+}\times V^{+}-t_2{\sigma }^{-}\times V^{-}),X\wedge Y).$$

$${\tilde{a}}_{\varkappa }^{\pm }=a^{\pm }(p)-g(a^{\pm }(p),{\sigma }^{\pm }){\sigma }^{\pm },\varkappa =({\sigma }^{+},{\sigma }^{-}),p=\pi (\varkappa ).$$

$$[X^h,\tilde{a}{]}_{\varkappa }={\widetilde{({\nabla }_{X}a)}}_{\varkappa }$$

$$\tilde{a}=\sum _{i=1}^3[{\tilde{a}}_i^{+}\frac{\partial }{\partial y_i^{+}}+{\tilde{a}}_i^{-}\frac{\partial }{\partial y_i^{-}}],$$

$${\tilde{a}}^{\pm }=\sum _{i=1}^3[a_i^{\pm }\circ \pi -y_i^{\pm }\sum _{j=1}^3{y}_j^{\pm }(a_j^{\pm }\circ \pi )]\frac{\partial }{\partial y_i^{\pm }}$$

$$X_{\varkappa }^h=\sum _{\alpha =1}^4{X}^{\alpha }(p)\frac{\partial }{\partial {\tilde{x}}_{\alpha }}(\varkappa ),[X^h,\frac{\partial }{\partial y_i^{\pm }}{]}_{\varkappa }=0,i=1,2,3,$$

$$\begin{array}[]{r}[X^{h},\widetilde{a}]_{\varkappa}=\sum\limits_{i=1}^{3}[X(a^{+}_{i})-y^{+}_{i}(\varkappa)\sum\limits_{j=1}^{3}y^{+}_{j}(\varkappa)X_{p}(a^{+}_{j})]\displaystyle{\frac{\partial}{\partial y^{+}_{j}}(\varkappa)}\\[10.0pt] +\sum\limits_{i=1}^{3}[X(a^{-}_{i})-y^{-}_{i}(\varkappa)\sum\limits_{j=1}^{3}y^{-}_{j}(\varkappa)X_{p}(a^{-}_{j})]\displaystyle{\frac{\partial}{\partial y^{-}_{j}}(\varkappa)}.\end{array}$$

$$\begin{array}[]{r}D_{X}\widetilde{a}=\sum\limits_{i=1}^{3}[X(a^{+}_{i})-y^{+}_{i}(\varkappa)\sum\limits_{j=1}^{3}y^{+}_{j}(\varkappa)X_{p}(a^{+}_{j})]s_{i}^{+}(p)\\[8.0pt] +\sum\limits_{i=1}^{3}[X(a^{-}_{i})-y^{-}_{i}(\varkappa)\sum\limits_{j=1}^{3}y^{-}_{j}(\varkappa)X_{p}(a^{-}_{j})]s_{i}^{-}(p).\end{array}$$

$$(D_{X^h}{Y}^h{)}_{\varkappa }=({\nabla }_{X}Y{)}_{\varkappa }^h+\frac{1}{2}R(X,Y)\varkappa .$$

$$(D_V{X}^h{)}_{\varkappa }=\mathcal{H}(D_{X^h}V{)}_{\varkappa }=-\frac{1}{2}(R_p(t_1{\sigma }^{+}\times V^{+}-t_2{\sigma }^{-}\times V^{-})X{)}_{\varkappa }^h$$

$$(D_{X^h}{Y}^h{)}_{\varkappa }=({\nabla }_{X}Y{)}_{\varkappa }^h+\frac{1}{2}R(X,Y)\varkappa .$$

Full text

2010 Mathematics Subject Classification. Primary 53C15, Secondary 53C28

*Key words: Riemannian almost product manifolds, twistor spaces *

The author is partially supported by the National Science Fund, Ministry of Education and Science of Bulgaria under contract DN 12/2

1. Introduction

Recall that a Riemannian almost product manifold is a Riemannian manifold $(N,h)$ endowed with a pair of orthogonal distributions $V$ and $H$ on $N$ such that $TN=V\oplus H$, $rank\,V<dim\,N$. A Riemannian manifold of dimension $n$ admits an almost product structure $(V,H)$ with $rank\,V=d$ if and only if the structure group of the manifold can be reduced to the group $O(d)\times O(n-d)$. The decomposition $TN=V\oplus H$ determines an orthogonal isomorphism $P$ of the tangent bundle $TN$ with $P|V=Id$, $P|H=-Id$, hence $P^{2}=Id$, $P\neq\pm Id$ at every point of $N$. Conversely, an orthogonal isomorphism $P$ of $TN$ with $P_{x}^{2}=Id$, $P_{x}\neq\pm Id$ for every $x\in N$, defines an almost product structure on $(N,h)$ provided the dimension $d(x)$ of the $(+1)$-eigenspaces $V_{x}$ of $P_{x}$ is constant. An isomorphism $P$ with these properties is also called an almost product structure on $(N,h)$. The distribution $V$ on which $P$ is the identity map is usually called vertical, while the orthogonal distribution $H$ is called horizontal.

Similar to the Gray-Hervella classification of almost Hermitian manifolds [9], A.M. Naveira [15] has introduced 36 classes of Riemannian almost product manifolds. These come from an orthogonal invariant decomposition under the action of the group $O(d)\times O(n-d)$ on the space of covariant $3$-tensor on an Euclidean vector space having the same symmetries as the the covariant derivative of the fundamental form $\Phi(X,Y)=h(PX,Y)$ of a Riemannian almost product manifold. This decomposition have been found by Naveira [ibid.] and it has been proved by F.J. Carreras [3] that it is irreducible.

Naveira [ibid.], Gil-Medrano [12] and A. Montesinos [14] have given geometric interpretations of the Naveira classes. V. Miquel [13] has constructed examples for each class.

Gil-Medrano [ibid] has introduced algebraic conditions for the covariant derivative of $P$ restricted to the distributions $V$ and $H$ (see also Sec. 4) and has given their geometric characterization. Combining one of these conditions on $V$ with one on $H$, we can cover the 36 classes of Naveira.

A trivial example of a Riemannian almost product manifold is the product $N=M_{1}\times M_{2}$ of Riemannian manifolds with $V=TM_{1}$ and $H=TM_{2}$. In this paper, we use twistor theory to provide non-trivial examples of Riemannina almost product manifolds. Let $(M,g)$ be an oriented four-dimensional Riemannian manifold, and let ${\mathcal{Z}}_{\pm}$ be the twistor spaces of $(M,g)$, the bundles over $M$ whose sections are almost complex structures on $M$ compatible with the metric and $\pm$ the orientation. These are $S^{2}$-bundles over $M$. The product bundle ${\mathscr{P}}={\mathcal{Z}}_{+}\times{\mathcal{Z}}_{-}$ admits a natural $2$-parameter family $G_{t_{1},t_{2}}$, $t_{1},t_{2}>0$, of Riemannian metrics and four compatible almost product structures ${\mathcal{K}}_{\nu}$. We show that these structures are not integrable, so they are not trivial products even locally. We also find the Gil-Medrano types of $({\mathscr{P}},G_{t_{1},t_{2}},{\mathcal{K}}_{\nu})$, $\nu=1,...,4$, in terms of the curvature of the base manifold $(M,g)$ and specific values of the parameters $t_{1},t_{2}$. Using this, we determine the Naveira classes of $({\mathscr{P}},G_{t_{1},t_{2}},{\mathcal{K}}_{\nu})$. Finally we give a geometric interpretation of the obtained results.

2. Preliminaries

2.1. The twistor space of a four-manifold

Let $(M,g)$ be an oriented Riemannian manifold of dimension four. The metric $g$ induces a metric on the bundle of two-vectors $\pi:\Lambda^{2}TM\to M$ by the formula

[TABLE]

The Levi-Civita connection of $(M,g)$ determines a connection on the bundle $\Lambda^{2}TM$, both denoted by $\nabla$, and the corresponding curvatures are related by

[TABLE]

for $X,Y,Z,T\in TM$. The curvature operator ${\cal R}$ is the self-adjoint endomorphism of $\Lambda^{2}TM$ defined by

[TABLE]

Let us note that we adopt the following definition for the curvature tensor $R$ : $R(X,Y)=\nabla_{[X,Y]}-[\nabla_{X},\nabla_{Y}]$.

The Hodge star operator defines an endomorphism $\ast$ of $\Lambda^{2}TM$ with $\ast^{2}=Id$. Hence we have the orthogonal decomposition

[TABLE]

where $\Lambda^{2}_{\pm}TM$ are the subbundles of $\Lambda^{2}TM$ corresponding to the $(\pm 1)$-eigenvalues of the operator $\ast$.

Let $(E_{1},E_{2},E_{3},E_{4})$ be a local oriented orthonormal frame of $TM$. Set

[TABLE]

Then $(s_{1}^{\pm},s_{2}^{\pm},s_{3}^{\pm})$ is a local orthonormal frame of $\Lambda^{2}_{\pm}TM$ defining an orientation on $\Lambda^{2}_{\pm}TM$, which does not depend on the choice of the frame $(E_{1},E_{2},E_{3},E_{4})$ (see, for example, [6]).

For every $a\in\Lambda^{2}TM$, define a skew-symmetric endomorphism $K_{a}$ of $T_{\pi(a)}M$ by

[TABLE]

It is easy to check that:

[TABLE]

Note also that, denoting by $\gamma$ the standard metric $-\frac{1}{2}Trace\,PQ$ on the space of skew-symmetric endomorphisms, we have $\gamma(K_{a},K_{b})=2g(a,b)$ for $a,b\in\Lambda^{2}TM$. If $\sigma\in\Lambda^{2}_{\pm}TM$ is a unit vector, then $K_{\sigma}$ is a complex structure on the vector space $T_{\pi(\sigma)}M$ compatible with the metric and $\pm$ the orientation of $M$. Conversely, the $2$-vector $\sigma$ dual to one half of the fundamental $2$-form of such a complex structure is a unit vector in $\Lambda^{2}_{\pm}TM$. Thus, the unit sphere subbunlde ${\cal Z}_{\pm}$ of $\Lambda^{2}_{\pm}TM$ parametrizes the complex structures on the tangent spaces of $M$ compatible with its metric and $\pm$ orientation. The subbundles ${\mathcal{Z}}_{+}$ and ${\mathcal{Z}}_{-}$ are called the postive and the negative twistor space of $M$. They are the two connected components of the bundle over $M$ whose fibre at a point $p\in M$ consists of all complex structures on $T_{p}M$ compatible with the metric.

The connection $\nabla$ on $\Lambda^{2}TM$ induced by the Levi-Civita connection of $M$ preserves the bundles $\Lambda^{2}_{\pm}TM$, so it induces a metric connection on each of them denoted again by $\nabla$. The horizontal distribution of $\Lambda^{2}_{\pm}TM$ with respect to $\nabla$ is tangent to the twistor space ${\mathcal{Z}}_{\pm}$. Thus, we have the decomposition $T{\mathcal{Z}}_{\pm}={\mathcal{H}}\oplus{\mathcal{V}}$ of the tangent bundle of ${\mathcal{Z}}_{\pm}$ into horizontal and vertical components. The vertical space ${\mathcal{V}}_{\tau}=\{V\in T_{\tau}{\mathcal{Z}}_{\pm}:~{}\pi_{\ast}V=0\}$ at a point $\tau\in{\mathcal{Z}}_{\pm}$ is the tangent space to the fibre of ${\mathcal{Z}}_{\pm}$ through $\tau$. Considering $T_{\tau}{\mathcal{Z}}_{\pm}$ as a subspace of $T_{\tau}(\Lambda^{2}_{\pm}TM)$, ${\mathcal{V}}_{\tau}$ is the orthogonal complement of $\tau$ in $\Lambda^{2}_{\pm}T_{\pi(\tau)}M$. The map $V\ni{\cal V}_{\tau}\to K_{V}$ gives an identification of the vertical space with the space of skew-symmetric endomorphisms of $T_{\pi(\tau)}M$ which anti-commute with $K_{\tau}$. Let $s$ be a local section of ${\mathcal{Z}}_{\pm}$ such that $s(p)=\tau$ where $p=\pi(\tau)$. Considering $s$ as a section of $\Lambda^{2}_{\pm}TM$, we have $\nabla_{X}s\in{\mathcal{V}}_{\tau}$ for every $X\in T_{p}M$ since $s$ has a constant length. Moreover, $X^{h}_{\tau}=s_{\ast}X-\nabla_{X}s$ is the horizontal lift of $X$ at ${\tau}$.

Denote by $\times$ the usual vector cross product on the oriented $3$-dimensional vector space $\Lambda^{2}_{\pm}T_{p}M$, $p\in M$, endowed with the metric $g$. Then it is easy to check that

[TABLE]

for $a\in\Lambda^{2}T_{p}M$, $b,c\in\Lambda^{2}_{\pm}T_{p}M$. Also

[TABLE]

Denote by ${\cal B}:\Lambda^{2}TM\to\Lambda^{2}TM$ the endomorphism corresponding to the traceless Ricci tensor. If $s$ denotes the scalar curvature of $(M,g)$ and $\rho:TM\to TM$ is the Ricci operator, $g(\rho(X),Y)=Ricci(X,Y)$, we have

[TABLE]

Note that ${\mathcal{B}}$ sends $\Lambda^{2}_{\pm}TM$ into $\Lambda^{2}_{\mp}TM$. Let ${\cal W}:\Lambda^{2}TM\to\Lambda^{2}TM$ be the endomorphism corresponding to the Weyl conformal tensor. Denote the restriction of ${\cal W}$ to $\Lambda^{2}_{\pm}TM$ by ${\cal W}_{\pm}$, so ${\cal W}_{\pm}$ sends $\Lambda^{2}_{\pm}TM$ to $\Lambda^{2}_{\pm}TM$ and vanishes on $\Lambda^{2}_{\mp}TM$. Moreover, $Trace\,{\mathcal{W}}_{\pm}=0$.

It is well known that the curvature operator decomposes as ([17], see e.g. [2, Chapter 1 H])

[TABLE]

Note that this differs by a factor $1/2$ from [2] because of the factor $1/2$ in our definition of the induced metric on $\Lambda^{2}TM$. Note also that changing the orientation of $M$ interchanges the roles of $\Lambda^{2}_{+}TM$ and $\Lambda^{2}_{-}TM$, correspondingly the roles of ${\mathcal{W}}_{+}$ and ${\mathcal{W}}_{-}$.

The Riemannian manifold $(M,g)$ is Einstein exactly when ${\cal B}=0$. It is called anti-self-dual (self-dual), if ${\mathcal{W}}_{+}=0$ (resp. ${\mathcal{W}}_{-}=0$). By a famous result of Atiyah-Hitchin-Singer [1], the anti-self-duality (self-duality) condition is necessary and sufficient for integrability of a naturally defined almost complex structure on ${\mathcal{Z}}_{+}$ (resp., ${\mathcal{Z}}_{-}$).

3. Riemannian almost product structure on the product bundle ${\mathcal{Z}}_{+}\times{\mathcal{Z}}_{-}$

Let ${\mathscr{P}}={\mathcal{Z}}_{+}\times{\mathcal{Z}}_{-}$ be the product bundle over $M$ of the bundles ${\mathcal{Z}}_{\pm}$.

The projection to $M$ of the vector bundle $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ will be denoted by $\pi$ and we shall use the same symbol for its restriction to the subbundle ${\mathscr{P}}={\mathcal{Z}}_{+}\times{\mathcal{Z}}_{-}$. By abuse of notation, the direct sum of the connections $\nabla|\Lambda^{2}_{\pm}$ will also be denoted by $\nabla$.

Let $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$ and $p=\pi(\varkappa)$. Take sections $\phi^{\pm}$ of $\Lambda^{2}_{\pm}$ such that $\phi^{\pm}(p)=\sigma^{\pm}$ and $\nabla\phi^{\pm}|_{p}=0$. Then $\Phi=(\frac{\phi^{+}}{||\phi^{+}||},\frac{\phi^{-}}{||\phi^{-}||})$ is a section of $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ taking values in ${\mathscr{P}}$ and such that $\Phi(p)=\varkappa$, $\nabla\Phi|_{p}=0$. Hence the horizontal space ${\mathcal{H}}_{\varkappa}=\Phi_{\ast}(T_{p}M)$ of connection $\nabla$ on $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ at $\varkappa$ is tangent to the submanifold ${\mathscr{P}}$. Thus, we have the decomposition $T_{\varkappa}{\mathscr{P}}={\mathcal{H}}_{\varkappa}\oplus{\mathcal{V}}_{\varkappa}$ into horizontal and vertical parts, where the vertical space ${\mathcal{V}}_{\varkappa}$ of the bundle ${\mathscr{P}}\to M$ is clearly the product of the vertical spaces ${\mathcal{V}}_{\sigma^{\pm}}$ of the bundles ${\mathcal{Z}}_{\pm}\to M$. This decomposition allows one to define four almost product structures ${\mathcal{K}}_{\nu}$ on ${\mathscr{P}}$, $\nu=1,..,4$, setting

[TABLE]

It is convenient to set $\varepsilon_{1}=\varepsilon_{2}=1$, $\varepsilon_{3}=\varepsilon_{4}=-1$, so that

[TABLE]

Clearly ${\mathcal{K}}_{\nu}^{2}=Id$. The endomorphism $P_{\varkappa}=K_{\sigma^{+}}\circ K_{\sigma^{-}}$ of $T_{p}M$ is an involution different form $\pm Id$ and its $\pm 1$ eigenspaces are invariant under $K_{\sigma^{+}}$ and $K_{\sigma^{-}}$. Hence we can find an oriented orthonormal basis $E_{1},...,E_{4}$ of $T_{p}M$ such that $K_{\sigma^{+}}E_{1}=K_{\sigma^{-}}E_{1}=E_{2}$ and $K_{\sigma^{+}}E_{3}=-K_{\sigma^{-}}E_{3}=E_{4}$. Then $P_{\varkappa}E_{i}=-E_{i}$ for $i=1,2$ and $P_{\varkappa}E_{j}=E_{j}$ for $j=3,4$. Therefore the dimensions of the $(+1)$ and $(-1)$-eigenspaces of ${\mathcal{K}}_{1},...,{\mathcal{K}}_{4}$ are $(6,2)$, $(4,4)$, $(2,6)$, $(4,4)$, respectively. Thus, ${\mathcal{K}}_{\nu}$, $\nu=1,...,4$, are almost product structures on the manifold ${\mathscr{P}}$.

For ${\bf t}=(t_{1},t_{2})$ with $t_{1}>0$, $t_{2}>0$, define a $2$-parameter family of Riemannian metrics $G_{\bf t}$ on ${\mathscr{P}}$ by

[TABLE]

where $X,Y\in T_{\pi(\varkappa)}M$ and $V=(V^{+},V^{-}),W=(W^{+},W^{-})\in{\mathcal{V}}_{\varkappa}$.

Clearly, the projection $\pi:({\mathscr{P}},G_{\bf t})\to(M,g)$ is a Riemannian submersion. Moreover, the almost product structures ${\mathcal{K}}_{\nu}$ are compatible with every metric $G_{\bf t}$.

Let $({\mathscr{U}},x_{1},...,x_{4})$ be a local coordinate system of $M$, and let $(E_{1},...,E_{4})$ be an oriented orthonormal frame of $TM$ on ${\mathscr{U}}$. If $s_{i}^{\pm}$, $i=1,2,3$, are the local frames of $\Lambda^{2}_{\pm}TM$ define by (1), for $\varkappa=(\sigma^{+},\sigma^{-})\in\pi^{-1}({\mathscr{U}})$, set $\widetilde{x}_{\alpha}=x_{\alpha}\circ\pi$, $y^{\pm}_{j}(\varkappa)=g(\sigma^{\pm},(s_{j}^{\pm}\circ\pi)(\varkappa))$, $1\leq\alpha\leq 4$, $1\leq j\leq 3$. Then $\{\widetilde{x}_{\alpha},y^{+}_{j},y^{-}_{j}\}$ are local coordinates of the manifold $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ on $\pi^{-1}({\mathscr{U}})$.

The horizontal lift $X^{h}$ on $\pi^{-1}({\mathscr{U}})$ of a vector field

[TABLE]

is given by

[TABLE]

Hence

[TABLE]

for every vector fields $X,Y$ on ${\mathscr{U}}$. Using the standard identification

[TABLE]

we obtain from (8) the well-known formula

[TABLE]

where $R_{p}(X\wedge Y)\varkappa=(R_{p}(X\wedge Y)\sigma^{+},R_{p}(X\wedge Y)\sigma^{-})\in{\mathcal{V}}_{\varkappa}={\mathcal{V}}_{\sigma^{+}}\times{\mathcal{V}}_{\sigma^{-}}$.

Note also that it follows from (4) that if $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathcal{Z}}_{+}\times{\mathcal{Z}}_{-}$ and $V=(V^{+},V^{-})\in{\mathcal{V}}_{\varkappa}$, $X,Y\in T_{\pi(\varkappa)}M$,

[TABLE]

For any (local) section $a=(a^{+},a^{-})$ of $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$, denote by $\widetilde{a}=(\widetilde{a}^{+},\widetilde{a}^{-})$ the vertical vector field on ${\mathscr{P}}$ defined by

[TABLE]

Note that for every $\varkappa\in{\mathscr{P}}$ we can find sections $a_{1},...,a_{4}$ of $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ near the point $p=\pi(\varkappa)$ such that $\widetilde{a}_{1},...,\widetilde{a}_{4}$ form a basis of the vertical vector space at each point in a neighbourhood of $\varkappa$.

The next lemma is a kind of folklore appearing in different contexts (cf, for example, [7, 5]).

Lemma 1**.**

Let $X$ be a vector field on $M$ and let $a=(a^{+},a^{-})$ be a section of $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ defined on a neighbourhood of the point $p=\pi(\varkappa)$, $\varkappa\in{\mathscr{P}}$. Then:

[TABLE]

Proof.

Fix a point $p\in M$, take an oriented orthonormal frame $(E_{1},...,E_{4})$ of $TM$ such that $\nabla E_{i}|_{p}=0$, and define $s_{i}^{\pm}$, $i=1,2,3$, by (1). Set $a^{\pm}=\sum\limits_{i=1}^{3}a_{i}^{\pm}s_{i}^{\pm}$. Then, in the local coordinates of $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ introduced above,

[TABLE]

where

[TABLE]

Let us also note that for every vector field $X$ on $M$ near the point $p$, we have by (7)

[TABLE]

since $\nabla s_{i}^{\pm}|_{p}=0$, $i=1,2,3$. Hence

[TABLE]

On the other hand, considering $\widetilde{a}$ as a section of $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$, we have for $X\in T_{p}M$

[TABLE]

This proves the lemma. ∎

Denote by $D$ the Levi-Civita connection of $({\mathcal{P}},G_{\bf t})$.

Let $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$ and $p=\pi(\varkappa)$. As we have noticed, we can find an oriented orthonormal basis $(E_{1},...,E_{4})$ of $T_{p}M$ such that $\sigma^{\pm}=E_{1}\wedge E_{2}\pm E_{3}\wedge E_{4}$. Extend this basis to an oriented orthonormal frame of vector fields in a neighbourhood of $p$ such that $\nabla E_{\alpha}|_{p}=0$, ${\alpha}=1,...,4$. Define $s^{\pm}_{i}$, $i=1,2,3$, by (1), so that $s_{1}^{\pm}(p)=\sigma^{\pm}$ and $\nabla s^{\pm}_{i}|_{p}=0$. The vertical vector fields $\widetilde{a}_{1},...,\widetilde{a}_{4}$ determined by the sections $a_{1}=(s_{2}^{+},0)$, $a_{2}=(s_{3}^{+},0)$, $a_{3}=(0,s_{2}^{-})$, $a_{4}=(0,s_{3}^{-})$ of $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ form a frame of the vertical bundle ${\mathcal{V}}$ of ${\mathscr{P}}$ in a neighbourhood of $\varkappa$. Let $V\in{\mathcal{V}}_{\varkappa}$ and let $v$ be a section of $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ such that $v(p)=V$ and $\nabla v|_{p}=0$. Denote by $\widetilde{v}$ the vertical vector field corresponding to this section. By Lemma 1, $[X^{h},\widetilde{a}_{l}]_{\varkappa}=[X^{h},\widetilde{v}]_{\varkappa}=0$, $l=1,...,4$, for every vector field $X$ in a neighbourhood of $p$. It follows from the Koszul formula for the Levi-Civita connection that the vectors $(D_{\widetilde{v}}\widetilde{a}_{l})_{\varkappa}$ for all $l=1,...,4$ are $G_{\bf t}$-orthogonal to every horizontal vector $X^{h}_{\varkappa}$. Hence $D_{V}\widetilde{a}_{l}$ are vertical tangent vectors of ${\mathscr{P}}$ at $\varkappa$. It follows that, for every vertical vector field $W$, $D_{V}W$ is a vertical vector field. Thus, the fibres of ${\mathscr{P}}$ are totally geodesic submanifolds. This, of course, follows also from the Vilms theorem (see, for example, [2, Theorem 9.59].

The proof of the following lemma is practically given in [4, 5]) and we present it here just for completeness.

Lemma 2**.**

If $X,Y$ are vector fields on $M$ and $V=(V^{+},V^{-})$ is a vertical vector field on ${\mathscr{P}}$, then

[TABLE]

[TABLE]

where $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$, $p=\pi(\varkappa)$, and ${\cal H}$ means ”the horizontal component”.

Proof.

The Koszul formula, identity (8), and Lemma 1 imply

[TABLE]

Next, $D_{V}X^{h}$ is orthogonal to any vertical vector field $W$ since $D_{V}W$ is a vertical vector field. Thus $D_{V}X^{h}$ is a horizontal vector field. Hence $D_{V}X^{h}={\cal H}D_{X^{h}}V$ since $[V,X^{h}]$ is a vertical vector field. Therefore

[TABLE]

Thus, (13) follows from (10). ∎

Set

[TABLE]

Corollary 1**.**

Let $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$, $X,Y\in T_{\pi(\varkappa)}M$, $V\in{\mathcal{V}}_{\varkappa}$. Then

[TABLE]

where $P_{\varkappa}=K_{\sigma^{+}}\circ K_{\sigma^{-}}$.

Proof.

This follows from the identity

[TABLE]

and identities (12), and (14).

∎

Lemma 3**.**

Let $\varkappa=(\sigma^{+},\sigma^{{}_{-}})\in{\mathscr{P}}$, $X,Y,Z\in T_{\pi(\varkappa)}M$, and $U,V,W\in{\mathcal{V}}_{\varkappa}$. Then:

• (i)

$(D_{X^{h}_{\varkappa}}F_{\bf t,\nu})(Y^{h},Z^{h})=0$;

• (ii)

\begin{array}[]{c}(D_{X^{h}_{\varkappa}}F_{\bf t,\nu})(Y^{h},U)=-\frac{1}{2}\varepsilon_{\nu}g({\mathcal{R}}(t_{1}\sigma^{+}\times U^{+}+(-1)^{\nu}t_{2}\sigma^{-}\times U^{-}),X\wedge Y)\\[6.0pt] +\frac{1}{2}g({\mathcal{R}}(t_{1}\sigma^{+}\times U^{+}-t_{2}\sigma^{-}\times U^{-}),X\wedge K_{\sigma^{+}}K_{\sigma^{-}}Y);\end{array}**

• (iii)

\begin{array}[]{c}(D_{U}F_{\bf t,\nu})(Y^{h},Z^{h})_{\varkappa}=g_{p}((K_{\sigma^{-}}K_{U^{+}}+K_{\sigma^{+}}K_{U^{-}})Y,Z)\\[6.0pt] +\frac{1}{2}g({\mathcal{R}}(t_{1}\sigma^{+}\times U^{+}-t_{2}\sigma^{-}\times U^{-}),Y\wedge K_{\sigma^{+}}K_{\sigma^{-}}Z-K_{\sigma^{+}}K_{\sigma^{-}}Y\wedge Z);\end{array}**

• (iv)

$(D_{X^{h}_{\varkappa}}F_{\bf t,\nu})(U,V)=0$;

• (v)

$(D_{U}F_{\bf t,\nu})(X^{h},V)=0$;

• (vi)

$(D_{U}F_{\bf t,\nu})(V,W)=0$;

Proof.

Take an oriented orthonormal basis $E_{1},...,E_{4}$ of $T_{p}M$ such that $\sigma^{\pm}=E_{1}\wedge E_{2}\pm E_{3}\wedge E_{4}$. Extend the basis $E_{1},...,E_{4}$ to an oriented orthormal frame in a neighbourhood of the point $p$ such that $\nabla E_{\alpha}|_{p}=0$, ${\alpha}=1,...,4$. Using this frame, define sections $s_{i}^{\pm}$, $i=1,2,3$, of $\Lambda^{2}_{\pm}TM$ by (1); clearly $\nabla s^{\pm}_{i}|_{p}=0$. Also, extend $Y$ and $Z$ to vector fields such that $\nabla Y|_{p}=\nabla Z|_{p}=0$. Then

[TABLE]

since ${\mathcal{K}}_{\nu}D_{X^{h}_{\varkappa}}Y^{h}$ and ${\mathcal{K}}_{\nu}D_{X^{h}_{\varkappa}}Z^{h}$ are vertical vectors by (12). Setting $S=(s_{1}^{+},s_{1}^{-})$, we get a section of ${\mathscr{P}}$ with $S(p)=\varkappa$, $\nabla S|_{p}=0$. Hence

[TABLE]

since $\nabla E_{\alpha}|_{p}=\nabla Y|_{p}=\nabla Z|_{p}=0$. This proves identity (i).

Extending the vector $U$ to a vertical vector field in a neighbourhood of $\varkappa$, we see that

[TABLE]

since the vector ${\mathcal{K}}_{\nu}Y^{h}$ is horizontal, while U and $[X^{h},U]$ are vertical. Thus, the second formula of the lemma follows from (12), (13), and (10).

Formula (ii) follows from Corollary 1 and (10).

Formula (iii) follows from (13) and the identity

[TABLE]

To prove (iv), take sections $a=(a^{+},a^{-})$ and $b=(b^{+},b^{-})$ of $\Lambda^{2}_{+}TM\oplus\Lambda^{2}_{-}TM$ such that $a(p)=U$, $b(p)=V$ and $\nabla a|_{p}=\nabla b|_{p}=0$. Let $\widetilde{a}=(\widetilde{a}^{+},\widetilde{a}^{-})$ and $\widetilde{b}=(\widetilde{b}^{+},\widetilde{b}^{-})$ be the vertical vector fields on ${\mathscr{P}}$ defined by means of $a$ and $b$ via (11). Then $\widetilde{a}(\varkappa)=U$, $\widetilde{b}(\varkappa)=V$, and $[X^{h},\widetilde{a}]_{\varkappa}=[X^{h},\widetilde{b}]_{\varkappa}=0$ by Lemma 1. Hence $D_{X^{h}_{\varkappa}}\widetilde{a}$ and $D_{X^{h}_{\varkappa}}\widetilde{b}$ are horizontal vectors by (13). Thus,

[TABLE]

We have $X^{h}_{\varkappa}(y^{\pm}_{i})=0$, $i=1,2,3$, by (7). Moreover,

[TABLE]

Hence $X^{h}_{\varkappa}(g(\widetilde{a}^{+},\widetilde{b}^{+}))=0$. Similarly $X^{h}_{\varkappa}(g(\widetilde{a}^{-},\widetilde{b}^{-}))=0$. Therefore

[TABLE]

This proves (iv).

Next,

[TABLE]

since ${\mathcal{K}}_{\nu}X^{h}$ and ${\mathcal{K}}_{\nu}D_{U}X^{h}$ are horizontal vectors and $D_{U}\widetilde{b}$ is vertical. This is identity (v).

Since $D=\nabla$ for vertical vector fields, identity (vi) is a straightforward consequence from the definition of ${\mathcal{K}}_{\nu}$ and the fact that $\nabla$ is a metric connection.

∎

Let $(N,h)$ be a Riemannian almost product manifold with almost product structure $P$. Its Nijenhuis tensor is defined by

[TABLE]

As usual, the structure $P$ is called integrable if the Nijenhuis tensor vanishes. This condition is equivalent to the integrability of both the vertical and horizontal distributions on the manifold $N$. In this case $N$ is locally the product of two Riemannian manifolds and $P$ is the trivial product structure determined by these manifolds.

Denote by ${\mathcal{N}}_{\nu}$ the Nijenhuis tensor of the endomorphism ${\mathcal{K}}_{\nu}$ of $T{\mathscr{P}}$. It can be written in terms of the form $F_{\bf t,\nu}$ as

[TABLE]

This identity, Corollary 1, and Lemma 3 imply:

Corollary 2**.**

Let $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$, $X,Y\in T_{\pi(\varkappa)}M$, $U,V\in{\mathcal{V}}_{\varkappa}$. Set $P_{\varkappa}=K_{\sigma^{+}}\circ K_{\sigma^{-}}$. Then

[TABLE]

[TABLE]

[TABLE]

Proposition 1**.**

The almost product structures ${\mathcal{K}}_{\nu}$ are never integrable.

Proof.

Take an oriented orthonormal basis $E_{1},...,E_{4}$ of a tangent space $T_{p}M$ and define $s_{i}^{\pm}$, $i=1,2,3$, by (1). Set $\varkappa=(s_{1}^{+},s_{1}^{-})$, $U=(s_{2}^{+},0)$. Then ${\mathcal{N}}_{1}(E_{3},U)={\mathcal{N}}_{2}(E_{3},U)=2(E_{2})^{h}_{\varkappa}$ and ${\mathcal{N}}_{3}(E_{1},U)={\mathcal{N}}_{4}(E_{1},U)=-2(E_{4})^{h}_{\varkappa}$. ∎

4. Gil-Medrano conditions on the manifold ${\mathscr{P}}$

Let $(N,h)$ be a Riemannian almost product manifold with almost product structure $P$ and Levi-Civita connection $\nabla$. Let $\mathfrak{D}$ be one of its vertical or horizontal distribution. Denote the dimension of $\mathfrak{D}$ by $m$. Define an $1$-form on $N$ setting

[TABLE]

where $\{E_{l}\}$ is an orthonormal basis of $\mathfrak{D}_{p}$.

Following [12], we shall say that:

$(a)$ $\mathfrak{D}$ has the property $F$ if $(\nabla_{A}P)(B)=(\nabla_{B}P)(A)$ for every $A,B\in\mathfrak{D}$;

$(b)$ $\mathfrak{D}$ has the property $D_{1}$ if $(\nabla_{A}P)(B)=-(\nabla_{B}P)(A)$ for $A,B\in\mathfrak{D}$ (equivalently, $(\nabla_{A}P)(A)=0$);

$(c)$ $\mathfrak{D}$ has the property $D_{2}$ if $\alpha(X)=0$ for every $X\in\mathfrak{D}^{\perp}$;

$(d)$ $\mathfrak{D}$ has the property $D_{3}$ if

[TABLE]

$(e)$ $\mathfrak{D}$ has the property $F_{i}$, $i=1,2,3$, if it has the properties $F$ and $D_{i}$.

Remark 2. Note that $\mathfrak{D}$ has the property $D_{1}$ if and only if it has the properties $D_{2}$ and $D_{3}$.

For the geometric interpretations of these conditions given in [12], see Section 6.

Combining conditions $F$, $D_{i}$, $F_{i}$ for the vertical and the horizontal distributions on $(N,h)$, and eliminating their duality, we obtain the 36 Naveira classes.

Lemma 4**.**

Let $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$, $X,Y,Z\in T_{\pi(\varkappa)}M$, and $U,V,W\in{\mathcal{V}}_{\varkappa}$. Set $A=(X^{h}_{\varkappa}+U)+{\mathcal{K}}_{\nu}(X^{h}_{\varkappa}+U)$, $B=(Y^{h}_{\varkappa}+V)+{\mathcal{K}}_{\nu}(Y^{h}_{\varkappa}+V)$. Then

[TABLE]

[TABLE]

where $P_{\varkappa}=K_{\sigma^{+}}\circ K_{\sigma^{-}}$.

Proof.

These formulas follow from Lemma 3 and the identity

[TABLE]

by a simple computation. ∎

Let $\mathcal{D}_{\nu}$ be the distribution on the manifold ${\mathscr{P}}$ for which ${\mathcal{K}}_{\nu}|\mathcal{D}_{\nu}=Id$, $\nu=1,2$.

Proposition 2**.**

$(i)$*   The distribution ${\mathcal{D}}_{\nu}$ of the almost product structure ${\mathcal{K}}_{\nu}$ does not have the property F for $\nu=1,2,4$.*

$(ii)$*   The distribution ${\mathcal{D}}_{3}$ has the property F if and only if $(M,g)$ is of constant curvature.*

Proof.

$(i)$ Let $E_{1},...,E_{4}$ be an oriented orthonormal basis of a tangent space $T_{p}M$. Define $s_{i}^{\pm}$, $i=1,2,3$, by (1), and set $\varkappa=(s_{1}^{+},s_{1}^{-})$, $X=0$, $U=(s_{2}^{+},s_{2}^{-})$, $Y=E_{3}$, $V=0$, $Z=E_{2}$. Then the identity $G_{\bf t}((D_{A}{\mathcal{K}}_{\nu})(B),Z^{h}_{\varkappa})=G_{\bf t}((D_{B}{\mathcal{K}}_{\nu})(A),Z^{h}_{\varkappa})$ becomes $[\varepsilon_{\nu}+1]-[\varepsilon_{\nu}(-1)^{\nu}-1]=0$, an identity, which does not hold for $\nu=1,2,4$.

$(ii)$ By Lemma 4, the distribution ${\mathcal{D}}_{3}$ has the property F if and only if

[TABLE]

for every $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$, $W^{\pm}\in\Lambda^{2}_{\pm}T_{\pi(\varkappa)}M$ with $W^{\pm}\perp\sigma^{\pm}$ and $X,Y\in T_{\pi(\varkappa)}M$. Applying this identity for $(W^{+},-W^{-})$, we see that condition (17) is equivalent to

[TABLE]

Replacing $\sigma^{-}$ and $W^{-}$ by $-\sigma^{-}$ and $-W^{-}$, we observe that the latter equations are equivalent to

[TABLE]

Let $E_{1},...,E_{4}$ be an oriented orthonormal basis of a tangent space $T_{p}M$ and define $s_{i}^{\pm}$, $i=1,2,3$, by (1). We apply(18) with (a) $\varkappa=(s_{1}^{+},s_{1}^{-})$, $W^{+}=s_{2}^{+},s_{3}^{+}$, $(X,Y)=(E_{1},E_{2}),(E_{3},E_{4})$, (b) $\varkappa=(s_{1}^{+},s_{2}^{-})$, $W^{+}=s_{3}^{+}$, $(X,Y)=(E_{1},E_{3})$, (c) $\varkappa=(s_{3}^{+},s_{1}^{-})$, $W^{+}=s_{2}^{+}$, $(X,Y)=(E_{1},E_{2})$. This gives

[TABLE]

Replacing the basis $E_{1},E_{2},E_{3},E_{4}$ by $E_{1},E_{3},E_{4},E_{2}$ and $E_{1},E_{4},E_{2},E_{3}$, we see that

[TABLE]

Therefore ${\mathcal{W}}_{+}={\mathcal{B}}=0$. In the same way, we get ${\mathcal{W}}_{-}=0$ from (18). This shows that $(M,g)$ is of constant curvature.

Conversely, if $(M,g)$ is of constant curvature, the identity (18) is satisfied by (3).

∎

Proposition 3**.**

(I)* The distribution ${\mathcal{D}}_{\nu}$, $\nu=1,2,4$, of the almost product structure ${\mathcal{K}}_{\nu}$ has the property $D_{1}$ if and only if:*

$(i)$*  $(M,g)$ is of positive constant sectional curvature $\chi$ and $t_{1}=t_{2}=\displaystyle{\frac{3}{8\chi}}$, in the case $\nu=1$;*

$(ii)$*  $(M,g)$ is anti-self-dual and Einstein with positive scalar curvature $s$, and $t_{1}=\displaystyle{\frac{6}{s}}$, in the case $\nu=2$ (no condition on $t_{2}>0$).*

$(iii)$*   $(M,g)$ is self-dual and Einstein with positive scalar curvature $s$, and $t_{2}=\displaystyle{\frac{6}{s}}$, in the case $\nu=4$ (no condition on $t_{1}>0$).*

(II)*   The distribution ${\mathcal{D}}_{3}$ has the property $D_{1}$.*

Proof.

By Lemma 4, ${\mathcal{D}}_{\nu}$ has the property $D_{1}$ if and only if

[TABLE]

for every $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$, $U^{\pm}\in\Lambda^{2}_{\pm}T_{\pi(\varkappa)}M$ with $U^{\pm}\perp\sigma^{\pm}$ and $X,Z\in T_{\pi(\varkappa)}M$.

As in the proof of the preceding proposition, it is easy to see that this condition is equivalent to the identities

[TABLE]

Clearly both identities are satisfied if $\nu=3$. Note also that, by (5), $K_{\sigma^{+}}K_{U^{+}}=K_{\sigma^{+}\times U^{+}}$ and $K_{\sigma^{-}}K_{U^{-}}=-K_{\sigma^{-}\times U^{-}}$. Thus, if $\nu=1$, changing the orientation of $M$ interchanges the identities in (20). If $\nu=2$, the second identity in (20) is trivially satisfied and if $\nu=4$, so does the second one.

Now, suppose that $\varepsilon_{\nu}+1\neq 0$ and the first identity in (20) holds. Let $E_{1},...,E_{4}$ be an oriented orthonormal basis of a tangent space $T_{p}M$ of $M$ and define $s^{\pm}_{i}$, $i=1,2,3$, by (1). Taking $\varkappa=(s^{+}_{1},s^{-}_{1})$, $U^{+}=s^{+}_{2}$, we get from the first identity of (20)

[TABLE]

Therefore

[TABLE]

Similarly, taking $U^{+}=s^{+}_{3}$, we obtain

[TABLE]

Replacing the basis $E_{1},E_{2},E_{3},E_{4}$ by $E_{1},E_{3},E_{4},E_{2}$ and $E_{1},E_{4},E_{2},E_{3}$, we see from (21) and (22) that

[TABLE]

Now, the curvature decomposition (6) and the fact that $Trace\,{\mathcal{W}}_{+}=0$ imply $\displaystyle{t_{1}=\frac{6}{s}}$. Then the first identity of (23) gives $g({\mathcal{W}}_{+}(s_{i}^{+}),s_{i}^{+})=0$, $i=1,2,3$. Hence ${\mathcal{W}}_{+}=0$. The second identity of (23) means that ${\mathcal{B}}=0$.

Conversely, if $t_{1}=\displaystyle{\frac{6}{s}}$ and ${\mathcal{B}}={\mathcal{W}}_{+}=0$, it is easy to check, using (2), (3) and (5), that the first identity of (20) is fulfilled. This proves the result for $\nu=2$.

If $\nu=1$ or $\nu=4$, the second identity of (20) holds if and only if $t_{2}=\displaystyle{\frac{6}{s}}$ and ${\mathcal{B}}={\mathcal{W}}_{-}=0$.

∎

Proposition 4**.**

The distribution ${\mathcal{D}}_{\nu}$, $\nu=1,...,4$, has the property $D_{2}$.

Proof.

Denote by $\alpha_{\nu}$ the $1$-form corresponding to the distribution ${\mathcal{D}}_{\nu}$ via (16). Let $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$ and set $P_{\varkappa}=K_{\sigma^{+}}\circ K_{\sigma^{-}}$. Take an oriented orthonormal basis $E_{1},...,E_{4}$ of $T_{\pi(\varkappa)}M$ such that $P_{\varkappa}E_{i}=-E_{i}$ for $i=1,2$ and $P_{\varkappa}E_{j}=E_{j}$ for $j=3,4$. Let $V^{\pm}_{i}$, $i=1,2$, be a $g$-orthonormal basis of ${\mathcal{V}}_{\pm}$. Then $E^{h}_{3},E^{h}_{4},\frac{1}{\sqrt{t}_{1}}(V^{+}_{1},0),\frac{1}{\sqrt{t}_{1}}(V^{+}_{2},0),\frac{1}{\sqrt{t}_{2}}(0,V^{-}_{1}),\frac{1}{\sqrt{t}_{2}}(0,V^{-}_{2})$ is a $G_{\bf t}$-orthonormal basis of the fibre of ${\mathcal{D}}_{1}$ at $\pi({\varkappa})$, $E^{h}_{3},E^{h}_{4},\frac{1}{\sqrt{t}_{1}}(V^{+}_{1},0),\frac{1}{\sqrt{t}_{1}}(V^{+}_{2},0)$ of ${\mathcal{D}}_{2}$, $E^{h}_{3},E^{h}_{4}$ of ${\mathcal{D}}_{3}$, and $E^{h}_{3},E^{h}_{4},\frac{1}{\sqrt{t}_{2}}(0,V^{-}_{1}),\frac{1}{\sqrt{t}_{2}}(0,V^{-}_{2})$ is a $G_{\bf t}$-orthonormal basis of the fibre of ${\mathcal{D}}_{4}$ . Using these bases, we get $\alpha_{\nu}=0$ by Lemma 3. ∎

Remark 2 and Proposition 4 imply:

Proposition 5**.**

The distribution ${\mathcal{D}}_{\nu}$, $\nu=1,...,4$, has the property $D_{3}$ exactly when it has the property $D_{1}$.

Lemma 3 imply the following.

Lemma 5**.**

Let $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$, $X,Y,Z\in T_{\pi(\varkappa)}M$, and $U,V,W\in{\mathcal{V}}_{\varkappa}$. Set $\widetilde{A}=(X^{h}_{\varkappa}+U)-{\mathcal{K}}_{\nu}(X^{h}_{\varkappa}+U)$, $\widetilde{B}=(Y^{h}_{\varkappa}+V)-{\mathcal{K}}_{\nu}(Y^{h}_{\varkappa}+V)$. Then

[TABLE]

[TABLE]

where $P_{\varkappa}=K_{\sigma^{+}}\circ K_{\sigma^{-}}$.

Proposition 6**.**

$(i)$*   The distribution ${\mathcal{D}}^{\perp}_{1}$ has the property F if and only if the manifold $(M,g)$ is of constant curvature.*

$(ii)$*   The distribution ${\mathcal{D}}^{\perp}_{\nu}$ does not have the property F for $\nu=2,3,4$.*

Proof.

By Lemma 5, the distribution ${\mathcal{D}}^{\perp}_{\nu}$ has the property F if and only if the following two identities hold:

[TABLE]

for every $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$, $X,Y,Z\in T_{\pi(\varkappa)}M$, $U,V,W\in{\mathcal{V}}_{\varkappa}$.

Let $E_{1},...,E_{4}$ be an oriented orthonormal basis of a tangent space $T_{p}M$ and define $s_{i}^{\pm}$, $i=1,2,3$, by (1).

If $\nu=2,3,4$, the first identity of (24) does not hold for $\varkappa=(s_{1}^{+},s_{1}^{-})$, $X=0$, $U=(s_{2}^{+},s_{2}^{-})$, $Y=E_{2}$, $Z=E_{3}$.

If $\nu=1$, (24) reduces to

[TABLE]

This is equivalent to the identities

[TABLE]

As we have seen in the proof of Proposition  2, the latter identities are satisfied if and only if the manifold $(M,g)$ is of constant curvature.

∎

Proposition 7**.**

(I)* The distribution ${\mathcal{D}}_{\nu}^{\perp}$, $\nu=2,3,4$, of the almost product structure ${\mathcal{K}}_{\nu}$ has the property $D_{1}$ if and only if:*

$(i)$*  $(M,g)$ is self-dual and Einstein with positive scalar curvature $s$, and $t_{2}=\displaystyle{\frac{6}{s}}$, in the case $\nu=2$ (no condition on $t_{1}>0$);*

$(ii)$*  $(M,g)$ is of positive constant sectional curvature $\chi$ and $t_{1}=t_{2}=\displaystyle{\frac{3}{8\chi}}$, in the case $\nu=3$;*

$(iii)$*  $(M,g)$ is anti-self-dual and Einstein with positive scalar curvature $s$, and $t_{1}=\displaystyle{\frac{6}{s}}$, in the case $\nu=4$ (no condition on $t_{2}>0$).*

(II)*  The distribution ${\mathcal{D}}_{1}^{\perp}$ has the property $D_{1}$.*

Proof.

By Lemma 5, ${\mathcal{D}}_{\nu}^{\perp}$ has the property $D_{1}$ if and only if

[TABLE]

for every $\varkappa=(\sigma^{+},\sigma^{-})\in{\mathscr{P}}$, $U^{\pm}\in\Lambda^{2}_{\pm}T_{\pi(\varkappa)}M$ with $U^{\pm}\perp\sigma^{\pm}$, and $X,Z\in T_{\pi(\varkappa)}M$.

This condition is equivalent to

[TABLE]

Obviously, if $\nu=1$ these conditions are satisfied, if $\nu=2$ the first identity trivially hods and if $\nu=4$ the second one holds. The result follows from the proof of Proposition 3.

∎

Lemma 3 easily implies:

Proposition 8**.**

The distribution ${\mathcal{D}}_{\nu}^{\perp}$ has the property $D_{2}$.

Proposition 9**.**

The distribution ${\mathcal{D}}_{\nu}^{\perp}$ has the property $D_{3}$ exactly when it has the property $D_{1}$.

5. The Naveira classes of the manifold ${\mathscr{P}}$

The results in the preceding section allow one to determine the Naveira classes of $({\mathscr{P}},{\mathcal{K}}_{\nu},G_{\bf t})$, ${\bf t}=(t_{1},t_{2})$, as follows.

Theorem 1**.**

The Riemannian almost product manifold $({\mathscr{P}},{\mathcal{K}}_{1},G_{\bf t})$ belongs to the Naveira class ${\mathscr{W}}_{1}\oplus{\mathscr{W}}_{2}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$ or to the class ${\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}$.

$({\mathscr{P}},{\mathcal{K}}_{1},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}$* if and only if $(M,g)$ is of positive constant curvature $\chi$ and $t_{1}=t_{2}=\frac{3}{8\chi}$.*

Theorem 2**.**

The Riemannian almost product manifold $({\mathscr{P}},{\mathcal{K}}_{2},G_{\bf t})$ belongs to the Naveira classes ${\mathscr{W}}_{1}\oplus{\mathscr{W}}_{2}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$, ${\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$, or ${\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}$.

$(i)$* $({\mathscr{P}},{\mathcal{K}}_{2},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$ if and only if $(M,g)$ is anti-self-dual and Einstein with positive scalar curvature $s$, and $t_{1}=\displaystyle{\frac{6}{s}}$ (no condition on $t_{2}>0$).*

$(ii)$*  $({\mathscr{P}},{\mathcal{K}}_{2},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}$ if and only if $(M,g)$ is of positive constant curvature $\chi$ and $t_{1}=t_{2}=\frac{3}{8\chi}$.*

Theorem 3**.**

The Riemannian almost product manifold $({\mathscr{P}},{\mathcal{K}}_{3},G_{\bf t})$ belongs to the Naveira classes ${\mathcal{W}}_{1}\oplus{\mathcal{W}}_{4}\oplus{\mathcal{W}}_{5}$, ${\mathcal{W}}_{4}\oplus{\mathcal{W}}_{5}$, or ${\mathcal{W}}_{4}$.

$(i)$*  $({\mathscr{P}},{\mathcal{K}}_{3},G_{\bf t})\in{\mathcal{W}}_{4}\oplus{\mathcal{W}}_{5}$ if and only if $(M,g)$ is of constat curvature.*

$(ii)$* $({\mathscr{P}},{\mathcal{K}}_{3},G_{\bf t})\in{\mathcal{W}}_{4}$ if and only if $(M,g)$ is of positive constant sectional curvature $\chi$ and $t_{1}=t_{2}=\displaystyle{\frac{3}{8\chi}}$*

Theorem 4**.**

The Riemannian almost product manifold $({\mathscr{P}},{\mathcal{K}}_{4},G_{\bf t})$ belongs to the Naveira classes ${\mathscr{W}}_{1}\oplus{\mathscr{W}}_{2}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$, ${\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$, or ${\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}$.

$(i)$* $({\mathscr{P}},{\mathcal{K}}_{4},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$ if and only if $(M,g)$ is self-dual and Einstein with positive scalar curvature $s$, and $t_{2}=\displaystyle{\frac{6}{s}}$ (no condition on $t_{1}>0$).*

$(ii)$*  $({\mathscr{P}},{\mathcal{K}}_{4},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}$ if and only if $(M,g)$ is of positive constant curvature $\chi$ and $t_{1}=t_{2}=\frac{3}{8\chi}$.*

6. Geometric interpretation

In this section, we restate the results obtained in preceding sections in geometric terms.

Recall the geometric characterizations of the Gil-Medrano conditions for the vertical or horizontal distribution $\mathfrak{D}$ of a Riemannian almost product manifold ([12]). First, condition $F$ is equivalent to $\mathfrak{D}$ being integrable. Next, a second fundamental form $T$ of a distribution on a Riemannian manifold has been proposed by B. Reinhart in [16]. It is a symmetric $2$-form with values in the normal bundle. If the distribution is integrable, $T$ coincides with the usual second fundamental form of the leaves as immersed submanifolds. A distribution is called minimal if the trace of $T$ vanishes; it is called totally geodesic if $T=0$. It has been proved in [12] and [16] that a distribution is totally geodesic if and only if every geodesic, which is tangent to the distribution at one point, is tangent to it at all points. Now, condition $D_{1}$ means that $\mathfrak{D}$ is totally geodesic, while condition $D_{2}$ is equivalent to $\mathfrak{D}$ being minimal.

Theorem 5**.**

(I)*  $(i)$  The distributions ${\mathcal{D}}_{\nu}$, $\nu=1,2,4$, are not integrable.*

$(ii)$*  The distribution ${\mathcal{D}}_{3}$ is integrable if and only if $(M,g)$ is of constat curvature.*

(II)* All distributions ${\mathcal{D}}_{\nu}$ are minimal, $\nu=1,...,4$.*

(III)*  $(i)$  The distribution ${\mathcal{D}}_{3}$ is totally geodesic.*

$(ii)$*  The distribution ${\mathcal{D}}_{\nu}$, $\nu=1,2,4$, is totally geodesic if and only if:*

$\bullet$*  $(M,g)$ is of positive constant sectional curvature $\chi$ and $t_{1}=t_{2}=\displaystyle{\frac{3}{8\chi}}$, in the case $\nu=1$;*

$\bullet$*  $(M,g)$ is anti-self-dual and Einstein with positive scalar curvature $s$, and $t_{1}=\displaystyle{\frac{6}{s}}$, in the case $\nu=2$ (no condition on $t_{2}>0$).*

$\bullet$*   $(M,g)$ is self-dual and Einstein with positive scalar curvature $s$, and $t_{2}=\displaystyle{\frac{6}{s}}$, in the case $\nu=4$ (no condition on $t_{1}>0$).*

Theorem 6**.**

(I)*  $(i)$  The distributions ${\mathcal{D}}_{\nu}^{\perp}$, $\nu=2,3,4$, are not integrable.*

$(ii)$*  The distribution ${\mathcal{D}}_{1}^{\perp}$ is integrable if and only $(M,g)$ is of constat curvature.*

(II)* All distributions ${\mathcal{D}}_{\nu}^{\perp}$ are minimal, $\nu=1,...,4$.*

(III)*  $(i)$   The distribution ${\mathcal{D}}_{1}^{\perp}$ is totally geodesic.*

$(ii)$*  The distribution ${\mathcal{D}}_{\nu}^{\perp}$, $\nu=2,3,4$, is totally geodesic if and only if:*

$\bullet$*  $(M,g)$ is self-dual and Einstein with positive scalar curvature $s$, and $t_{2}=\displaystyle{\frac{6}{s}}$, in the case $\nu=2$ (no condition on $t_{1}>0$).*

$\bullet$*  $(M,g)$ is of positive constant sectional curvature $\chi$ and $t_{1}=t_{2}=\displaystyle{\frac{3}{8\chi}}$, in the case $\nu=3$;*

$\bullet$*   $(M,g)$ is anti-self-dual and Einstein with positive scalar curvature $s$, and $t_{1}=\displaystyle{\frac{6}{s}}$, in the case $\nu=4$ (no condition on $t_{2}>0$).*

In the next theorem, we give a geometric interpretation of the Naveira classes of the Riemannian almost product manifolds $({\mathscr{P}},{\mathcal{K}}_{\nu},G_{\bf t})$ determined in Theorems 1-4.

Theorem 7**.**

(I)* $(i)$  The distributions ${\mathcal{D}}_{1}$ and ${\mathcal{D}}_{1}^{\perp}$ are both minimal *($({\mathcal{K}_{1}},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{2}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$).

$(ii)$ ${\mathcal{D}}_{1}$ and ${\mathcal{D}}_{1}^{\perp}$ are totally geodesic distributions ($({\mathcal{K}_{1}},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}$) if and only if $(M,g)$ is of positive constant curvature $\chi$ and $t_{1}=t_{2}=\frac{3}{8\chi}$.

(II) $(i)$  The distributions ${\mathcal{D}}_{2}$ and ${\mathcal{D}}_{2}^{\perp}$ are both minimal ($({\mathcal{K}_{2}},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{2}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$).

$(ii)$  ${\mathcal{D}}_{2}$ is totally geodesic and ${\mathcal{D}}_{2}^{\perp}$ is minimal ($({\mathcal{K}}_{2},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$) if and only if $(M,g)$ is anti-self-dual and Einstein with positive scalar curvature $s$, and $t_{1}=\displaystyle{\frac{6}{s}}$ (no condition on $t_{2}>0$).

$(iii)$  ${\mathcal{D}}_{2}$ and ${\mathcal{D}}_{2}^{\perp}$ are totally geodesic distributions ($({\mathcal{K}}_{2},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4})$ if and only if $(M,g)$ is of positive constant curvature $\chi$ and $t_{1}=t_{2}=\frac{3}{8\chi}$.

(III) $(i)$ The distribution ${\mathcal{D}}_{3}$ is totally geodesic and ${\mathcal{D}}_{3}^{\perp}$ is minimal

((${\mathcal{K}_{3}},G_{\bf t})\in{\mathcal{W}}_{1}\oplus{\mathcal{W}}_{4}\oplus{\mathcal{W}}_{5}$).

$(ii)$  ${\mathcal{D}}_{3}$ is integrable and totally geodesic, and ${\mathcal{D}}_{3}^{\perp}$ is minimal ($({\mathcal{K}}_{3},G_{\bf t})\in{\mathcal{W}}_{4}\oplus{\mathcal{W}}_{5}$) if and only if $(M,g)$ is of constat curvature

$(iii)$ ${\mathcal{D}}_{3}$ is integrable and totally geodesic, and ${\mathcal{D}}_{3}^{\perp}$ is totally geodesic ($({\mathcal{K}}_{3},G_{\bf t})\in{\mathcal{W}}_{4}$) if and only if $(M,g)$ is of positive constant sectional curvature $\chi$ and $t_{1}=t_{2}=\displaystyle{\frac{3}{8\chi}}$

(IV) $(i)$  The distributions ${\mathcal{D}}_{4}$ and ${\mathcal{D}}_{4}^{\perp}$ are both minimal ($({\mathcal{K}_{4}},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{2}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$).

$(ii)$  ${\mathcal{D}}_{4}$ is totally geodesic and ${\mathcal{D}}_{4}^{\perp}$ is minimal ($({\mathcal{K}}_{4},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4}\oplus{\mathscr{W}}_{5}$) if and only if $(M,g)$ is self-dual and Einstein with positive scalar curvature $s$, and $t_{2}=\displaystyle{\frac{6}{s}}$ (no condition on $t_{1}>0$).

$(iii)$  ${\mathcal{D}}_{4}$ and ${\mathcal{D}}_{4}^{\perp}$ are totally geodesic distributions ($({\mathcal{K}}_{4},G_{\bf t})\in{\mathscr{W}}_{1}\oplus{\mathscr{W}}_{4})$ if and only if $(M,g)$ is of positive constant curvature $\chi$ and $t_{1}=t_{2}=\frac{3}{8\chi}$.

It is a result of Hitchin (see [2, Theorem 13.30]) that every compact self-dual (anti-self-dual) Einstein manifold with positive scalar curvature is isometric to $S^{4}$ or ${\mathbb{C}}{\mathbb{P}}^{2}$ with their standard metrics and orientations (resp. the opposite orientations) (cf. also [8, 10]).

It is well known [1] that the twistor spaces ${\mathcal{Z}}_{\pm}$ of $S^{4}$ and ${\mathbb{C}}{\mathbb{P}}^{2}$ can be identified as smooth manifolds with ${\mathbb{C}}{\mathbb{P}}^{3}$ and the flag complex manifold $F_{3}$. The sphere $S^{4}$ is conformally flat, so the Atiyah-Hitchin-Singer almost complex structure on both twistor spaces ${\mathcal{Z}}_{+}$ and ${\mathcal{Z}}_{-}$ of $S^{4}$ is integrable. It coincides with the complex structure of ${\mathbb{C}}{\mathbb{P}}^{3}$. The manifold ${\mathbb{C}}{\mathbb{P}}^{2}$ with the orientation induced by its complex structure is self-dual, but not anti-self-dual. The Atiyah-Hitchin-Singer almost complex structure is integrable only on ${\mathcal{Z}}_{-}$ and it coincides on this twistor space with the complex structure of $F_{3}$. We recall now how the points of ${\mathbb{C}}{\mathbb{P}}^{3}$ and $F_{3}$ determine complex structures on the tangent spaces of the corresponding base manifolds compatible with the metric and $\pm$ the orientation.

In order to deal with the twistor space of $S^{4}$, we identify $S^{4}$ with quaternionic projective line ${\mathbb{H}}{\mathbb{P}}^{1}$. Writing quaternions as $z_{1}+z_{2}j$ with $z_{1},z_{2}\in{\mathbb{C}}$, the projection map $\pi:{\mathbb{C}}{\mathbb{P}}^{3}\to{\mathbb{H}}{\mathbb{P}}^{1}$ is given in homogeneous coordinates by $[z_{1},z_{2},z_{3},z_{4}]\to[z_{1}+z_{2}j,z_{3}+z_{4}j]$. We orient the space of quaternions by means of the basis $(1,i,j,k)$. Consider the sphere $S^{7}$ as a submanifold of ${\mathbb{H}}^{2}$. The group $Sp(1)$ of unit quaternions acts on $S^{7}$ by left multiplication and ${\mathbb{H}}{\mathbb{P}}^{1}$ is the quotient space of $S^{7}$ under this action of $Sp(1)$. Denote the quotient map by $\rho:S^{7}\to{\mathbb{H}}{\mathbb{P}}^{1}$. Let $\zeta=[z]\in{\mathbb{C}}{\mathbb{P}}^{3}$, where $z\in{\mathbb{C}}^{4}={\mathbb{H}}^{2}$ and $||z||=1$. Then $\rho(z)=\pi([z])$. Moreover $Ker\,\rho_{\ast\,z}=span\{iz,jz,kz\}\subset T_{z}S^{7}$. Let ${\mathcal{H}}_{z}$ be the orthogonal complement of $span\{iz,jz,kz\}$ in the tangent space $T_{z}S^{7}$. Then ${\mathcal{H}}_{z}$ is invariant under multiplications by $i,j,k$ and $\rho_{\ast\,z}|{\mathcal{H}}_{z}$ is an isomorphism onto $T_{\pi(\zeta)}{\mathbb{H}}{\mathbb{P}}^{1}$. Let $I$ be the complex structure on the vector space ${\mathcal{H}}_{z}$ defined by multiplication by $i$. Then $\widetilde{I}=\rho_{\ast\,z}\circ I\circ(\rho_{\ast\,z}|{\mathcal{H}}_{z})^{-1}$ is a complex structure on $T_{\pi(\zeta)}{\mathbb{H}}{\mathbb{P}}^{1}$ compatible with the metric and the orientation. If we consider the space ${\mathbb{H}}$ with the opposite orientation, then $\widetilde{I}$ is compatible with the metric and the opposite orientation of $T_{\pi(\zeta)}{\mathbb{H}}{\mathbb{P}}^{1}$. The complex structure $\widetilde{I}$ does not depend on the choice of a representative $z$ of the point $\zeta\in{\mathbb{C}}{\mathbb{P}}^{3}$. We refer to [18, Sec. 5.12] for details.

Now, consider the complex flag manifold $F=F_{3}$. Recall that its points are pairs $(l,m)$ of a complex line $l$ and a complex plane $m$ in ${\mathbb{C}}^{3}$ such that $l\subset m$. In this setting, the projection map $\pi:{\mathcal{Z}}_{\pm}=F\to{\mathbb{C}}{\mathbb{P}}^{2}$ is $(l,m)\to l^{\perp}\cap m$, where $l^{\perp}$ is the orthogonal complement of $l$ in ${\mathbb{C}}^{3}$ with respect to the standard Hermitian metric of ${\mathbb{C}}^{3}$. It is convenient to set $E_{1}=l$, $E_{2}=l^{\perp}\cap m$, $E_{3}=m^{\perp}$ so that to identify the points of $F$ with the triples $(E_{1},E_{2},E_{3})$ of mutually orthogonal complex lines in ${\mathbb{C}}^{3}$ with $\oplus_{i=1}^{3}E_{i}={\mathbb{C}}^{3}$. Then the projection map $\pi$ sends $\sigma=(E_{1},E_{2},E_{3})$ to $E_{2}$. Its fibre is $\{E_{1}:~{}E_{1}~{}\rm{a~{}complex~{}line~{}in}~{}E_{2}^{\perp}\}\cong{\mathbb{C}}{\mathbb{P}}^{1}$. The tangent space of the flag manifold $F$ at $\sigma$ is isomorphic to $Hom(E_{1},E_{2})\oplus Hom(E_{1},E_{3})\oplus Hom(E_{2},E_{3})$ (see, for example, [11]). The embedding of, say, $Hom(E_{1},E_{3})$ is defined as follows. For $f\in Hom(E_{1},E_{3})$ and $t\in{\mathbb{R}}$, let $\Gamma_{f}(t)=\{x+tf(x):~{}x\in E_{1}\}$ be the graph of the map $tf$ in $E_{1}\oplus E_{3}$. Then $c_{f}(t)=(\Gamma_{f}(t),E_{2},\Gamma_{f}(t)^{\perp})$ is a smooth curve in $F$ passing through $\sigma$, and the map $f\to\dot{c}_{f}(0)$ is an embedding of $Hom(E_{1},E_{3})$ into $T_{\sigma}F$. Similarly for $Hom(E_{1},E_{2})$ and $Hom(E_{2},E_{3})$. Clearly $\pi\circ c_{f}(t)\equiv E_{2}$. Therefore $Ker\,\pi_{\ast\,\sigma}=Hom(E_{1},E_{3})$ and the restriction of $\pi_{\ast\,\sigma}$ to $Hom(E_{1},E_{2})\oplus Hom(E_{2},E_{3})$ is a vector space isomorphism onto $T_{\pi(\sigma)}{\mathbb{C}}{\mathbb{P}}^{2}=Hom(E_{2},E_{1}\oplus E_{3})=Hom(E_{2},E_{1})\oplus Hom(E_{2},E_{3})$. In particular, we see that the map $\pi$ is neither holomorphic nor anti-holomorphic. The multiplication by $i$ in both $Hom(E_{1},E_{2})$ and $Hom(E_{2},E_{3})$ defines a complex structure on the vector space $Hom(E_{1},E_{2})\oplus Hom(E_{2},E_{3})$. Transferring this complex structure to $T_{\pi(\sigma)}{\mathbb{C}}{\mathbb{P}}^{2}$ by means of the map $\pi_{\ast\,\sigma}$, we obtain a complex structure compatible with the metric and the opposite orientation of ${\mathbb{C}}{\mathbb{P}}^{2}$. In order to obtain a complex structure compatible with the metric and the standard orientation of ${\mathbb{C}}{\mathbb{P}}^{2}$, we transfer the complex structure on $Hom(E_{1},E_{2})\oplus Hom(E_{2},E_{3})$, which is multiplication by $-i$ on $Hom(E_{1},E_{2})$ and by $i$ on $Hom(E_{2},E_{3})$.

Acknowledgements

The author would like to thank the referee for his/her remarks.