---
abstract: |
    We study the basic properties of an indefinite locally conformal Kähler (l.c. K.) manifold.  Any indefinite l.c. K. manifold $M$ with a parallel Lee form $\omega$ is shown to possess two canonical foliations $\mathcal F$ and ${\mathcal
    F}_c$, the first of which is given by the Pfaff equation $\omega =
    0$ and the second is spanned by the Lee and the anti-Lee vectors of $M$. We build an indefinite l.c. K. metric on the noncompact complex manifold $\Omega_+ = (\Lambda_+ \setminus \Lambda_0
    )/G_\lambda$ (similar to the Boothby metric on a complex Hopf manifold) and prove a CR extension result for CR functions on the leafs of $\mathcal F$ when $M = \Omega_+$ (where $\Lambda_+
    \setminus \Lambda_0 \subset {\mathbb C}^n_s$ is $- |z_1 |^2 -
    \cdots - |z_s |^2 + |z_{s+1}|^2 + \cdots + |z_n |^2 > 0$).  We study the geometry of the second fundamental form of the leaves of $\mathcal F$ and ${\mathcal F}_c$. In the degenerate cases (corresponding to a lightlike Lee vector) we use the technique of screen distributions and (lightlike) transversal bundles developed by A. Bejancu et al., [ @kn:BeDu]. 
---

Indefinite locally conformal Kähler manifolds

Sorin Dragomir[^1] Krishan L. Duggal[^2]

The first canonical foliation
=============================

Let $M$ be a complex $n$-dimensional indefinite Hermitian manifold of index $0 < \nu < 2n$, with the complex structure $J$ and the semi-Riemannian metric $g$. As well known $\nu$ must be even, $\nu
= 2s$. $M$ is an [*indefinite Kähler manifold*]{} if $\nabla J =
0$, where $\nabla$ is the Levi-Civita connection of $(M , g)$, cf.  M. Baros & A. Romero, [@kn:BaRo].  An indefinite Hermitian manifold $M$ is an [*indefinite locally conformal Kähler*]{} (l.c. K.) [*manifold*]{} if for any point $x \in M$ there is an open neighborhood $U$ of $x$ in $M$ and a $C^\infty$ function $f :
U \to {\mathbb R}$ such that $(U , e^{-f} g)$ is an indefinite Kähler manifold. 

Note that any two conformally related indefinite Kähler metrics are actually homothetic.  Indeed, let $(U , z^1 , \cdots , z^n )$ be a local system of complex coordinates on $M$ and set $g_{j\overline{k}} = g(Z_j , \overline{Z}_k )$, where $Z_j$ is short for $\partial /\partial z^j$ (and overbars denote complex conjugates).  If $\hat{g} = e^f g$ then the Levi-Civita connections $\nabla$ and $\hat{\nabla}$ (of $(M , g)$ and $(M , \hat{g})$, respectively) are related by $$\hat{\nabla}_{Z_j} \overline{Z}_k = \nabla_{Z_j} \overline{Z}_k
- \frac{1}{2} \{ (Z_j f) \overline{Z}_k + (\overline{Z}_k f) Z_j -
g_{j\overline{k}} \nabla f \}$$ where $\nabla f$ is the gradient of $f$ with respect to $g$, i.e. $g(\nabla f , X) = X(f)$, for any $X \in T(M)$. Let $T(M) \otimes {\mathbb C}$ be the complexified tangent bundle.  Let $Z^{1,0}$ denote the $(1,0)$-component of $Z
\in T(M) \otimes {\mathbb C}$ with respect to the direct sum decomposition $T(M) \otimes {\mathbb C} = T^{1,0}(M) \oplus
T^{0,1}(M)$, where $T^{1,0}(M)$ is the holomorphic tangent bundle over $M$ and $T^{0,1}(M) = \overline{T^{1,0}(M)}$. When $g$ and $\hat{g}$ are indefinite Kähler metrics, both $\nabla$ and $\hat{\nabla}$ descend to connections in $T^{0,1}(M)$ hence $$(\overline{Z}_k f) Z_j - g_{j\overline{k}} (\nabla f)^{1,0} = 0$$ or $\delta_j^s \overline{Z}_k f - g_{j\overline{k}}
g^{s\overline{r}} \overline{Z}_r f = 0$. Contraction of $s$ and $j$ leads to $(n-1) \overline{Z}_k f = 0$, i.e. $f$ is a real valued holomorphic function, hence a constant. 

Let $(M , J, g)$ be an indefinite l.c. K. manifold.  Let then $\{
U_i \}_{i \in I}$ be an open cover of $M$ and $\{ f_i \}_{i \in
I}$ a family of $C^\infty$ functions $f_i : U_i \to {\mathbb R}$ such that $g_i := e^{-f_i} g$ is an indefinite Kähler metric on $U_i$. Then $g_j = e^{f_i - f_j} g_i$ on $U_i \cap U_j$, i.e. $g_i$ and $g_j$ are conformally related indefinite Kähler metrics on $U_i \cap U_j$, hence $f_i - f_j = c_{ij}$, for some $c_{ij} \in {\mathbb R}$. In particular $d f_i = d f_j$, i.e. the local $1$-forms $d f_i$, $i \in I$, glue up to a globally defined closed $1$-form $\omega$ on $M$. By analogy with the positive definite case (cf.  e.g. S. Dragomir & L. Ornea, [@kn:DrOr]) we shall refer to $\omega$ as the [*Lee form*]{} of $M$. An indefinite l.c. K. metric $g$ is called [*globally conformal Kähler*]{} (g.c. K.) if the Lee form $\omega$ is exact (cf.  e.g. [@kn:Vai2] for the Riemannian case).  The tangent vector field $B$ on $M$ defined by $g(X , B) = \omega (X)$, for any $X \in
T(M)$, is the [*Lee field*]{}.  Let us set $c = g(B,B) \in C^\infty
(M)$ and ${\rm Sing}(B) = \{ x \in M : B_x = 0 \}$. Note that (opposite to the positive definite case) it may be $c = 0$ and ${\rm Sing}(B) = \emptyset$ (when $B$ is lightlike). 

Let $\nabla^i$ be the Levi-Civita connection of $(U_i , g_i )$, $i
\in I$. Then $$\nabla^i_X Y = \nabla_X Y - \frac{1}{2} \{ X(f_i ) Y + Y(f_i )
X - g(X,Y) \nabla f_i \}$$ for any $X, Y \in T(U_i )$, hence the local connections $\nabla^i$, $i \in I$, glue up to a globally defined linear connection $D$ on $M$ given by $$D_X Y = \nabla_X Y - \frac{1}{2}\{ \omega (X) Y + \omega (Y) X
- g(X,Y) B \} ,$$ the [*Weyl connection*]{} of $M$. Clearly $D J
= 0$.

Let us analyze indefinite l.c. K. manifolds with $\nabla \omega =
0$ (the indefinite counterpart of generalized Hopf manifolds, cf.  I. Vaisman, [@kn:Vai2]).  Such manifolds carry a natural foliation $\mathcal F$ defined by the Pfaff equation $\omega = 0$. Also $c \in {\mathbb R}$, so that $B$ is spacelike (respectively timelike, or lightlike) when $c > 0$ (respectively $c < 0$, or $c
= 0$).  We shall prove

Let $M$ be a complex $n$-dimensional indefinite l.c. K. manifold of index $2s$, $0 < s < n$, with a parallel Lee form and with ${\rm
Sing}(B) = \emptyset$. Then either [i)]{} $c \neq 0$ and then each leaf $L$ of $\mathcal F$ is a totally geodesic semi-Riemannian hypersurface of $(M , g)$ of index $$\label{e:1} {\rm ind}(L) =
\begin{cases} 2s & c
> 0 \cr 2s-1 & c < 0.  \cr
\end{cases}$$ or [ii)]{} $c = 0$ and then each leaf of $\mathcal F$ is a totally geodesic lightlike hypersurface of $(M , g)$. \[t:1\]

[*Proof*]{}.  Assume first that $c \neq 0$. Let us show that $$T(M) = T({\mathcal F}) \oplus {\mathbb R} B.
\label{e:2}$$ To this end, let $X \in T(M)$. Then $X - \frac{1}{c} \omega (X) B
\in T({\mathcal F})$. Moreover, if $X \in T({\mathcal F}) \cap
{\mathbb R}B$ then $X = \lambda B$, for some $\lambda \in C^\infty
(M)$, and $0 = \omega (X) = \lambda c$ yields $\lambda = 0$, i.e. $X = 0$. Therefore (\[e:2\]) holds.  Since ${\mathbb R}B$ is nondegenerate it follows that $T({\mathcal F}) = ({\mathbb
R}B)^\bot$ is nondegenerate, as well, hence each leaf of $\mathcal
F$ is a semi-Riemannian hypersurface of $(M , g)$ and (\[e:1\]) holds.  Let $L$ be a leaf of $\mathcal F$. Let $\nabla^L$ be the induced connection and $h^L$ the second fundamental form of $i : L
\hookrightarrow M$. If $X, Y \in T(L)$ then, by $\nabla \omega =
0$ and the Gauss formula $\nabla_X Y = \nabla^L_X Y + h^L (X,Y)$ $$0 = X(\omega (Y)) = \omega (\nabla_X Y ) = \omega (h^L (X,Y))$$ hence $h^L = 0$.

Let us assume now that $c = 0$, so that $B \in T({\mathcal F})$. Let us set as customary (cf.  e.g. [@kn:BeDu], p. 140) $${\rm Rad}\; T({\mathcal F})_x = T({\mathcal F})_x \cap
T({\mathcal F})^\bot_x \, , \;\;\; x \in M.$$ Clearly $B \in {\rm
Rad} \; T({\mathcal F})$. Note that ${\rm dim}_{\mathbb R} \;
T({\mathcal F})_x = 2n-1$ hence (cf.  e.g. Proposition 2.2 in [@kn:BeDu], p. 6) ${\rm dim}_{\mathbb R} \; T({\mathcal
F})^\bot_x = 1$, for any $x \in M$. Therefore, if ${\rm Sing}(B) =
\emptyset$ then $T({\mathcal F})^\bot = {\mathbb R}B$ and (by Proposition 1.1 in [@kn:BeDu], p. 78) each leaf of $\mathcal
F$ is a lightlike hypersurface of $(M , g)$.

If $Y \in T({\mathcal F})^\bot$ then $Y$ is orthogonal on the Lee field, hence $Y \in T({\mathcal F})$. It follows that $T({\mathcal
F})^\bot \subset T({\mathcal F})$, i.e. ${\rm Rad} \; T({\mathcal
F}) = T({\mathcal F})^\bot$. Let $S(T {\mathcal F})$ be a distribution on $M$ such that $$\label{e:3}
T({\mathcal F}) = S(T {\mathcal F}) \oplus_{\rm orth}
T({\mathcal F})^\bot .$$ If $V$ is a semi-Euclidean space and $W_a \subset V$, $a \in \{ 1
, 2 \}$, are two subspaces then we write $V = W_1 \oplus_{\rm
orth} W_2$ whenever $V = W_1 \oplus W_2$ and the subspaces $W_a$ are mutually orthogonal.  According to the terminology in [@kn:BeDu], p. 78, the portion of $S(T {\mathcal F})$ over a leaf $L$ of $\mathcal F$ is a [*screen distribution*]{} on $L$. The choice of $S(T {\mathcal F})$ is not unique, yet (by Proposition 2.1 in [@kn:BeDu], p. 5) $S(T {\mathcal F})$ is nondegenerate, hence $$T(M) = S(T{\mathcal F}) \oplus_{\rm orth} S(T{\mathcal
F})^\bot .  \label{e:4}$$ Note that $S(T{\mathcal F})^\bot$ has rank two and $T({\mathcal
F})^\bot \subset S(T{\mathcal F})^\bot$. The following result is an adaptation (to the foliation $\mathcal F$ of $M$, rather than a single lightlike hypersurface) of Theorem 1.1 in [@kn:BeDu], p. 79. 

Let $\pi : E \to M$ be a subbundle of $S(T{\mathcal F})^\bot \to M$ such that $S(T{\mathcal F})^\bot = T({\mathcal F})^\bot \oplus E$. Let $V \in \Gamma^\infty (U , E)$ be a locally defined nowhere zero section, defined on the open subset $U \subseteq M$. Then [i)]{} $\omega (V) \neq 0$ everywhere on $U$. Let us consider $N_V \in \Gamma^\infty (U , S(T{\mathcal F})^\bot )$ given by $$\label{e:5} N_V = \frac{1}{\omega (V)} \left\{ V - \frac{g(V,V)}{2
\omega (V)} \; B \right\} .$$ If $V^\prime \in \Gamma^\infty (U^\prime , E)$ is another nowhere zero section, defined on the open subset $U^\prime \subseteq M$ such that $U \cap U^\prime \neq \emptyset$, then [ii)]{} $N_V =
N_{V^\prime}$ on $U \cap U^\prime$. Moreover, let $x \in M$ and $U
\subseteq M$ an open neighborhood of $x$ such that $\left. 
E\right|_U = \pi^{-1}(U)$ is trivial.  Let $V$ us set $${\rm tr}(T {\mathcal F})_x = {\mathbb R} N_V (x) \, . 
\label{e:6}$$ Then [iii)]{} ${\rm tr}(T {\mathcal F})_x$ is well defined and gives a lightlike subbundle ${\rm tr}(T{\mathcal F}) \to M$ of $S(T{\mathcal F})^\bot \to M$ such that $$S(T{\mathcal F})^\bot = T({\mathcal F})^\bot \oplus {\rm tr}(T
{\mathcal F}).  \label{e:7}$$ Finally, [iv)]{} the definition of ${\rm tr}(T {\mathcal F})$ doesn’t depend upon the choice of complement $E$ to $T({\mathcal
F})^\bot$ in $S(T {\mathcal F})^\bot$. \[l:1\]

[*Proof*]{}.  The proof of (i) is by contradiction.  If $\omega (V)_{x_0} = 0$ for some $x_0 \in U$ then $V_{x_0} \in
T({\mathcal F})_{x_0}$ and then (by (\[e:3\])) $$V_{x_0} \in S(T{\mathcal F})_{x_0} \cap S(T{\mathcal
F})^\bot_{x_0} = (0),$$ a contradiction.  To prove (ii) let $V^\prime$ be a nowhere zero section in $E$ on $U^\prime$ (with $U
\cap U^\prime \neq \emptyset$).  Then $V^\prime = \alpha V$, for some $C^\infty$ function $\alpha : U \cap U^\prime \to {\mathbb R}
\setminus \{ 0 \}$, and an inspection of (\[e:5\]) leads to $N_V
= N_{V^\prime}$ on $U \cap U^\prime$. This also shows that the definition of ${\rm tr}(T {\mathcal F})_x$ doesn’t depend upon the particular local trivialization chart of $E$ at $x$. To check the remaining statement in (iii) note first that $$g(N_V , N_V ) = 0, \;\;\; \omega (N_V ) = 1.  \label{e:8}$$ The first relation in (\[e:8\]) shows that ${\rm tr}(T{\mathcal
F})$ is lightlike, while the second relation yields $T({\mathcal
F})^\bot \cap {\rm tr}(T {\mathcal F}) = (0)$. Yet both bundles have rank one, hence (\[e:7\]) holds.  Finally, if $F \to M$ is another complement to $T({\mathcal F})^\bot$ in $S(T{\mathcal
F})^\bot$ then an [*a priori*]{} new lightlike bundle, similar to ${\rm tr}(T{\mathcal F})$, may be build in terms of a local section $W \in \Gamma^\infty (U , F)$. Yet (by (\[e:7\])) $W =
\alpha N_V + \beta B$, for some $\alpha , \beta \in C^\infty (U)$, hence (by (\[e:8\])) $N_W = N_V$. $\square$ 0.1in According to the terminology in [@kn:BeDu], p. 79, the portion of ${\rm tr}(T{\mathcal F})$ over a leaf $L$ of $\mathcal F$ is the [*lightlike transversal vector bundle*]{} of $L$ with respect to the screen distribution $\left.  S(T{\mathcal F})\right|_L$. By (\[e:3\])-(\[e:4\]) and (\[e:7\]) we obtain the decomposition $$T(M) = S(T{\mathcal F}) \oplus_{\rm orth} \left[ T({\mathcal
F})^\bot \oplus {\rm tr}(T{\mathcal F})\right] = T({\mathcal F})
\oplus {\rm tr}(T{\mathcal F}).  \label{e:9}$$ Let ${\rm tan} : T(M) \to T({\mathcal F})$ and ${\rm tra} : T(M)
\to {\rm tr}(T {\mathcal F})$ be the projections associated with (\[e:9\]).  Next, we set $$\nabla^{\mathcal F}_X Y = {\rm tan} \left( \nabla_X Y \right) ,
\;\;\; h (X,Y) = {\rm tra}\left( \nabla_X Y \right),$$ $$A_V X = - {\rm tan}\left( \nabla_X V \right) ,
\;\;\; \nabla^{\rm tr}_X V = {\rm tra} \left( \nabla_X V \right) ,$$ for any $X, Y \in T({\mathcal F})$ and any $V \in {\rm
tr}(T{\mathcal F})$. Then $\nabla^{\mathcal F}$ is a connection in $T({\mathcal F}) \to M$, $h$ is a symmetric ${\rm tr}(T{\mathcal
F})$-valued bilinear form on $T({\mathcal F})$, $A_V$ is an endomorphism of $T({\mathcal F})$, and $\nabla^{\rm tr}$ is a connection in ${\rm tr}(T{\mathcal F}) \to M$. Also one has $$\nabla_X Y = \nabla^{\mathcal F}_X Y + h(X,Y), \;\;\; \nabla_X
V = - A_V X + \nabla^{\rm tr}_X V ,$$ the [*Gauss*]{} and [*Weingarten formulae*]{} of $\mathcal F$ in $(M , g)$. Clearly, the pointwise restrictions of $\nabla^{\mathcal F}$, $\nabla^{\rm
tr}$, $h$ and $A_V$ to a leaf $L$ of $\mathcal F$ are respectively the induced connections, the second fundamental form and the shape operator of $L$ in $(M , g)$, cf. [ @kn:BeDu], p. 83.  A leaf $L$ is [*totally geodesic*]{} if each geodesic of $\nabla^{\mathcal F}$ lying on $L$ is also a geodesic of the semi-Riemannian manifold $(M , g)$.

Let us prove the last statement in Theorem \[t:1\].  As $\nabla
\omega = 0$ it follows that $\omega (h(X,Y)) = 0$. Yet locally (with the notations in the proof of Lemma \[l:1\]) $h(X,Y) =
C(X,Y) N_V$, for some $C(X,Y) \in C^\infty (U)$, hence (by (\[e:8\])) $h = 0$ and then by a result in [@kn:BeDu1] (cf.  also Theorem 2.2 in [@kn:BeDu], p. 88) each leaf of $\mathcal
F$ is totally geodesic in $(M , g)$. $\square$ 0.1in We end this section by the following remark.  By Theorem 2.2 in [@kn:BeDu], p. 88, if $c = 0$ then $\nabla^{\mathcal F}$ is the Levi-Civita connection of the tangential metric induced by $g$ on $T({\mathcal F})$ and the distribution $T({\mathcal F})^\bot$ is Killing. 

Indefinite Hopf manifolds
=========================

Let ${\mathbb C}^n_s$ denote ${\mathbb C}^n$ together with the real part of the Hermitian form $$b_{s,n} (z,w) = - \sum_{j=1}^s z_j \overline{w}_j +
\sum_{j=s+1}^n z_j \overline{w}_j \, , \;\;\; z,w \in {\mathbb
C}^n .$$ Let $\Lambda = \{ z \in {\mathbb C}^n \setminus \{ 0 \}
: - \sum_{j=1}^s |z_j |^2 + \sum_{j=s+1}^n |z_j |^2 = 0 \}$ be the null cone in ${\mathbb C}^n_s$ and $\Lambda_0 = \Lambda \cup \{ 0
\}$. Given $\lambda \in {\mathbb C} \setminus \{ 0 \}$ $$F_\lambda (z) = \lambda z, \;\;\; z \in {\mathbb C}^n \setminus
\Lambda_0 \, ,$$ is a holomorphic transformation of ${\mathbb
C}^n \setminus \Lambda_0$. Let $G_\lambda = \{ F_\lambda^m : m \in
{\mathbb Z} \}$ be the discrete group generated by $F_\lambda$. Then

Let $n > 1$, $0 < s < n$ and $\lambda \in {\mathbb C} \setminus \{ 0 \}$, $|\lambda | \neq 1$. Then $G_\lambda$ acts freely on ${\mathbb C}^n \setminus
\Lambda_0$ as a properly discontinuous group of holomorphic transformations, hence the quotient space ${\mathbb C}H^n_s
(\lambda ) = ({\mathbb C}^n \setminus \Lambda_0 )/G_\lambda$ is a complex manifold and $$g_{s,n} = |z|_{s,n}^{-2} \left( - \sum_{j=1}^s d z^j \odot d
\overline{z}^j + \sum_{j=s+1}^n d z^j \odot d \overline{z}^j
\right)
\label{e:10}$$ [(]{}where $|z|_{s,n} =
|b_{s,n}(z,z)|^{1/2}$[)]{} is a globally defined semi-Riemannian metric, making ${\mathbb C}H^n_s (\lambda )$ into an indefinite locally conformal Kähler manifold.  Moreover, if $0 < \lambda <
1$ then ${\mathbb C}H^n_s (\lambda ) \approx \Sigma^{2n-1} \times
S^1$ [(]{}a diffeomorphism[)]{}, where $\Sigma^{2n-1} = \{ z
\in {\mathbb C}^n : |z|_{s,n} = 1 \}$. In particular ${\mathbb
C}H^n_s (\lambda )$ is noncompact.  If $\Lambda_+ = \{ z \in
{\mathbb C}^n : b_{s,n}(z,z) \geq 0 \}$ and $\Lambda_- = \{ z \in
{\mathbb C}^n : b_{s,n}(z,z) \leq 0 \}$ [(]{}so that $\partial
\Lambda_\pm = \Lambda_0$[)]{} then ${\mathbb C}H^n_s (\lambda )$ consists of the two connected components $(\Lambda_+ \setminus
\Lambda_0 )/G_\lambda \approx S^{2n-1}_{2s} \times S^1$ and $(\Lambda_- \setminus \Lambda_0 )/G_\lambda \approx
H^{2n-1}_{2s-1} \times S^1$. \[t:2\]

If ${\mathbb R}^N_\nu = ({\mathbb R}^N , h_{\nu , N})$, with $h_{\nu , N} (x,y) = - \sum_{j=1}^\nu x_j y_j + \sum_{j=\nu
+1}^N x_j y_j$, then $S^N_\nu (r) = \{ x \in {\mathbb R}^{N+1} :
h_{\nu , N+1}(x,x) = r^2 \}$ ($r > 0$) is the pseudosphere in ${\mathbb R}^{N+1}_\nu$, while $H^N_\nu (r) = \{ x \in {\mathbb
R}^{N+1} : h_{\nu + 1, N+1} (x,x) = - r^2 \}$ ($r > 0$) is the pseudohyperbolic space in ${\mathbb R}^{N+1}_{\nu +1}$. When $r =
1$ we write simply $S^N_\nu$ and $H^N_\nu$. Also $\odot$ denotes the symmetric tensor product, e.g. $\alpha \odot \beta =
\frac{1}{2} (\alpha \otimes \beta + \beta \otimes \alpha )$ for any $1$-forms $\alpha$ and $\beta$. A construction similar to that in Theorem \[t:2\] was performed in [@kn:DrFr] (for metrics which are locally conformal to anti-Kählerian metics, cf.  Lemma 3, [*op.  cit.*]{},  p. 119). 

[*Proof of Theorem*]{} \[t:2\].  If $F_\lambda^m (z) = z$ for some $z \in {\mathbb C}^n \setminus \Lambda_0$ then $m = 0$, hence $G_\lambda$ acts freely on ${\mathbb C}^n \setminus
\Lambda_0$. Given $z_0 \in {\mathbb C}^n \setminus \Lambda_0$ let $B_r (z_0 )$ be the open Euclidean ball of center $z_0$ and radius $r$. Also, we set $\Omega_r (z_0 ) = B_r (z_0 ) \setminus
\Lambda_0$. As well known (cf. [ @kn:KoNo], Vol.  II, p. 137) $G_\lambda$ acts on ${\mathbb C}^n \setminus \{ 0 \}$ as a properly discontinuous group of holomorphic transformations, hence there is $r > 0$ such that $F^m_\lambda (B_r (z_0 )) \cap B_r (z_0
) = \emptyset$, and then $F^m_\lambda (\Omega_r (z_0 )) \cap
\Omega_r (z_0 ) = \emptyset$, for any $m \in {\mathbb Z} \setminus
\{ 0 \}$. As ${\mathbb C}^n \setminus \Lambda_0$ is an open subset of ${\mathbb C}^n$ it follows (cf.  e.g. [@kn:Boo], p. 97) that ${\mathbb C}H^n_s (\lambda ) := ({\mathbb C}^n \setminus \Lambda_0
)/G_\lambda$ is a complex manifold. 

Assume from now on that $0 < \lambda < 1$. Let $\pi : {\mathbb
C}^n \setminus \Lambda_0 \to {\mathbb C}H^n_s (\lambda )$ be the projection.  To prove the last statement in Theorem \[t:2\] we consider the $C^\infty$ diffeomorphism $$F : {\mathbb C}H^n_s (\lambda ) \to \Sigma^{2n-1} \times S^1 ,$$ $$\label{e:diff}
F(\pi (z)) = \left( |z|_{s,n}^{-1} \, z \; , \; \exp \left(
\frac{2\pi i \log |z|_{s,n}}{\log \lambda} \right) \right) ,$$ with the obvious inverse $$F^{-1}(\zeta , w) = \pi \left( \lambda^{\arg (w)/(2\pi )} \,
\zeta \right) , \;\;\; \zeta \in \Sigma^{2n-1} , \;\; w \in S^1 ,$$ where $\arg : {\mathbb C} \to [0, 2 \pi )$.  Finally, note that $\Sigma^{2n-1} \cap \Lambda_+ = S^{2n-1}_{2s}$ and $\Sigma^{2n-1}
\cap \Lambda_- = H^{2n-1}_{2s-1}$. $\square$

The Lee form of $({\mathbb C}H^n_s (\lambda ),
g_{s,n})$ is locally given by $$\omega = - d \log |z|^2_{s,n} .  \label{e:11}$$ In particular ${\mathbb C}H^n_s (\lambda )$ has a parallel Lee form.  Let $\Omega_{\pm} = (\Lambda_\pm \setminus \Lambda_0
)/G_\lambda$ be the connected components of ${\mathbb C}H^n_s
(\lambda )$ and $a(z) = {\rm sign}(b_{s,n}(z,z)) = \pm 1$ for $z
\in \Lambda_\pm \setminus \Lambda_0$. Then the Lee field $B$ of $({\mathbb C}H^n_s (\lambda ) , g_{s,n})$ is given by $$B =  - 2 a(z) \left( z^j \frac{\partial}{\partial
z^j} + \overline{z}^j \frac{\partial}{\partial \overline{z}^j}
\right) .  \label{e:12}$$ Finally, if $\left.  B_\pm = B\right|_{\Omega_\pm}$ then $B_+$ is spacelike while $B_-$ is timelike.  \[p:1\]

[*Proof*]{}.  An inspection of (\[e:10\]) leads to (\[e:11\]) and hence to $$\omega = b_{s,n}(z,z)^{-1} \left\{ \sum_{j=1}^s (\overline{z}_j d z^j +
z_j d \overline{z}^j ) - \sum_{j= s+1}^n (\overline{z}_j d z^j +
z_j d \overline{z}^j ) \right\} ,$$ with the convention $z_j =
z^j$. Next $$g_{j\overline{k}} = \frac{1}{2} \, |z|_{s,n}^{-2} \,
\epsilon_j\,  \delta_{jk}$$ (where $\epsilon_j = -1$ for $1 \leq
j \leq s$ and $\epsilon_j = 1$ for $s+1 \leq j \leq n$) yields (\[e:12\]).  Since $$Z_j \left( |z|^{-2}_{s,n} \right) = - a(z)
|z|_{s,n}^{-4} \, \epsilon_j \, \overline{z}_j$$ the identity $$2 g_{AD} \Gamma_{BC}^A  = Z_B (g_{CD}) + Z_C (g_{BD}) - Z_D
(g_{BC})$$ leads to $$\Gamma^\ell_{jk} = - \frac{a(z)}{2 |z|^2_{s,n}}
\left( \epsilon_j \overline{z}_j \delta_k^\ell + \epsilon_k
\overline{z}_k \delta_j^\ell \right) , \;\;\;
\Gamma^{\overline{\ell}}_{jk} = 0,$$ $$\Gamma_{j\overline{k}}^\ell =
\frac{a(z)}{2 |z|^2_{s,n}} \left( \epsilon_j \delta_{jk} z^\ell -
\epsilon_k z_k \delta_j^\ell \right) , \;\;
\Gamma^{\overline{\ell}}_{j\overline{k}} =  \frac{a(z)}{2
|z|^2_{s,n}} \left( \epsilon_j \delta_{jk} \overline{z}^\ell -
\epsilon_j \overline{z}_j \delta^\ell_k \right) ,$$ hence a calculation shows that $\nabla_{Z_j} B = 0$. Finally $g_{s,n}(B,B)
= 4 a(z)$. $\square$

0.1in Given two semi-Riemannian manifolds $M$ and $N$ and a $C^\infty$ submersion $\Pi : M \to N$, we say $\Pi$ is a [*semi-Riemannian submersion*]{} if 1) $\Pi^{-1} (y)$ is a semi-Riemannian submanifold of $M$ for each $y \in N$, and 2) $d_x
\Pi : {\mathcal H}_x \to T_{\Pi (x)} (N)$ is a linear isometry of semi-Euclidean spaces, where ${\mathcal H}_x = {\rm Ker}(d_x \Pi
)^\bot$, for any $x \in M$ (cf. [ @kn:O'Ne], p. 212).  Also we recall (cf. [ @kn:BaRo]) the indefinite complex projective space ${\mathbb C}P^{n-1}_s (k)$. Its underlying complex manifold is the open subset of the complex projective space $${\mathbb C}P^{n-1}_s (k)  = (\Lambda_+ \setminus \Lambda_0
)/{\mathbb C}^* \subset {\mathbb C}P^{n-1} \;\;\;\;\; ({\mathbb
C}^* = {\mathbb C} \setminus \{ 0 \} ).$$ As to the semi-Riemannian metric of ${\mathbb C}P^{n-1}_s (k)$, let $$\Pi : S^{2n-1}_{2s} ( 2/\sqrt{k}) \to
{\mathbb C}P^{n-1}_s (k), \;\;\; \Pi (z) = z \cdot {\mathbb C}^*
\;\;\; (k > 0)$$ be the [*indefinite Hopf fibration*]{}.  It is a principal $S^1$-bundle and $S^1$ acts on $S^{2n-1}_{2s}(2/\sqrt{k})$ as a group of isometries, hence (by slightly adapting the proof of Proposition E.3 in [@kn:BGM], p. 7, to the semi-Riemannian context) there is a unique semi-Riemannian metric of index $2s$ on ${\mathbb C}P^{n-1}_s (k)$ such that $\Pi$ is a semi-Riemannian submersion and ${\mathbb C}P^{n-1}_s (k)$ is an indefinite complex space form of (constant) holomorphic sectional curvature $k$. Again by [@kn:BaRo], p. 57, ${\mathbb C}P^{n-1}_s (k)$ is homotopy equivalent to ${\mathbb C}P^{n-1-s}$, hence ${\mathbb
C}P^{n-1}_s (k)$ is simply connected.  We shall prove

Let $D = \{ 2\pi i a + (\log \lambda ) b : a , b
\in {\mathbb Z} \}$ [(]{}$0 < \lambda < 1$[)]{} and consider the torus $T^1_{\mathbb C} = {\mathbb C}/D$. Then $T^1_{\mathbb
C}$ acts freely on ${\mathbb C}H^n_s (\lambda )$ and $$p : \Omega_+ \to {\mathbb C}P^{n-1}_s (4), \;\;\; p(\pi (z)) =
z \cdot {\mathbb C}^* ,$$ is a principal $T^1_{\mathbb C}$-bundle and a semi-Riemannian submersion of $\Omega_+$ [(]{}carrying the indefinite l.c. K. metric $g_{s,n}$[)]{} onto ${\mathbb
C}P^{n-1}_s (4)$. Moreover the complex Hopf manifold ${\mathbb
C}H^{n-s}(\lambda )$ [(]{}respectively ${\mathbb C}H^s (\lambda
)$[)]{} is a strong deformation retract of $\Omega_+$ [(]{}respectively of $\Omega_-$[)]{} hence $$H_k (\Omega_+ ; {\mathbb Z}) = \begin{cases} {\mathbb Z}
\otimes {\mathbb Z}, & k = 2(n-s), \cr {\mathbb Z}, & k \neq
2(n-s), \cr
\end{cases}$$ $$H_k (\Omega_- ; {\mathbb Z}) = \begin{cases} {\mathbb Z}
\otimes {\mathbb Z} , & k = 2s, \cr {\mathbb Z}, & k \neq 2s, \cr
\end{cases}$$ and $\Omega_{\pm}$ are not simply connected $$\pi_1 (\Omega_+ ) = \begin{cases} {\mathbb Z} \oplus {\mathbb
Z}, & s = n-1, \cr {\mathbb Z}, & s \neq n-1, \cr \end{cases}
\;\;\; \pi_1 (\Omega_- ) = \begin{cases} {\mathbb Z} \oplus
{\mathbb Z}, & s = 1, \cr {\mathbb Z}, & s \neq 1.  \cr \end{cases}$$ Each fibre $p^{-1}(z \cdot {\mathbb C}^* )$, $z \in S^{2n-1}_{2s}$, is tangent to the Lee field $B$ of $\Omega_+$ hence $p : \Omega_+ \to
{\mathbb C}P^{n-1}_s (4)$ is a harmonic map.  \[t:3\]

[*Proof*]{}.  The action of $T^1_{\mathbb C}$ on ${\mathbb C}H^n_s (\lambda )$ is given by $$\pi (z) \cdot (\zeta + D) = \pi \left( e^\zeta \; z \right) ,
\;\;\; z \in {\mathbb C}^n \setminus \Lambda_0 , \;\; \zeta \in
{\mathbb C} .$$ As $b_{s,n}(e^\zeta z , e^\zeta z) = e^{2 {\rm
Re}(\zeta )} b_{s,n}(z,z) \neq 0$ the action is well defined.  To see that the action is free, let us assume that $\pi (e^\zeta z_0
) = \pi (z_0 )$, for some $z_0 \in {\mathbb C}^n \setminus
\Lambda_0$. Then $e^\zeta _0 = \lambda^m z_0$, for some $m \in
{\mathbb Z}$, hence $\zeta = m \log \lambda + 2k \pi i$, for some $k \in {\mathbb Z}$, i.e. $\zeta + D = 0$.

To see that $S^1 \to \Omega_+ \stackrel{p}{\to} {\mathbb
C}^{n-1}_s (4)$ is a principal bundle let us assume that $p(\pi (z
)) = p( \pi (z^\prime ))$, with $z, z^\prime \in {\mathbb C}^n
\setminus \Lambda_0$. Then $z^\prime = \alpha z$, for some $\alpha
\in {\mathbb C}^*$. We wish to show that there is a unique $\zeta
+ D \in T^1_{\mathbb C}$ such that $\pi (z^\prime ) = \pi (z)
\cdot (\zeta + D)$. Indeed we may consider $\zeta = \log |\alpha |
+ i \arg (\alpha )$. 

Let $F_t : \Lambda_+ \setminus \Lambda_0 \to {\mathbb C}^n$, $0
\leq t \leq 1$, be given by $$F_t (z) = ((1-t) z^\prime \, , \, z^{\prime\prime}), \;\;\; z =
(z^\prime , z^{\prime\prime}) \in \Lambda_+ \setminus \Lambda_0 ,$$ where $z^\prime = (z_1 , \cdots , z_s )$ and $z^{\prime\prime} =
(z_{s+1} , \cdots , z_n )$.  Then $$b_{s,n}(F_t (z), F_t (z)) = - (1-t)^2 |z^\prime |^2 +
|z^{\prime\prime}|^2 \geq b_{s,n}(z,z) > 0$$ hence $F_t$ is $(\Lambda_+ \setminus \Lambda_0 )$-valued.  Therefore $F_t$ induces a homotopy $$H^+_t : \Omega_+ \to \Omega_+ , \;\;\; H^+_t (\pi (z)) = \pi \left(
F_t (z) \right) , \;\;\; 0 \leq t \leq 1.$$ Let us consider (cf.  e.g. [@kn:KoNo], Vol.  II, p. 137) the complex Hopf manifold ${\mathbb C}H^n (\lambda ) = ({\mathbb C}^n \setminus \{ 0 \}
)/G_\lambda$ and denote by $\pi_0 : {\mathbb C}^n \setminus \{ 0
\} \to {\mathbb C}H^n (\lambda )$ the projection.  Let ${\mathbb
C}H^{n-s}(\lambda )$ be thought of as identified to $$\{ \pi_0 (z) \in
{\mathbb C}H^n (\lambda ) : z_1 = 0, \cdots , z_s = 0 \} .$$ Note that ${\mathbb C}H^{n-s}(\lambda ) \subset \Omega_+$. Also $$H^+_0 = 1_{\Omega_+} \, , \;\;\;
H^+_1 \left( \Omega_+ \right) \subset {\mathbb C}H^{n-s}(\lambda
),$$ $$\left.  H^+_t \right|_{{\mathbb C}H^{n-s}(\lambda )} = i, \;\;\; 0 \leq t
\leq 1,$$ (where $i : {\mathbb C}H^{n-s} (\lambda ) \to \Omega_+$ is the inclusion) hence ${\mathbb C}H^{n-s}(\lambda )$ is a strong deformation retract of $\Omega_+$. Also $H^+_1 \circ i =
1_{{\mathbb C}H^{n-s}(\lambda )}$ and $H^+ : 1_{\Omega_+} \simeq i
\circ H^+_1$ (i.e. the maps $1_{{\mathbb C}H^{n-s}(\lambda )}$ and $i \circ H^+_1$ are homotopic) so that $i, H^+_1$ are reciprocal homotopy equivalences, i.e. $$\Omega_+ \simeq {\mathbb C}H^{n-s}(\lambda ),
\label{e:13}$$ (a homotopy equivalence).  As well known, (\[e:13\]) implies that $$i_* : H_k ({\mathbb C}H^{n-s}(\lambda ) ; {\mathbb Z}) \approx
H_k (\Omega_+ ; {\mathbb Z}),$$ (a group isomorphism).  Therefore, to compute $H_k (\Omega_+ ; {\mathbb Z})$ it suffices to compute the singular homology of the complex Hopf manifold.  This is an easy exercise in algebraic topology (based on the Künneth formula).  Indeed $$H_k ({\mathbb C}H^n (\lambda ) ; {\mathbb Z}) =
\sum_{p+q=k} H_p (S^{2n-1}; {\mathbb Z}) \otimes H_q (S^1 ;
{\mathbb Z}) =$$ $$= H_{k-1}(S^{2n-1} ; {\mathbb Z}) \otimes H_1 (S^1 ; {\mathbb
Z}) = \begin{cases} {\mathbb Z} \otimes {\mathbb Z} , & k = 2n,
\cr {\mathbb Z}, & k \neq 2n, \cr \end{cases}$$ yielding (\[e:11\]).  As to the homotopy groups, again by (\[e:13\]) $$\pi_k (\Omega_+ ) \approx \pi_k ({\mathbb C}H^{n-s})$$ (a group isomorphism) and if $n > 1$ $$\pi_k ({\mathbb C}H^n (\lambda )) = \pi_k (S^{2n-1}) \oplus
\pi_k (S^1 ) = \begin{cases} {\mathbb Z}, & k \in \{ 1 , 2n-1 \} ,
\cr 0, & k \not\in \{ 1 , 2n-1 \} , \cr \end{cases}$$ while if $n
= 1$ $$\pi_k ({\mathbb C}H^1 (\lambda )) = \pi_k (S^1 ) \oplus \pi_k
(S^1 ) = \begin{cases} {\mathbb Z} \oplus {\mathbb Z}, & k = 1,
\cr 0, & k \neq 1.  \cr \end{cases}$$ Let us show now that $p :
\Omega_+ \to {\mathbb C}P^{n-1}_s (4)$ is a semi-Riemannian submersion.  Let $i : S^{2n-1}_{2s} \to \Lambda_+ \setminus
\Lambda_0$ be the inclusion and $V_{z_0} = {\rm Ker}(d_{z_0} \Pi
)$ the vertical space, $z_0 \in S^{2n-1}_{2s}$. As $\Pi :
S^{2n-1}_{2s} \to {\mathbb C}P^{n-1}_s (4)$ is a semi-Riemannian submersion $V_{z_0}$ is nondegenerate, hence the perp space $H_{z_0}$ of $V_{z_0}$ is also nondegenerate and $$T_{z_0} (S^{2n-1}_{2s}) = H_{z_0} \oplus_{\rm orth} V_{z_0} .$$ Let $V_{0, \pi (z_0 )} = {\rm Ker}(d_{\pi (z_0 )} p)$ and let $N(S^{2n-1}_{2s})_{z_0}$ be the normal space of $i$ at $z_0$ (as $S^{2n-1}_{2s}$ has index $2s$ it follows that $N(S^{2n-1}_{2s})_{z_0}$ has index zero). 

For any $z_0 \in S^{2n-1}_{2s}$ $$V_{0, \pi (z_0 )} = (d_{z_0} \pi ) \{ N(S^{2n-1}_{2s})_{z_0}
\oplus (d_{z_0} i ) V_{z_0} \} .$$ \[l:2\]

[*Proof*]{}.  Let $N = z^j Z_j + \overline{z}^j
\overline{Z}_j$ be the unit normal on $S^{2n-1}_{2s}$ in $\Lambda_+ \setminus \Lambda_0$, with the flat indefinite Kähler metric $$g_0 = \sum_{j=1}^n \epsilon_j d z^j \odot d \overline{z}^j .$$ Let $U$ be the tangent vector field on $S^{2n-1}_{2s}$ defined by $(d i) U = - J N$, where $J$ is the complex structure on ${\mathbb
C}^n$. Let $$\Pi_0 : \Lambda_+ \setminus \Lambda_0 \to {\mathbb
C}P^{n-1}_s (4)$$ be the canonical projection (so that $\Pi =
\left.  \Pi_0 \right|_{S^{2n-1}_{2s}}$).  Then $(d \Pi_0 ) U = 0$ hence $V_{z_0} = {\mathbb R} U_{z_0}$. In particular, by the commutativity of the diagram $$\begin{array}{ccccc} {\mathbb C}H^n_s (\lambda ) & \supset &
\;\;\;\;\; \Omega_+ & \stackrel{p}{\longrightarrow} & {\mathbb
C}P^{n-1}_s (4)
\\
\empty & \empty & \pi \uparrow & \empty & \uparrow \Pi \\
{\mathbb C}^n_s & \supset & \;\;\; \Lambda_+ \setminus \Lambda_0 &
\stackrel{i}{\longleftarrow} & S^{2n-1}_{2s} \end{array}$$ it follows that $$(d_{z_0} \pi )(d_{z_0} i) V_{z_0} \subseteq V_{0,
\pi (z_0 )} .  \label{e:14}$$ On the other hand $\Pi_0$ is a holomorphic map hence $$(d_{\pi (z_0 )} p)(d_{z_0} \pi ) N_{z_0} = (d_{z_0} \Pi_0 )
J_{z_0} (d_{z_0} i) U_{z_0} =$$ $$= J^\prime_{\Pi_0 (z_0 )} (d_{z_0} \Pi_0 )(d_{z_0} i) U_{z_0} =
J^\prime_{\Pi (z_0 )} (d_{z_0} \Pi )U_{z_0} = 0,$$ where $J^\prime$ denotes the complex structure on ${\mathbb C}P^{n-1}$. We obtain $$(d_{z_0} \pi ) N(S^{2n-1}_{2s})_{z_0} \subseteq V_{0, \pi (z_0 )}
.  \label{e:15}$$ At this point Lemma \[l:2\] follows from (\[e:14\])-(\[e:15\]) and an inspection of dimensions.  $\square$

As $|z_0 |_{s,t} = 1$ the indefinite scalar product $(\; , \;
)_{z_0}$ induced on $T_{z_0} (\Lambda_+ \setminus \Lambda_0 )$ by $g_{0, z_0}$ coincides with that induced by $|z|_{s,t}^{-2} \sum_j
\epsilon_j d z^j \odot d \overline{z}^j$ at $z_0$.

$V_{0, \pi (z)}$ is nondegenerate, for any $z \in \Lambda_+ \setminus
\Lambda_0$. \[l:3\]

[*Proof*]{}.  Let $A = - J B$, where $J$ is the complex structure on $\Omega_+$. Then $\{ A_{\pi (z)} , B_{\pi (z)} \}$ span $V_{0, \pi (z)}$ and (by Proposition \[p:1\]) both $A$ and $B$ are spacelike.  $\square$

By Lemma \[l:3\] the perp space $H_{0, \pi (z_0 )}$ of $V_{0,
\pi (z_0 )}$ is also nondegenerate and $$T_{\pi (z_0 )} (\Omega_+ ) = H_{0, \pi (z_0 )} \oplus_{\rm
orth} V_{0, \pi (z_0 )} .$$ Let $v \in H_{z_0}$. Then $(d_{z_0}
i)v$ is perpendicular on $N(S^{2n-1}_{2s})_{z_0} \oplus (d_{z_0} i
) V_{z_0}$. On the other hand $d_{z_0} \pi : T_{z_0}(\Lambda_+
\setminus \Lambda_0 ) \to T_{\pi (z_0 )} (\Omega_+ )$ is a linear isometry hence (by Lemma \[l:2\]) $(d_{z_0} \pi )(d_{z_0} i )v$ is perpendicular on $V_{0, \pi (z_0 )}$, and then it lies on $H_{0, \pi (z_0 )}$.  Again by inspecting dimensions we obtain $$H_{0, \pi (z_0 )} = (d_{z_0} \pi )(d_{z_0} i) H_{z_0} . 
\label{e:16}$$ Next, by (\[e:16\]) and by $d_{z_0} \Pi : H_{z_0} \approx T_{\Pi
(z_0 )} ({\mathbb C}P^{n-1}_s (4))$ (a linear isometry), it follows that $$\label{e:17} d_{\pi (z )} p : H_{0, \pi (z )} \approx T_{\Pi (z )}
({\mathbb C}P^{n-1}_s (4)),$$ a linear isometry for any $z \in S^{2n-1}_s$. We wish to show that (\[e:17\]) actually holds for any $z \in \Lambda_+ \setminus \Lambda_0$.

The torus $T^1_{\mathbb C}$ acts on $\Omega_+$ as a group of isometries of the semi-Riemannian manifold $(\Omega_+ , g_{s,n})$. \[l:4\]

[*Proof*]{}.  If $g = w + D \in T^1_{\mathbb C}$ ($w \in
{\mathbb C}$) then the right translation $R_g : \Omega_+ \to
\Omega_+$ is a holomorphic map locally given by $z \mapsto z e^w$ hence $$|z e^w |^{-2}_{s,n} \; \left( (d R_g ) Z_j \; , \; (d R_g ) \overline{Z}_k
\right)_{z e^w} =$$ $$= e^{2 {\rm Re}(w)} \; |z|^{-2}_{s,n} \;
b_{s,n} (e^w e_j , e^w e_k ) = |z|^{-2}_{s,n} \; \epsilon_j \;
\delta_{jk}$$ where $\{ e_j : 1 \leq j \leq n \}$ is the canonical basis in ${\mathbb C}^n$. $\square$

As $p \circ R_g =$ const.,  each $R_g$ preserves the vertical spaces $V_0$. Then (by Lemma \[l:4\]) $R_g$ preserves the horizontal spaces $H_0$, as well.  Therefore, to complete the proof we must show that for any $z \in \Lambda_+ \setminus \Lambda_0$ there is $g \in T^1_{\mathbb C}$ and $z_0 \in S^{2n-1}_{2s}$ such that $\pi (z) = \pi (z_0 ) g$. Indeed we may consider $g = \log
|z|_{s,t} + D$ and $z_0 = |z|_{s,t}^{-1} z$.

To prove the last statement in Theorem \[t:2\] we establish

Let $z_0 \in \Lambda_+ \setminus \Lambda_0$ and let $j : T^1_{\mathbb C} \to \Omega_+$ be the immersion given by $j (\zeta + D) = \pi (z_0 ) \cdot (\zeta + D)$. Then $$\left. ( d j) \frac{\partial}{\partial u} \right|_{\zeta + D} = -
\frac{1}{4}\; e^\zeta \; B_{\pi (z_0 )} \, ,$$ where $\zeta = u +
i v$, hence $j(T^1_{\mathbb C})$ is tangent to the Lee field of $\Omega_+$.

This follows easily from (\[e:12\]).  Then (by Proposition \[p:3\]) $j : T^1_{\mathbb C} \to \Omega_+$ is a minimal isometric immersion.  Therefore $p$ is a semi-Riemannian submersion with minimal fibres, hence a harmonic map (in the sense of [@kn:Fug]).  Compare to Theorem 3 in [@kn:Dra], p. 375.  $\square$

An indefinite l.c. K. metric with nonparallel Lee form
=====================================================

Let ${\mathbb C}_+ = \{ w \in {\mathbb C} : {\rm Im}(w) > 0 \}$ be the upper half space and consider the [*Tricerri metric*]{} (cf. [ @kn:DrOr], p. 24) $$g_{0,1} = {\rm Im}(w)^{-2} \; d w \odot
d \overline{w} + {\rm Im}(w) \; d z \odot d \overline{z} .$$ $g_{0,1}$ is (by a result in [@kn:Tri]) a (positive definite) g.c. K. metric on ${\mathbb C}_+ \times {\mathbb C}$. We build a family of indefinite l.c. K. metrics of index $0 \leq s < n$ containing the Tricerri metric as a limiting case (for $s = 0$ and $n=1$). 

Let $0 \leq s < n$ and $n \geq 1$. Let $g_{s,n}$ be the indefinite Hermitian metric on ${\mathbb C}_+
\times {\mathbb C}^n_s$ given by $$g_{s,n} = {\rm Im}(w)^{-2} \; d w \odot d \overline{w} +
{\rm Im}(w) \sum_{j=1}^n \epsilon_j d z^j \odot d \overline{z}^j . 
\label{e:19}$$ Then $g_{s,n}$ is an indefinite globally conformal Kähler metric with a nonparallel Lee form and its Lee field is spacelike.  Moreover, let $a \in {\rm SL}(3 , {\mathbb Z})$ be a unimodular matrix with ${\rm Spec}(a) = \{ \alpha , \beta , \overline{\beta}
\}$, where $\alpha > 1$ and $\beta \in {\mathbb C} \setminus
{\mathbb R}$. Let $G_{\alpha , \beta}$ be the group of holomorphic transformations of ${\mathbb C}_+ \times {\mathbb C}^n_s$ generated by $F_0 (w , z) = (\alpha w , \beta z)$, $w \in {\mathbb
C}_+$, $z \in {\mathbb C}^n$. Then $g_{s,n}$ is $G_{\alpha ,
\beta}$-invariant.  \[p:2\]

[*Proof*]{}.  Note that we may write (\[e:19\]) as $g_{s,n} = {\rm Im}(w) g_0$ where $$g_0 = {\rm Im}(w)^{-3} \; d w \odot d \overline{w} +
\sum_{j=1}^n \epsilon_j d z^j \odot d \overline{z}^j$$ is an indefinite Hermitian metric whose Kähler $2$-form $\Omega_0$ is $$- i \{ {\rm Im}(w)^{-3} \; d w \wedge d \overline{w} +
\sum_{j=1}^n \epsilon_j \; d z^j \wedge d \overline{z}^j \} .$$ Hence $d \Omega_0 = 0$, i.e. $g_0$ is an indefinite Kähler metric.  Thus (\[e:19\]) is an indefinite l.c. K. metric whose Lee form $$\omega = d f = \frac{1}{w - \overline{w}} \, (d w - d
\overline{w}) \;\;\;\; (f = \log {\rm Im}(w))$$ is exact.  Raising indices we obtain the Lee field $$B = i \, {\rm Im}(w) \, \left( \frac{\partial}{\partial w} -
\frac{\partial}{\partial \overline{w}} \right)$$ so that $g_{s,n}(B,B) = 1$, i.e. $B$ is spacelike.  Next, the only surviving coefficients of the Levi-Civita connection $\nabla$ of $({\mathbb C}_+ \times {\mathbb C}^n_s , g_{s,n})$ are $$\Gamma_{k0}^j = - \Gamma_{k\overline{0}}^j = - \frac{i}{4} \,
{\rm Im}(w)^{-1} \, \delta_k^j \, ,$$ hence $\nabla_{Z_j} B =
\frac{1}{2} \, Z_j \neq 0$. Finally, the $G_{\alpha ,
\beta}$-invariance of $g_{s,n}$ follows from $F_0^* d w = \alpha
\; d w$, $F^*_0 d z^j = \beta \; d z^j$, $1 \leq j \leq n$, and from $\alpha \beta \overline{\beta} = \det (a) = 1$. $\square$

The second canonical foliation
==============================

Let $(M , J, g)$ be an indefinite l.c. K. manifold, of index $\nu =
2 s$. Let $A = - J B$ be the [*anti-Lee field*]{}.  We also set $\theta (X) = g(X , A)$ (the [*anti-Lee form*]{}), so that $\theta
= \omega \circ J$. Also, let $\Omega (X,Y) = g(X , J Y)$ be the Kähler $2$-form.  Since $D J = 0$ it follows that $$(\nabla_X J) Y = \frac{1}{2} \{ \theta (Y) X - \omega (Y) J X -
g(X,Y) A - \Omega (X,Y) B \} \label{e:20}$$ for any $X,Y \in T(M)$. As an immediate application of (\[e:20\]) we have

Let $(M , J , g)$ be an indefinite l.c. K. manifold and $i : N \hookrightarrow M$ a complex submanifold of $M$ such that $i^* g$ is a semi-Riemannian metric.  Let $h$ be the second fundamental form of $i$. Then $$h(J X , J Y) = -
h(X,Y) - g(X,Y) B^\bot \, , \label{e:18}$$ for any $X,Y \in T(N)$, where $B^\bot$ is the normal component of the Lee field of $M$. Then the mean curvature vector of $i$ is given by $H = - \frac{1}{2} B^\bot$. In particular $i$ is minimal if and only if $N$ is tangent to the Lee field.  \[p:3\]

This extends a result of I. Vaisman (cf. [ @kn:Vai2]) to the case of semi-Riemannian complex submanifolds of an indefinite l.c. K. manifold. [* Proof of Proposition*]{} \[p:3\].  As $N$ is a complex manifold $T(N)$ admits a local orthonormal frame of the form $\{ E_\alpha , J E_\alpha : 1 \leq \alpha \leq m \}$, i.e. $g(E_\alpha , E_\beta ) = \epsilon_\alpha \delta_{\alpha\beta}$, and then (by (\[e:18\])) the mean curvature vector $H$ of $i$ is given by $$H = \frac{1}{2m} \; \sum_\alpha \epsilon_\alpha \{ h(E_\alpha , E_\alpha ) + h(J
E_\alpha , J E_\alpha ) \} = - \frac{1}{2} \; B^\bot .$$ It remains that we prove (\[e:18\]).  Let ${\rm tan}$ and ${\rm
nor}$ be the projections associated with the decomposition $T(M) =
T(N) \oplus T(N)^\bot$ and let us set $$t \xi = {\rm tan}(\xi ), \;\;\; f \xi = {\rm nor}(\xi ), \;\;\;
\xi \in T(N)^\bot .$$ Then $J \xi = t \xi + f \xi$ and by applying $J$ once more we get $f^2 = - I$. Let $A^\bot = {\rm
nor}(A)$ and $B^\bot = {\rm nor}(B)$. Then (by (\[e:20\]) and the Gauss formula) $$h(X , J Y) = f \; h(X,Y) - \frac{1}{2} \{ g(X, Y) A^\bot +
\Omega (X,Y) B^\bot \} ,$$ for any $X,Y \in T(N)$. Finally, using $A^\bot = - f \; B^\bot$ and $f^2 = - I$ we obtain (\[e:18\]).  $\square$ 0.1in As another application of (\[e:20\]) we shall prove

Let $M$ be a complex $n$-dimensional [(]{}$n > 1$[)]{} indefinite l.c. K. manifold with a parallel Lee form $(\nabla
\omega = 0)$ and $c = g(B,B) \in {\mathbb R}$. Let $M(c) = M
\setminus {\rm Sing}(B)$, an open subset of $M$. Then $$P : x \in M(c) \mapsto {\mathbb R} A_x \oplus {\mathbb R}B_x
\subset T_x (M)$$ is an integrable distribution, hence $P$ determines a foliation ${\mathcal F}_c$ of $M(c)$ by real surfaces such that [i)]{} either $c \neq 0$ and then each leaf $L \in
M/{\mathcal F}_c$ is Riemannian [(]{}with the metric ${\rm
sign}(c) \; i^* g$, $i : L \hookrightarrow M$[)]{} and a totally geodesic surface in $(M , g)$, or [ii)]{} $c = 0$ and then each leaf $L \in M(c)/{\mathcal F}_c$ is either an isotropic surface [(]{}when $n \geq 3$[)]{} or a totally lightlike surface [(]{}when $n=2$[)]{}.  Assume that $n \geq 3$. Then the second fundamental form of a leaf $L \in M(0)/{\mathcal F}_0$ with respect to any transversal vector bundle ${\rm tr}(T(L)) \to L$ vanishes.  \[t:4last\]

[*Proof*]{}.  Note that $c \neq 0$ yields $M(c) = M$. Moreover (by the very definition of the anti-Lee field) $\{ A_x ,
B_x \}$ are linearly dependent if and only if $x \in {\rm
Sing}(B)$. Hence the sum ${\mathbb R} A_x + {\mathbb R}B_x$ is direct, for any $x \in M(c)$. To see that $P$ is involutive it suffices to check that $[A, B] \in P$. Let $X \in P^\bot$. Then $$g([A,B] , X) = g(\nabla_A B - \nabla_B A , X) = \;\;\;\;\;\;\;
({\rm as} \; \nabla B = 0, \; \nabla g = 0)$$ $$= - B(g(A,X)) + g(A , \nabla_B X ) = \;\;\;\;\;\;\;\;\;\;\;\;
({\rm as} \; \theta (X) = 0)$$ $$= - g(J B , \nabla_B X ) = g(B , J \nabla_B X) = \;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\; ({\rm by} \; (\ref{e:20}))$$ $$= g(B , \nabla_B J X ) = B(g(B , J X)) - g(\nabla_B B , J X) =
B(\theta (X)) = 0,$$ hence $[A,B] \in (P^\bot )^\bot = P$. By the classical Frobenius theorem there is a foliation ${\mathcal F}_c$ of $M(c)$ such that $P = T({\mathcal F}_c )$.  Assume that $c \neq
0$. Then either $P$ is spacelike (when $c > 0$) or timelike (when $c < 0$).  Let $L \in M/{\mathcal F}_c$ and let $h$ be the second fundamental form of $L \hookrightarrow M$. Then $\nabla B = 0$ yields $h(A,B) = h(B,B) = 0$. Finally (by (\[e:20\])) $$\nabla_A A = - \nabla_A J B =$$ $$= - J \nabla_A B - \frac{1}{2} \{
\theta (B) A - \omega (B) J A - g(A,B) A - \Omega (A,B) B \} = 0$$ so that $h(A,A) = 0$. We may conclude that $h = 0$. $\square$

Assume now that $c = 0$, so that both the Lee and anti-Lee fields are lightlike.  Let us set $${\rm Rad} \; P_x = P_x \cap P^\bot_x \, , \;\;\; x \in M(0).$$ We have ${\rm dim}_{\mathbb R} P_x = 2$ and ${\rm dim}_{\mathbb R}
P^\bot_x = 2(n-1)$, hence ${\rm Rad} \; P = P$. Therefore each leaf $L \in M(0)/{\mathcal F}_0$ is a $2$-lightlike submanifold (surface) in $M(0)$ and in particular (according to the terminology in [@kn:BeDu], p. 149-150) an isotropic submanifold (when $n
> 2$) or a totally lightlike submanifold (when $n = 2$) of $M(0)$. We shall need the following adaptation of Lemma 1.4 and Theorem 1.6 in [@kn:BeDu], p. 149-150 (to the case of the lightlike foliation ${\mathcal F}_0$, rather than a single istropic submanifold)

Assume that $n \geq 3$. Let $S(P^\bot ) \to M(0)$ be a vector subbundle of $P^\bot \to M(0)$ such that $$P^\bot = P \oplus_{\rm orth} S(P^\bot ).  \label{e:21}$$ Then for any $x \in M(0)$ there exist an open neighborhood $U
\subseteq M(0)$ and a system of linearly independent tangent vector fields $\{ N_1 , N_2 \}$ on $U$ such that $$\theta (N_1 ) = \omega (N_2 ) = 1, \;\;\; \theta (N_2 ) = \omega
(N_1 ) = 0, \label{e:22}$$ $$g(N_i , N_j ) = 0, \;\;\; g(N_i , W) = 0, \label{e:23}$$ for any $W \in S(P^\bot )$.  Moreover, if $\left. { \rm
ltr}(P)\right|_U$ is given by $${\rm ltr}(P)_x = {\mathbb R}N_{1, x} \oplus {\mathbb R} N_{2,
x} \, , \;\;\; x \in U,$$ then the vector bundles $\left. { \rm
ltr}(P)\right|_U$ glue up to a vector bundle ${\rm ltr}(P) \to
M(0)$ such that $$S(P^\bot )^\bot = P \oplus {\rm ltr}(P).  \label{e:24}$$ \[l:6\]

According to the terminology in [@kn:BeDu], $\left. 
{\rm ltr}(P)\right|_L$ is the [*lightlike transversal vector bundle*]{} of $(L , \left.  S(P^\bot )\right|_L )$, for each leaf $L
\in M(0)/{\mathcal F}_0$. By Proposition 2.1 in [@kn:BeDu], p. 5, $S(P^\bot )$ is nondegenerate.  Let $S(P^\bot )^\bot$ be the orthogonal complement of $S(P^\bot )$.  Of course, this is also nondegenerate and $$\label{e:25} T(M(0)) = S(P^\bot ) \oplus_{\rm orth} S(P^\bot
)^\bot .$$ Note that $S(P^\bot )^\bot$ has rank $4$ and (by (\[e:21\])) $P
\subset S(P^\bot )^\bot$. To prove Lemma \[l:6\] let $E \to
M(0)$ be a subbundle of $S(P^\bot )^\bot \to M(0)$ such that $$S(P^\bot )^\bot = P \oplus E. \label{e:26}$$ For any $x \in M(0)$ there is an open neighborhood $U \subseteq
M(0)$ of $x$ and a local frame $\{ V_1 , V_2 \} \subset
\Gamma^\infty (U, E)$. Let $D \in C^\infty (U)$ be given by $$D = \theta (V_1 ) \omega (V_2 ) - \omega (V_1 ) \theta (V_2 ).$$ We claim that $D(x) \neq 0$, for any $x \in U$. Indeed, if $D(x_0
) = 0$ for some $x_0 \in U$ then $$\theta (V_2 )_{x_0} = \lambda \theta (V_1 )_{x_0} \, , \;\;\;
\omega (V_2 )_{x_0} = \lambda \omega (V_1 )_{x_0} \, ,$$ for some $\lambda \in {\mathbb R}$, hence $v_\lambda := V_{2, x_0} -
\lambda V_{1, x_0}$ is orthogonal to both the Lee and anti-Lee vectors, i.e. $v_\lambda \in P^\bot_{x_0}$. Yet $v_\lambda \in
E_{x_0} \subset S(P^\bot )^\bot_{x_0}$, i.e. $v_\lambda$ is orthogonal to $S(P^\bot )_{x_0}$. Then (by (\[e:21\])) $v_\lambda \in P_{x_0} \cap E_{x_0} = (0)$, i.e. $\{ V_{1, x_0} ,
V_{2 , x_0} \}$ are linearly dependent, a contradiction.  Let us set $$N_1 = \lambda_{11} A + \lambda_{12} B + \frac{1}{D} \{ \omega
(V_2 ) V_1 - \omega (V_1 ) V_2 \} ,$$ $$N_2 = \lambda_{21} A + \lambda_{22} B - \frac{1}{D} \{ \theta
(V_2 ) V_1 - \theta (V_1 ) V_2 \} ,$$ where $\lambda_{ij} \in
C^\infty (U)$ are given by $$\lambda_{11} = - \frac{1}{D^2} \{ \omega (V_2 )^2 g(V_1 , V_1 )
- 2 \omega (V_1 ) \omega (V_2 ) g(V_1 , V_2 ) + \omega (V_1 )^2
g(V_2 , V_2 ) \} ,$$ $$\lambda_{22} = - \frac{1}{D^2} \{ \theta (V_2 )^2 g(V_1 , V_1 )
- 2 \theta (V_1 ) \theta (V_2 ) g(V_1 , V_2 ) + \theta (V_1 )^2
g(V_2 , V_2 ) \} ,$$ $$\lambda_{12} = \lambda_{21} = \frac{1}{2 D^2} \{ \omega (V_2 )
\theta (V_2 ) g(V_1 , V_1 ) +$$ $$+ [\theta (V_1 ) \omega (V_2 )
+ \omega (V_1 ) \theta (V_2 ) ] g(V_1 , V_2 ) - \omega (V_1 )
\theta (V_1 ) g(V_2 , V_2 ) \} .$$ A calculation shows that $N_i$ are linearly independent at each $x \in U$ and satisfy (\[e:22\])-(\[e:23\]).  Therefore $\left. { \rm
ltr}(P)\right|_U$ is well defined.  Let $U^\prime \subseteq M(0)$ be another open neighborhood of $x$ and $\{ V_1^\prime ,
V_2^\prime \}$ a local frame of $E$ on $U^\prime$, so that $V^\prime_i = f_i^j V_j$, for some $f^j_i \in C^\infty (U \cap
U^\prime )$.  A calculation shows that $\lambda^\prime_{ij} =
\lambda_{ij}$ and then $N^\prime_i = N_i$ on $U \cap U^\prime$, hence $\left. { \rm ltr}(P)\right|_U$ and $\left. { \rm
ltr}(P)\right|_{U^\prime}$ glue up over $U \cap U^\prime$. Finally, one may check that $P_{x_0} \cap {\rm ltr}(P)_{x_0} \neq
(0)$ at some $x_0 \in U$ yields $D(x_0 ) = 0$, a contradiction.  Hence the sum $P + {\rm ltr}(P)$ is direct and (\[e:24\]) must hold.  $\square$ 0.1in With the choices in Lemma \[l:6\] we set $${\rm tr}(P) = {\rm ltr}(P) \oplus_{\rm orth} S(P^\bot ),$$ so that (cf. [ @kn:BeDu1]) $\left. { \rm tr}(P)\right|_L$ is the [*transversal bundle*]{} of $L \in M(0)/{\mathcal F}_0$. Then (\[e:25\]) yields $T(M(0)) = P \oplus {\rm tr}(P)$ and we may decompose $\nabla_X Y = \nabla^P_X Y + h^P (X,Y)$, for any $X,Y
\in P$, such that $\nabla^P$ is a connection in $P \to M(0)$ and $h^P$ is a $C^\infty (M(0))$-bilinear symmetric ${\rm
tr}(P)$-valued form on $P$ (compare to (2.1) in [@kn:BeDu1], p. 154).  Once again we may use $\nabla B = 0$, $\nabla_A A = 0$ to conclude that $h^P = 0$. $\square$

A CR extension result
=====================

Let $M$ be a complex $n$-dimensional indefinite l.c. K. manifold of index $2s$, $0 < s < n$, with a parallel Lee form.  Let $\mathcal
F$ be the first canonical foliation (given by $\omega = 0$).  Each leaf $L$ of $\mathcal F$ is a real hypersurface in $M$, hence a CR manifold with the CR structure $$T_{1,0}(L) = T^{1,0}(M) \cap
[T(L) \otimes {\mathbb C}]$$ induced by the complex structure of $M$. There is a natural first order differential operator $$\overline{\partial}_L : C^1 (L) \to \Gamma^\infty (T_{0,1}(L)^*)$$ given by $(\overline{\partial}_L f) \overline{Z} =
\overline{Z}(f)$, for any $C^1$ function $f : L \to {\mathbb C}$ and any $Z \in T_{1,0}(L)$. Here $T_{0,1}(L) =
\overline{T_{1,0}(L)}$. The solutions to $\overline{\partial}_L f
= 0$ are the CR functions on $L$ (and $\overline{\partial}_L f =
0$ are the tangential Cauchy-Riemann equations on $L$, cf.  e.g. [@kn:Bog], p. 124).  Let $CR^k (L)$ be the space of all CR functions on $L$, of class $C^k$. It is a natural question whether a CR function on a leaf $L$ of $\mathcal F$ extends to a holomorphic function on $M$ (at least locally).  We answer this question for the canonical foliation of $\Omega_+$ (a similar result holds for $\Omega_-$) where an explicit description of the leaves is available.  Precisely

Let $n \geq 2$ and $0 < s < n$ such that $s \neq (n-1)/2$ when $n$ is odd.  Let $w \in S^1$ and $N_w^+ = F^{-1} (S^{2n-1}_{2s}
\times \{ w \} )$, where $F : \Omega_+ \to S^{2n-1}_{2s} \times
S^1$ is the diffeomorphism [(\[e:diff\])]{}.  Let $\mathcal F$ be the foliation of $\Omega_+$ given by the Pfaff equation $d \log
|z|^2_{s,n} = 0$. Then the leaf space is $$\label{e:28}
\Omega_+ /{\mathcal F} = \{ N^+_w : w \in S^1 \}$$ and for any point $x \in N_w^+$ there is an open neighborhood $U$ of $x$ in $M$ such that for any $f \in CR^1 (N_w^+ )$ there is a holomorphic function $F \in {\mathcal O}(U)$ such that $\left.  F
\right|_{U \cap N_w^+} = f$. \[t:5\]

[*Proof*]{}.  Note that $$N^+_w = \{ \pi (\lambda^{{\rm arg}(w)/(2\pi )} \zeta ) : \zeta
\in S^{2n-1}_{2s} \} \;\;\; (w \in S^1 ).$$ Let $z \in \Lambda_+
\setminus \Lambda_0$ and let us consider $\zeta = |z|^{-1}_{s,t} z
\in S^{2n-1}_{2s}$ and $w = \exp (2\pi i \log |z|_{s,n} / \log
\lambda ) \in S^1$. Then $\arg (w) = 2\pi \log |z|_{s,n} /\log
\lambda + 2 m \pi$ for some $m \in {\mathbb Z}$, so that $$\pi (z) = \pi (\lambda^m |z|_{s,t} \zeta ) = \pi
(\lambda^{\arg (w)/(2 \pi )} \zeta ) \in N_w^+ \, ,$$ that is through each point $\pi (z) \in \Omega_+$ passes at least one hypersurface of the form $N_w^+$. Next, let us assume that $\pi
(z) \in N^+_w \cap N_{w^\prime}^+$. Then $$e^{\arg (w^\prime )/(2\pi )} \zeta^\prime = \lambda^m e^{\arg
(w)/(2 \pi )} \zeta$$ for some $\zeta , \zeta^\prime \in
S^{2n-1}_{2s}$ and some $m \in {\mathbb Z}$. Then $b_{s,n}(\zeta ,
\zeta ) = b_{s,n}(\zeta^\prime , \zeta^\prime ) = 1$ imply $$\arg (w^\prime ) = \arg (w) + 2 m \pi \log \lambda$$ hence $N_w^+ = N^+_{w^\prime}$, that is through each $\pi (z ) \in
\Omega_+$ passes a unique hypersurface of the form $N_w^+$. To emphasize, $N^+_w = N^+_{w^\prime}$ if and only if $w^\prime =
e^{2m\pi i \log \lambda} w$, for some $m \in {\mathbb Z}$. Therefore, to prove (\[e:28\]) it suffices to check that the Lee field $B$ of $\Omega_+$ is orthogonal to each $N_w^+$. We set $$D(0, r) = \{ z \in \Lambda_+ \setminus \Lambda_0 : |z|_{s,n} <
r \} \;\;\;\; (r > 0)$$ and consider the annulus $A_k = D(0 ,
\lambda^k ) \setminus \overline{D}(0, \lambda^{k+1})$, $k \in
{\mathbb Z}$. If $U_k = \pi (A_k )$ then $\phi_k = (\pi : A_k \to
U_k )^{-1}$ are local charts on $\Omega_+$. Note that the holomorphic transformation $F_\lambda$ maps the pseudosphere $S^{2n-1}_{2s}(\lambda^k )$ onto $S^{2n-1}_{2s}(\lambda^{k+1})$, for any $k \in {\mathbb Z}$. In other words, when building $\Omega_+$ one identifies the points where a generic complex line through the origin intersects the pseudospheres $S^{2n-1}_{2s}(\lambda^k )$.  In particular $\pi
(S^{2n-1}_{2s}(\lambda^k )) = \pi (S^{2n-1}_{2s})$ and $U_k =
U_0$, for any $k \in {\mathbb Z}$.

Let $w \in S^1$ and $a = \arg (w)/(2\pi \log \lambda
)$.  If $a \in {\mathbb R} \setminus {\mathbb Z}$ then $N_w^+
\subset U_0$, while if $a \in {\mathbb Z}$ then $N_w^+ = \pi
(S^{2n-1}_{2s})$. In particular, for any $w \in S^1 \setminus \{
e^{2m\pi i \log \lambda} : m \in {\mathbb Z} \}$ $$\label{e:phi0}
\phi_0 (N_w^+ ) = S^{2n-1}_{2s}\left( \lambda^{- [a]} e^{\arg
(w)/(2\pi )} \right) .$$ \[l:7\]

Here $[a]$ is the integer part of $a \in {\mathbb R}$. Note that (\[e:phi0\]) doesn’t apply to the leaf $L_0 = \pi
(S^{2n-1}_{2s}) \in \Omega_+ /{\mathcal F}$ (corresponding to $a
\in {\mathbb Z}$).  However, in this case one may consider $\lambda
< \epsilon < 1$ and the annulus $A = D(0, \lambda^{-1} \epsilon )
\setminus \overline{D}(0, \epsilon )$ and then $L_0$ is contained in $U = \pi (A)$ and $(\pi : A \to U)^{-1}$ is a local chart on $\Omega_+$. To prove Lemma \[l:7\] let $x$ be a point of $N_w^+$, $x = \pi (e^{\arg (w)/(2\pi )} \zeta )$, and let us set $z = \lambda^{k - [a]} e^{\arg (w)/(2 \pi )} \zeta$. Then $a - 1 <
[a] \leq a$ yields $\lambda^{k+1} < |z|_{s,n} \leq \lambda^k$ and the second inequality becomes an equality if and only if $a \in
{\mathbb Z}$, that is if $w = e^{2m\pi i \log \lambda}$, for some $m \in {\mathbb Z}$. Finally, if $a \in {\mathbb R} \setminus
{\mathbb Z}$ then $N_w^+ \subset U_0$ and $$\phi_0 (N^+_w ) = \{ \phi_0 (\pi (^{\arg (w)/(2 \pi )} \zeta )) :
\zeta \in S^{2n-1}_{2s} \} =$$ $$= \{ \lambda^{-[a]} e^{\arg
(w)/(2 \pi )} \zeta : \zeta \in S^{2n-1}_{2s} \}  =
S^{2n-1}_{2s}\left( \lambda^{-[a]} e^{\arg (w)/(2\pi )} \right)$$ and $B = - 2(z^j Z_j + \overline{z}^j \overline{Z}_j )$ is orthogonal to any $S^{2n-1}_{2s}(r)$. The Cayley transform $${\mathcal C}(z) = \left( \frac{z^\prime}{r + z_n} \, , \,
\frac{i(r - z_n )}{r + z_n} \right) , \;\;\; z = (z^\prime , z_n )
\in {\mathbb C}^n \setminus \{ z_n + r = 0 \},$$ is a CR isomorphism of $S^{2n-1}_{2s}(r)$ onto $\partial {\mathcal
S}_{s,n} \setminus \{ \zeta_n + i = 0 \}$, where $${\mathcal S}_{s,n} = \{ \zeta \in {\mathbb C}^n : {\rm
Im}(\zeta_n ) > \sum_{\alpha =1}^{n-1} \epsilon_\alpha
|\zeta_\alpha |^2 \} .$$ The CR structure $T_{1,0}(\partial
{\mathcal S}_{s,n})$ is the span of $\{ \partial /\partial
\zeta^\alpha + 2 i \epsilon_\alpha \overline{\zeta}_\alpha
\partial /\partial \zeta^n : 1 \leq \alpha \leq n - 1 \}$ hence the Levi form has signature $(s , n-s-1)$. Yet $s \geq 1$ hence (by H. Lewy’s CR extension theorem, cf.  e.g. Theorem 1 in [@kn:Bog], p. 198) for any $x \in \partial {\mathcal S}_{s,n}$ there is an open neighborhood $U \subseteq {\mathbb C}^n$ of $x$ such that for any $f \in CR^1 (\partial {\mathcal S}_{s,n})$ there is a unique $F \in {\mathcal O}(U \cap {\mathcal S}_{s,n}) \cap C^0 (U \cap
\overline{\mathcal S}_{s,n})$ such that $\left.  F\right|_{U \cap
\partial {\mathcal S}_{s,n}} = f$. Then the last statement in Theorem \[t:5\] holds for any $x \in N_w^+ \setminus \{ \pi (e^{\arg (w)/(2\pi )}
\zeta ) : \zeta \in \Lambda^{n-1}_0 \times \{ - 1 \} \}$, where $\Lambda^{n-1}_0 = \Lambda^{n-1} \cup \{ 0 \}$ and $\Lambda^{n-1}$ is the null cone in ${\mathbb C}^{n-1}_s$ (so that $\phi_0 (x)$ satisfies $z_n + r \neq 0$ ($r = \lambda^{-[a]} e^{\arg (w)/(2\pi
)}$)).  For arbitrary $x \in N_w^+$ the argument requires that $z_j
+ r \neq 0$, for some $1 \leq j \leq n$ (the remaining case is ruled out by our assumption that $n \neq 2s+1$).  $\square$

Levi foliations
===============

Let $(M , J , g)$ be a complex $n$-dimensional indefinite l.c. K. manifold and $B$, $A$ its Lee and anti-Lee fields, respectively.  Let us set $Z := B + i A \in T^{1,0}(M)$. Clearly $\omega (Z ) =
c$. Let us assume that $\nabla \omega = 0$ and ${\rm Sing}(B) =
\emptyset$ and set $$T_{1,0}({\mathcal F}) = T^{1,0}(M) \cap [T({\mathcal F})
\otimes {\mathbb C}]$$ so that the portion of $T_{1,0}({\mathcal
F})$ over a leaf $L \in M/{\mathcal F}$ is the CR structure of $L$. Also the portion of $H({\mathcal F}) := {\rm Re}\{
T_{1,0}({\mathcal F}) \oplus \overline{T_{1,0}({\mathcal F})} \}$ over $L$ is the Levi distribution $H(L)$ of $L$. The distribution $H({\mathcal F})$ carries the complex structure $$J : H({\mathcal F}) \to H({\mathcal F}), \;\;\; J(V +
\overline{V}) = i(V - \overline{V}), \;\;\; V \in
T_{1,0}({\mathcal F}).$$ See also [@kn:DrNi].  Let us set $${\mathcal L}(V , \overline{W}) = i \; \pi \; [V ,
\overline{W}], \;\;\; V , W \in T_{1,0}({\mathcal F}),$$ where $\pi : T({\mathcal F}) \to T({\mathcal F}) /H({\mathcal F})$ is the natural projection, so that ${\mathcal L}$ is the Levi form of each leaf of $\mathcal F$. The [*null space*]{} of $\mathcal L$ is $${\rm Null}({\mathcal L}) = \{ V \in T_{1,0}({\mathcal F}) : {\mathcal L}(V , \overline{V}) = 0 \} .$$ We may state the following corollary of Theorems \[t:1\] and \[t:4last\]

If the Lee vector $B$ is lightlike then the Levi form of each leaf of $\mathcal F$ is degenerate $(Z \in {\rm
Null}({\mathcal L}))$ and ${\mathcal F}_0$ is a subfoliation of $\mathcal F$. Moreover if $n = 2$ then each leaf of $\mathcal F$ is Levi-flat and the Levi foliation of each leaf $L \in
M/{\mathcal F}$ extends to a unique holomorphic foliation of $M$. \[p:4\]

We recall that a CR manifold $L$ is [*Levi-flat*]{} if its Levi form vanishes identically (${\mathcal L} = 0$).  If this is the case $L$ is foliated by complex manifolds (whose complex dimension equals the CR dimension of $L$).  The resulting foliation (whose tangent bundle if the Levi distribution $H(L)$ of $L$) is the [*Levi foliation*]{} of $L$. If $L$ is embedded in some come complex manifold $M$ a problem raised by C. Rea, [@kn:Rea], is whether the Levi foliation of (a Levi-flat CR manifold) $L$ may extend to a holomorphic foliation of $M$. Proposition \[p:4\] exhibits a family of Levi foliations of class $C^\infty$ which extend holomorphically (while Rea’s extension theorem (cf. [* op.  cit.*]{})  requires real analytic data). 

[*Proof of Proposition*]{} \[p:4\].  Let us assume that $c = 0$. Then $Z \in T({\mathcal F}) \otimes {\mathbb C}$ hence $Z \in
T_{1,0}({\mathcal F})$. Moreover $Z + \overline{Z} \in H({\mathcal
F})$ yields $B \in H({\mathcal F})$ and by applying $J$ we may conclude that $A \in H({\mathcal F})$ as well.  Hence (with the notations of Theorem \[t:4last\]) $P \subseteq H({\mathcal F})$. Note that $[Z , \overline{Z}] = 2 i [A,B]$ and then (by the integrability of $P$) ${\mathcal L}(Z , \overline{Z}) = 0$. When $n = 2$ each leaf $L$ of $\mathcal F$ is a $3$-dimensional CR manifold hence $H({\mathcal F}) = P$ and $L$ is Levi-flat.  Finally the foliation induced by ${\mathcal F}_0$ on $L$ is precisely the Levi foliation of $L$. In other words, the Levi foliation of each leaf of $\mathcal F$ extends to a holomorphic foliation of $M$ which is precisely the second canonical foliation of $M$.

[.  The first author was partially supported by INdAM (Italy) within the interdisciplinary project [*Nonlinear subelliptic equations of variational origin in contact geometry*]{}.  The second author was supported by Natural Sciences and Engineering Council of Canada.  This paper was completed while the first author was a guest of the Department of Mathematics of the University of Windsor (Ontario, Canada).]{} 

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[^1]: Università degli Studi della Basilicata, Dipartimento di Matematica, Campus Macchia Romana, 85100 Potenza, Italy, e-mail: [dragomir@unibas.it]{}

[^2]: University of Windsor, Windsor, Ontario N9B3P4, Canada, e-mail: [yq8@uwindsor.ca]{}
