I am reading a paper, Light-like CR Hypersurfaces of Indefinite Kaehler Manifolds, by K. Duggal and A. Bejancu. I happen to have difficulty proving the following theorem:
Theorem 5: Let $M$ be a light-like CR hypersyrface of a Kaehler manifold $\bar{M}$. Then we have the following characterisation of integrability of distributions on $M$:
(i) $TM^{\perp} \oplus_{\perp} \bar{J}(TM^{\perp})$ is integrable iff $$B(X, Y) = 0, \forall X \in \Gamma (\bar{J}(TM^{\perp})), Y \in \Gamma (\bar{J}(TM^{\perp}) \oplus_{\perp} D_{0})$$.
My attempt at the problem is as follows; since the tangent bundle $TM = TM^{\perp} \oplus_{\perp} [\bar{J}(TM^{\perp}) \oplus \bar{J}(NM))] \oplus_{\perp} D_{0}$, then we know that if the distribution $TM^{\perp} \oplus_{\perp} \bar{J}(TM^{\perp})$ is integrable then $\forall x, y \in \Gamma (TM^{\perp} \oplus_{\perp} \bar{J}(TM^{\perp}))$ we have that $[x, y] \in \Gamma (TM^{\perp} \oplus{\perp} \bar{J}(TM^{\perp}))$. Then this means that the components of the bracket $[x, y]$ vanish along $\bar{J}(NM)$ and $D_{0}$, thus using the metric tensor and Levi-Civita connection on $\bar{M}$ we have that $$\bar{g}([x, y], \bar{J}(\xi)) = \bar{g}([x, y], W) = 0,$$ where $\xi \in \Gamma (TM^{\perp}), \bar{J}\xi \in \Gamma (TM^{\perp}), W \in \Gamma (D_{0})$. For the first equation, we have \begin{eqnarray*} \bar{g}([X, Y], \bar{J}\xi) &=& \bar{g}(\bar{\nabla}_{X}Y - \bar{\nabla}_{Y}X, \bar{J}\xi)\\ &=& \bar{g}(\bar{J}\bar{\nabla}_{X}Y, \bar{J}\bar{J}\xi) - \bar{g}(\bar{J}\bar{\nabla}_{Y}X, \bar{J}\bar{J}\xi)\\ &=& -\bar{g}(\bar{\nabla}_{X}\bar{J}Y, \xi) + \bar{g}(\bar{\nabla}_{Y}\bar{J}X,\xi)\\ &=& \bar{g}(\bar{J}Y, \bar{\nabla}_{X}\xi) - \bar{g}(\bar{J}X, \bar{\nabla}_{x}\xi)\\ &=& -g(\bar{J}Y, A^{*}_{\xi}X) + g(\bar{J}X, A^{*}_{\xi}Y)\\ &=& -B(X, \bar{J}Y) + B(Y, \bar{J}X), \end{eqnarray*} by setting $y = \bar{J}y$ we consequently have $B(x, y) = - B(\bar{J}y, \bar{J}x), \forall x, y \in \Gamma (TM^{\perp} \oplus_{\perp} \bar{J}(TM^{\perp}))$. In the paper, $B(x, y) = g(A_{\xi}x, y)$ is the second fundamental form, where $g$ is the induced degenerate metric tensor on the hypersurface $M$. I should also add that $TM^{\perp}, \bar{J}(TM^{\perp}), \bar{J}(NM)$ are degenerate as $g(\xi, \xi) = g(\bar{J}\xi, \bar{J}\xi) = g(\bar{J}N, \bar{J}N) = 0$, and $g(\xi, N) = g(\bar{J}\xi, \bar{J}N) = 1$, where $N \in \Gamma (NM)$, where $NM$ is the null transversal bundle of the hypersurface $M$.
We can also represent the Kaehler manifold $\bar{M}$ as $(\bar{M}, \bar{g}, \bar{J}, \bar{\nabla})$, where $\bar{g}$ is its metric tensor, $\bar{J}$ is the complex structure such that $\bar{J}^{2} = -I$, and thus $\bar{g}(JX, JY) = \bar{g}(X, Y)$.
From my approach, I do not see how the authors got the theorem in (i). Could this be elucidated to me more explicitly?
NOTE: The Gauss-Weingatern formula used is $\nabla_{X}\xi = -A^{*}_{\xi}X - \tau(X)\xi$, where $A^{*}_{\xi}$ is the screen shape operator.