By Aurel Bejancu

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**Example text**

2) b=1 where Sa are differentiable functions on u fl U. 1) for both basis we obtain that at each point the coefficients Sb(x) define an element of the proper orthogonal group SO(3) of dimension 3. By means of this result it follows that an almost quaternion manifold is orientable. 3) for any X, Y E f(TN) and any local section 4 of the vector bundle V. In this case we say that N is an almost quaternion metric manifold. Thus if {J1, J2, J3} is a canonical basis of V, each of J1, J2, J3 is almost Hermitian with respect to g.

9) we obtain [o, o](X, Y) _ [ox, $y] - P([X, Y]) - $([Ox, Y] + [x, 4Y]), for any X, Y E r(D). Thus, taking into account that D = Im. 6). 6) are satisfied. 5) we have Q([Jx, Y] + [X, JY]) = 0, which implies Q([JX, jy] - [x, Y]) = 0. CHAPTER! 26 Hence Q([J, J](X, obtain y)T) = 0. 2) we Q([J, J](X, Y)T) = Q([m; 4>](X, Y)) - Q([X, Y]). 6) we obtain Q([X, Y]) = 0, that is, D is integrable. 1. Let M be a CR-submanifold of a Hermitian manifold N. The distribution D is integrable if and only if the Nijenhuis tensor of 4> vanishes identically on D.

The exterior derivative of w is given by du(X, Y) = 2{ X(WY) - VY(wX) - w([X, Y])). 1. Let M be a CR-submanifold of a Kaehlerian 52 CHAPTER III manifold N. 10) for all vector fields X, Y, Z tangent to M. Proof. Since the Riemannian connection of M is given by 2g(VXY, Z) - Xg(Y, Z) + Yg(X, Z) - Zg(X, Y) + + g((X, Y3, Z) + g((Z, X], Y) - g((Y, Z], X), we have 2g((VX$)Y, Z) = 2g(VX$Y, Z) + 2g(VXY, $Z) = = Xg($Y, Z) + $Yg(X, Z) - Zg(X, $Y) + + g((X, $Y], Z) + g((Z, X], $Y) - g([$Y, Z], X) + + Xg(Y, $Z) + Yg(X, $Z) - $Zg(X, Y) + + g((X, Y], $Z) + g(($Z, X3, Y) - g((Y, $Z], X).