Warped product manifold with real and complex parts

Is possible to define a warped product manifold $$M=(N,g_N) \times f(F, g_F)$$ where $$(N, g_N)$$ is a Riemannian manifold with Riemannian metric (i.e., real manifold with real structure) and $$(F, g_F)$$ is a complex manifold with Hermitian metric?

Is possible to consider this warped product manifold as Hermitian manifold?

Its metric should be Hermitian because $$g=g_N + f^2(g_F)=Re_N + f^2(Re_F) + f^2(Im_F)$$, where with $$Re_N$$ and $$Re_F$$ I mean the real part of the $$N$$ and $$F$$ metrics, respectively, and with $$Im_F$$ the imaginary part of $$F$$-metric, being complex.

• I do not even see the (almost) complex structure on $M$. In particular, what happens if $M$ is odd-dimensional? Sep 28 '19 at 16:00
• @Sebastian Goette - Thanks for your answer, my question born because, for example, a Kenmotsu manifold is a warped product manifold where the base-manifold is an interval $I$ (real) and the fiber-manifold $N$ is a Kahler manifold. Then I wanted to know if, in general, it is possible to consider (and eventually the bibliographic references) a warped product manifold in which the fiber-manifold is Kahler and the base-manifold is a real-manifold where dimension of the base is $>1$. Oct 1 '19 at 13:12