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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.

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  • $\begingroup$ I do not even see the (almost) complex structure on $M$. In particular, what happens if $M$ is odd-dimensional? $\endgroup$
    – Sebastian Goette
    Commented Sep 28, 2019 at 16:00
  • $\begingroup$ @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 $>1$. $\endgroup$
    – MathDG
    Commented Oct 1, 2019 at 13:12

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