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I have a warped product $M=N_1\times_f N_2$ where $N_1$ and $N_2$ are Riemannian manifolds. The dimension of $N_2$ is $2n$ (for n integer) and $N_2$ is an almost Hermitian manifold, i.e., is compatibly equipped with Riemannian structure and almost complex structure $J$, with almost complex structure $J$ on $N_2$ I mean a linear complex structure (that is, a linear map which squares to $-1$) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field $J$ of degree $(1, 1)$ such that $J^2 = 1$ when regarded as a vector bundle isomorphism $J : TM \rightarrow TM $on the tangent bundle.

If $n=1$ then N_2 is 2-dimensional, and it is well know that every almost complex structure on a 2-dimensional manifold is integrable, hence is a complex structure; so we have $M$ with (real) Riemannian base and complex fiber.

Then, is possible consider $M$ as real-complex warped product manifolds?

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    $\begingroup$ I don't understand the question. What is a "real-complex warped product manifold"? $\endgroup$
    – Sebastian Goette
    Oct 18, 2019 at 8:25
  • $\begingroup$ @Sebastian Goette - thank you for your interest! If $M$ is a submanifold of Kaehler manifold, (where $N_1$ is totally real submanifold and $N_2$ is a holomorphic submanofold), B.-Y. Chen ( in "CR-submanifolds of a Kaehler manifold part II" (Theorem 3.1)), stated that $M$ exists only as product manifold (not warped manifold), but in this case (question case) with "real-complex warped product manifold" I mean a kind of CR-manifold $M$ that is not a submanifold of Kaehler manifold, and $N_1$ is not a totally real submanifold. $\endgroup$
    – MathDG
    Oct 18, 2019 at 8:55
  • $\begingroup$ Here B.-Y. Chen paper link.springer.com/article/10.1007/s006050170019#citeas $\endgroup$
    – MathDG
    Oct 18, 2019 at 9:07
  • $\begingroup$ Thanks, I will have a look! $\endgroup$
    – Sebastian Goette
    Oct 19, 2019 at 9:25
  • $\begingroup$ Thank very much! $\endgroup$
    – MathDG
    Oct 19, 2019 at 10:44

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