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There is a huge literature on complex manifolds and natural metrics over them, but I was unable to find references about Riemannian complex metric on real manifolds (for which we complexify tangent fibers).

Let's be (a little bit) more precise : Let $M$ be a smooth real manifold. Define a "Riemannian complex metric" on $M$ as a nondegenerate symmetric (0,2)-tensor over the complexified tangent space of $M$.

My question is : What do we know about such Riemannian complex metrics over real manifolds ? My guess is that we can still define a Levi-Civita connection, and a non-trivial notion of (complex) scalar curvature. Have they been studied, or used in the literature, or are they without interest for some reason?

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    $\begingroup$ Just a comment: as far as I know, on a complex manifold it's usually interesting to study either Hermitian forms (which give a Riemannian metric $g$ and a non-degenerate anti-symmetric $2$-form $\omega$ that satisfy something like $\omega(u,v)=g(u,Jv)$ where $J$ is the complec structure); or a complex symmetric section of the holomorphic cotangent bundle that is holomorphic (see mathoverflow.net/questions/19337/…). $\endgroup$ – Qfwfq Dec 25 '18 at 1:28
  • $\begingroup$ The metric is involved with the local coordinates that you choose on a chart of the manifold. For a complex metric, you need (anti) holomorphic local coordinates. Plus, the manifold should be of even (real) dimension. $\endgroup$ – Laptop Aug 5 '19 at 6:57

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