# Complex Riemannian metrics over real manifolds

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?

• 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/…). – Qfwfq Dec 25 '18 at 1:28