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?