6
$\begingroup$

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?

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ 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
    Commented Dec 25, 2018 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. $\endgroup$
    – Mishkaat
    Commented Aug 5, 2019 at 6:57
  • $\begingroup$ In the context of Physics, such metrices are considered in path integral quantum cosmology. See the paper Lorentzian Quantum Cosmology (2017) by Turok et al. However, there is no mathematical description of this kind of matrices. Therefore, an answer to this question would be extremely helpful. $\endgroup$
    – Faber Bosch
    Commented May 19, 2022 at 10:58

0

Reset to default

You must log in to answer this question.