A Hermitian manifold $(M,J,g)$ can be defined as a complex manifold $(M,J)$, with integrable complex structure $J$, equipped with a Riemannian metric satisfying $g\circ (J\otimes J) = g$. In this situation, one is interested in endowing $(M,g,J)$ with a connection compatible with the Hermitian structure, that is, a connection $\nabla$ satisfying:
$\nabla g= 0$ and $\nabla J = 0$
The issue is that the two conditions above do not specify a unique connection. Of course, we can require $\nabla$ to be torsion free (and hence unique) if and only if $(M,J,g)$ is Kahler, a condition which I do not want to assume. Instead, on a general Hermitan manifold one defines a line of canonical Hermitian connections as follows:
$2 \nabla^t = (1 + t) \nabla^C + (1 - t) \nabla^B$
for $t\in \mathbb{R}$, where $\nabla^C$ denotes the Chern connection and $\nabla^B$ denotes the Bismut connection. I am trying to find a book where this set up is explored in detail. I have found thousands of references on Kaehler geometry, but not many dealing with connections on more general Hermitian manifolds. I know Gauduchon's paper "Hermitian connections and Dirac operators", but I am looking for a more recent and pedagogical treatment, preferably in the form of a book. In particular, I am interested in a discussion of which Hermitian manifolds satisfy specific conditions on the curvature of the Bismut connection.
Thanks.
