Hermitian connections and Dirac operators. (English) Zbl 0876.53015

Gauduchon, Paul

Boll. Unione Mat. Ital., VII. Ser., B 11, No. 2, Suppl., 257-288 (1997).

These nice notes are intended to be a kind of vademecum for some basic material about almost Hermitian geometry. The content of the notes is described by the author in the following way: “The present paper principally includes:
(i) a unified presentation of a canonical class of (almost) Hermitian connections, already considered by P. Libermann, including the Lichnerowicz first and second canonical connections, the Libermann connection, the Bismut connection as well as a torsion-minimizing Hermitian connection, which seems not to have been considered so far;
(ii) explicit expressions for the corresponding Hermitian-Dirac operators, as well as the Riemannian-Dirac operator in the almost Hermitian framework, in particular for the canonical \(\text{Spin}^c\)-structure, which are well-known in the Kähler case, but seem to be lacking in the current literature in the general case.”
The notes are as self-contained as possible, but they do not include an exhaustive reference list and do not treat explicitly curvature properties of the Hermitian connections. They will certainly be very valuable for workers in this field.

Reviewer: L.Vanhecke (Heverlee)

MSC:

53C07

Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)

Keywords:

almost Hermitian connections; Hermitian-Dirac operators