In this paper we generalize special geometry to arbitrary signatures in
target space. We formulate the definitions in a precise mathematical setting
and give a translation to the coordinate formalism used in physics. For the
projective case, we first discuss in detail projective Kaehler manifolds,
appearing in N=1 supergravity. We develop a new point of view based on the
intrinsic construction of the line bundle. The topological properties are then
derived and the Levi-Civita connection in the projective manifold is obtained
as a particular projection of a Levi-Civita connection in a `mother' manifold
with one extra complex dimension. The origin of this approach is in the
superconformal formalism of physics, which is also explained in detail.
Finally, we specialize these results to projective special Kaehler manifolds
and provide explicit examples with different choices of signature.Comment: LaTeX, 83 pages; v2: typos corrected, version to be published in
Handbook of pseudo-Riemannian Geometry and Supersymmetry, IRMA Lectures in
Mathematics and Theoretical Physic
