INVARIANCE THEORY AND AFFINE DIFFERENTIAL GEOMETRY

NOVICA BLA\v{Z}I\'C and NEDA BOKAN

1991 Math. Subject Classification: 53B30, 53C50, 53A15.

Key words: Complex affine manifold, Chern classes,
Norden construction.

Abstract: The main purpose of this paper is to study
the characteristic classes of manifolds and submanifolds
endowed with a complex affine connection $D$ whose curvature
tensor $R$ and an almost complex structure $J$
satisfy the condition $R(JX,JY)$ $=-R(X,Y)$. These
connections are naturally appeared in the context
of pseudo-Riemannian manifolds with the metric of
the signature $(n,n)$ as well as in complex  affine
and projective differential geometry (see [1--5, 7--10],  etc.).
We show especially a complex equiaffine surface
endowed with a complex affine connection $D$ has the
vanishing first Chern class $c\sb{ 1}$.
We prove also the second Chern class $c\sb{ 2}$ and
$c\sb{ 1}\sp{ 2}$ vanish for all complex surfaces endowed with
a complex affine connection $D$. Some new and known examples
of manifolds endowed with a complex affine connection $D$
are given.