An analog of Chern's conjecture for the Euler-Satake characteristic of affine orbifolds
S.S. Chern conjectured that the Euler characteristic of every closed affine
manifold has to vanish. We present an analog of this conjecture stating that
the Euler-Satake characteristic of any compact affine orbifold is equal to zero.
We prove that Chern's conjecture is equivalent to its analog for
the Euler-Satake characteristic of compact affine orbifolds, and
orbifolds may be ineffective. This fact allowed us to extend to orbifolds the known results of B.~Klingler and
also results of B.~Kostant and D.~Sullivan
on sufficient conditions to fulfill Chern's conjecture.
Thus we prove that if an $n$-dimensional compact affine orbifold $\mathcal N$ is complete
or if its holonomy group belongs to the special linear group $SL(n,\mathbb R),$
then the Euler-Satake characteristic of $\mathcal N$ has to vanish. An application to
pseudo-Riemannian orbifolds is considered. Examples of orbifolds belonging to the investigated class are given. In particular, we construct an example of a compact incomplete affine orbifold with the vanishing
Euler characteristic, the holonomy group of which does not belong to $SL(n,\mathbb R).$