Affine manifold with measure preserving projective holonomy group
The term radiance was introduced by \textit{D. Fried}, \textit{W. Goldman} and \textit{M. W. Hirsch} [Comment. Math. Helv. 56, No. 4, 487-523 (1981; Zbl 0516.57014)] as an affine manifold whose affine holonomy has a fixed point. In this paper the author considers a compact affinely flat manifold \(M\) covered by a radiant manifold \(M\) and proves that the Euler characteristic \(\chi(M)\) is zero, in case the projective holonomy group of \(M\) leaves a probability Borel measure invariant. Otherwise the affine tangent bundle of \(\overline M\) contains a conformally flat vector subbundle, that is a linear subbundle with holonomy in the group of similarity transformations. Also in this case \(\chi(M)=0\). The paper gives another way of proving the theorem of \textit{M. W. Hirsch} and \textit{W. P. Thurston} [Ann. Math. (2) 101, 369-390 (1975; Zbl 0321.57015)] related particularly to amenability, a notion due to \textit{W. M. Goldman} and \textit{M. W. Hirsch} [Proc. Am. Math. Soc. 82, No. 3, 491-494 (1981; Zbl 0474.55014)].
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