Chern's conjecture for closed nilpotent affine manifolds: vanishing of the Euler characteristic

A nilpotent affine manifold is a manifold modelled on a simply connected nilpotent Lie group $\mathscr N$ with transition maps in the nilpotent affine group $\mathrm{Aff}(\mathscr N)=\mathrm{Aut}(\mathscr N)\ltimes\mathscr N$. In his 2020 work on geodesic convexity for nilpotent similarity structures, Raphaël Alexandre proved a Fried-type dichotomy for closed manifolds with such geometries under a rank-one dilatation hypothesis, and posed the analogue of Chern's conjecture: does every closed nilpotent affine manifold have vanishing Euler characteristic? Since the abelian case $\mathscr N=\mathbb R^n$ is exactly classical affine geometry, the question strictly generalizes the long-open Chern conjecture on affine manifolds.

Background

An affine manifold is a smooth manifold equipped with an atlas of charts in $\mathbb R^n$ whose transition maps lie in the affine group $\mathrm{Aff}(\mathbb R^n)=\mathrm{GL}(n,\mathbb R)\ltimes\mathbb R^n$; equivalently, it carries a flat, torsion-free linear connection. More generally, a $(G,X)$-structure on a manifold $M$ is an atlas of charts in a model space $X$ with transition maps in a Lie group $G$ acting on $X$; such a structure determines a developing map $D:\widetilde M\to X$, a local diffeomorphism from the universal cover of $M$ that is equivariant with respect to the holonomy representation $\rho:\pi_1(M)\to G$. Chern's conjecture asserts that every closed (compact, without boundary) affine manifold has vanishing Euler characteristic $\chi$. The conjecture is open in general; it is known for complete affine space forms (Kostant and Sullivan, 1975), for surfaces (Benzécri, 1960), for affine manifolds whose linear holonomy lies in $\mathrm{SL}(n,\mathbb R)$, that is, which carry a parallel volume form (Klingler, 2017), for higher-rank irreducible locally symmetric manifolds (Goldman and Hirsch, 1984), and for manifolds locally a product of hyperbolic planes (Bucher and Gelander, 2011); a survey of these cases appears in the introduction of Bucher–Connell–Lafont (2017).

Alexandre (2020) studies a nilpotent counterpart of this geometry. A simply connected nilpotent Lie group $\mathscr N$ — for instance a Carnot group, i.e. a stratified nilpotent Lie group such as the Heisenberg group, homogeneous under its one-parameter family of dilatations — admits, in suitable Lie-algebra coordinates, dilatations $x\mapsto\delta_t(x)$ rescaling each coordinate with weight $d_i\ge 1$, together with rotations and left translations. The group $\mathrm{Sim}(\mathscr N)$ generated by these transformations is a nilpotent analogue of the Euclidean similarity group, and an atlas with transition maps in $\mathrm{Sim}(\mathscr N)$ defines a nilpotent similarity structure on $M$. When the dilatation subgroup $A$ has rank one, a smooth subadditive homogeneous pseudo-norm of Hebisch and Sikora provides $\mathrm{Sim}(\mathscr N)$-invariant geodesically convex open balls, and Alexandre generalizes Fried's 1980 theorem on closed similarity manifolds: a closed $(\mathrm{Sim}(\mathscr N),\mathscr N)$-manifold is either complete, or radiant — the holonomy group fixes a point of $\mathscr N$ and the developing map is a covering onto $\mathscr N$ with that point removed.

If the similarity group is enlarged to the full nilpotent affine group $\mathrm{Aff}(\mathscr N)=\mathrm{Aut}(\mathscr N)\ltimes\mathscr N$, comprising all Lie-group automorphisms of $\mathscr N$ together with left translations, one obtains nilpotent affine manifolds, i.e. manifolds with an $(\mathrm{Aff}(\mathscr N),\mathscr N)$-structure. Theorem 4.1 of the source suggests that their theory parallels classical affine geometry, and Alexandre poses, as an explicitly recorded open question, the analogue of Chern's conjecture for this class. The analogy is tight: taking the abelian nilpotent group $\mathscr N=\mathbb R^n$ gives back exactly the classical affine structures, so the nilpotent question contains the classical Chern conjecture as the special case of an abelian model.

Problem Statement

Let $\mathscr N$ be a nilpotent Lie group (as in the source's geometric models, taken connected and simply connected), and let $\mathrm{Aff}(\mathscr N)=\mathrm{Aut}(\mathscr N)\ltimes\mathscr N$ be its group of affine transformations, generated by the Lie-group automorphisms of $\mathscr N$ and the left translations. Does every closed manifold $M$ carrying an $(\mathrm{Aff}(\mathscr N),\mathscr N)$-structure — that is, an atlas of charts in $\mathscr N$ whose transition maps lie in $\mathrm{Aff}(\mathscr N)$ — have vanishing Euler characteristic, $\chi(M)=0$ — equivalently, does the Euler class of the tangent bundle of every such $M$ evaluate trivially on the fundamental class (with the orientation local system used when $M$ is not orientable)? The question is posed at this full generality, over all nilpotent $\mathscr N$ — including the abelian case $\mathscr N=\mathbb R^n$, which recovers exactly the classical affine structures and hence the classical Chern conjecture — and over all closed $(\mathrm{Aff}(\mathscr N),\mathscr N)$-manifolds, complete or incomplete.

References

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