Chern’s conjecture for special affine manifolds

@article{Klingler2017ChernsCF,
  title={Chern’s conjecture for special affine manifolds},
  author={Bruno Klingler},
  journal={Annals of Mathematics},
  year={2017},
  volume={186},
  pages={69-95}
}
  • B. Klingler
  • Published 1 July 2017
  • Mathematics
  • Annals of Mathematics
An affine manifold X in the sense of differential geometry is a differentiable manifold admitting an atlas of charts with value in an affine space, with locally constant affine change of coordinates. Equivalently, it is a manifold whose tangent bundle admits a flat torsion free connection. Around 1955 Chern conjectured that the Euler characteristic of any compact affine manifold has to vanish. I will explain a proof of this conjecture in the case where X moreover admits a parallel volume form. 
View via Publisher
math.uni-bielefeld.de
12 Citations
Highly Influential Citations
2
Background Citations
6

12 Citations

Citation Type

Citation Type

Flat affine symplectic Lie groups
We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We prove that a left…
Lectures on the Euler characteristic of affine manifolds
These are lecture notes prepared for the summer school "Geometric, algebraic and topological methods in quantum field theory", held in Villa de Leyva in July 2017. Our goal is to provide an…
Vanishing simplicial volume for certain affine manifolds
We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some…
Flat vector bundles and open coverings
We establish a generic counting formula for the Euler number of a flat vector bundle of rank $2n$ over a $2n$ dimensional closed manifold, in terms of vertices of transversal open coverings of the…
An analog of Chern’s conjecture for the Euler–Satake characteristic of affine orbifolds
Complex surfaces and null conformal Killing vector fields
We study the relation between the existence of null conformal Killing vector fields and existence of compatible complex and para-hypercomplex structures on a pseudo-Riemannian manifold with metric of…
Bi-Lagrangian structures and Teichmüller theory
This paper has two purposes: the first is to study several structures on manifolds in the general setting of real and complex differential geometry; the second is to apply this study to Teichmuller…
On the structure of double complexes
  • J. Stelzig
  • Mathematics
    Journal of the London Mathematical Society
  • 2018
We study consequences and applications of the folklore statement that every double complex over a field decomposes into so‐called squares and zigzags. This result makes questions about the associated…
Bounded cohomology of discrete groups
Bounded cohomology of groups was first defined by Johnson and Trauber during the seventies in the context of Banach algebras. As an independent and very active research field, however, bounded…
GEOMETRIC STRUCTURES AND REPRESENTATIONS OF DISCRETE GROUPS
  • Fanny Kassel
  • Mathematics
    Proceedings of the International Congress of Mathematicians (ICM 2018)
  • 2019
We describe recent links between two topics: geometric structures on manifolds in the sense of Ehresmann and Thurston, and dynamics "at infinity" for representations of discrete groups into Lie…
...
1
2
...

References

SHOWING 1-10 OF 15 REFERENCES
The generalized Chern conjecture for manifolds that are locally a product of surfaces
Locally Homogeneous Geometric Manifolds
Motivated by Felix Klein’s notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally…
Complex Product Structures and Affine Foliations
A complex product structure on a manifold is an appropriate combination of a complex structure and a product structure. The existence of such a structure determines many interesting properties of the…
Foliated Bundles, Invariant Measures and Flat Manifolds
Let e = (p, E, M) be a smooth (= C1) bundle. A foliation of e is a foliation i of E where leaves are transverse to the fibres and of complementary dimension. The leaves are required to be smooth, and…
On the Geometry of the Tangent Bundle.
If M is a differentiable ra-dimensional manifold and V a linear connection for M, then the 2 rc-dimensional manifold TM, which is the total space of the tangent bündle of M, admits an almost complex…
Flat bundles with solvable holonomy
Let G be a solvable linear Lie group. We show that for every flat pnncipal G-bundle { over a CW-complex M, there is a finite-sheeted covering spacep: M-3 M such that p*C is trivial as a principal…
Autour de la conjecture de L. Markus sur les variétés affines
SummaryFor any subgroupG of (ℝn), we introduce some integer discG≦n called thediscompacity ofG. This number measures to what extent the closure ofG is not compact. The Markus' conjecture says that a…
Geometry of Foliations
1 Examples and Definition of Foliations.- 2 Foliations of Codimension One.- 3 Holonomy, Second Fundamental Form, Mean Curvature.- 4 Basic Forms, Spectral Sequence, Characteristic Form.- 5 Transversal…
ParaHermitian and paraquaternionic manifolds
A Survey on Paracomplex Geometry
We shall call paracomplex geometry the geometry related to the algebra of paracomplex numbers [70] and, mainly, the study of the structures on differentiable manifolds called paracomplex structures.…
...
1
2
...