Chern's conjecture for affinely flat manifolds was proposed by

Shiing-Shen Chern Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...

in 1955 in the field of

affine geometry In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of ''parallel lines'' is one of the main properties that is inde ...

. As of 2018, it remains an unsolved mathematical problem. Chern's conjecture states that the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

of a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

affine manifold In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset of ^n, with monodromy acting by affine t ...

vanishes.

Details

In case the connection ∇ is the

Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...

of a

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...

, the Chern–Gauss–Bonnet formula: :

\chi(M) = \left ( \frac \right )^n \int_M \operatorname(K)

implies that the Euler characteristic is zero. However, not all flat torsion-free connections on

T M

admit a compatible metric, and therefore, Chern–Weil theory cannot be used in general to write down the Euler class in terms of the curvature.

History

The conjecture is known to hold in several special cases: * when a compact affine manifold is

2-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...

(as shown by

Jean-Paul Benzécri Jean-Paul Benzécri was a French people, French mathematician and statistician. He studied at École Normale Supérieure and was professor at University of Rennes 1, Université de Rennes and later for most of his career at the Paris Institute of ...

in 1955, and later by

John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...

in 1957) * when a compact affine manifold is complete (i.e., affinely

diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...

to a quotient space of the

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...

under a proper action of a

discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...

affine transformations In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...

, then the conjecture is true; the result is shown by

Bertram Kostant Bertram Kostant (May 24, 1928 – February 2, 2017) was an American mathematician who worked in representation theory, differential geometry, and mathematical physics. Early life and education Kostant grew up in New York City, where he gradua ...

and

Dennis Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Ce ...

in 1975; the result would also immediately follow from the Auslander conjecture; Kostant and Sullivan showed that a closed manifold with nonzero Euler characteristic can't admit a complete affine structure) * when a compact affine manifold is a higher-rank irreducible locally symmetric manifold (as shown by

William Goldman William Goldman (August 12, 1931 – November 16, 2018) was an American novelist, playwright, and screenwriter. He first came to prominence in the 1950s as a novelist before turning to screenwriting. He won Academy Awards for his screenplays '' ...

and Morris Hirsch in 1984; they showed that a higher-rank irreducible locally symmetric manifold can never admit an affine structure) * when a compact affine manifold is locally a product of hyperbolic planes (as shown by Michelle Bucher and Tsachik Gelander in 2011) * when a compact affine manifold admits a parallel volume form (i.e., with linear holonomy in SL

(n, \mathbb)

; it was shown by Bruno Klingler in 2015; this weaker proven case was known as Chern's conjecture for special affine manifolds; a conjecture of Markus predicts this is equivalent to being complete) * when a compact affine manifold is a complex

hyperbolic surface In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...

(as shown by Hester Pieters in 2016) Additionally obtained related results: * In 1958, Milnor proved inequalities which completely characterise those oriented rank two bundles over a surface that admit a flat connection * In 1977, Smillie proved that the condition that the connection is torsion-free matters. For each even dimension greater than 2, Smillie constructed closed manifolds with non-zero Euler characteristic that admit a flat connection on their tangent bundleJ. Smillie, Flat manifolds with non-zero Euler characteristic, Commentarii Mathematici Helvetici, volume 52 (1977), pp. 453–456 For flat

pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

s or

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

affine manifolds, this follows from the

Chern–Gauss–Bonnet theorem In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating ...

. Also, as proven by M.W. Hirsch and

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...

in 1975 for incomplete affine manifolds, the conjecture holds if the holonomy group is a finite extension, a free product of amenable groups (however, their result applies to any flat bundles over manifolds). In 1977, John Smillie produced a manifold with the

tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

with nonzero-torsion flat connection and nonzero Euler characteristic, thus he disproved the strong version of the conjecture asking whether the Euler characteristic of a closed flat manifold vanishes. Later, Huyk Kim and Hyunkoo Lee proved for affine manifolds, and more generally projective manifolds developing into anaffine space with amenable

holonomy In differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus ...

by a different technique using nonstandard polyhedral

Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a ...

developed by Ethan Bloch and Kim and Lee. In 2002, Suhyoung Choi slightly generalized the result of Hirsch and Thurston that if the holonomy of a closed affine manifold is isomorphic to amenable groups amalgamated or HNN-extended along finite groups, then the Euler characteristic of the manifold is 0. He showed that if an even-dimensional manifold is obtained from a connected sum operation from ''K''(''π'', 1)s with amenable fundamental groups, then the manifold does not admit an affine structure (generalizing a result of Smillie). In 2008, after Smillie's simple examples of closed manifolds with flat tangent bundles (these would have affine connections with zero curvature, but possibly nonzero torsion), Bucher and Gelander obtained further results in this direction. In 2015, Mihail Cocos proposed a possible way to solve the conjecture and proved that the Euler characteristic of a closed even-dimensional affine manifold vanishes. In 2016, Huitao Feng ( zh, 冯惠涛) and Weiping Zhang, both of

Nankai University Nankai University (NKU or Nankai; ) is a national public research university located in Tianjin, China. It is a prestigious Chinese state Class A Double First Class University approved by the central government of China, and a member of the fo ...

, claimed to prove the conjecture in general case, but a serious flaw had been found, so the claim was thereafter retracted. After the correction, their current result is a formula that counts the Euler number of a flat vector bundle in terms of vertices of transversal open coverings. Notoriously, the intrinsic Chern–Gauss–Bonnet theorem proved by Chern that the Euler characteristic of a closed affine manifold is 0 applies only to orthogonal connections, not linear ones, hence why the conjecture remains open in this generality (affine manifolds are considerably more complicated than

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

s, where metric completeness is equivalent to geodesic completeness). There also exists a related conjecture by

Mikhail Leonidovich Gromov Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; russian: link=no, Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, ana ...

on the vanishing of bounded cohomology of affine manifolds.M. Gromov, Asymptotic invariants of infinite groups. Geometric group theory. Volume 2 (1993), 8.A

_4

Related conjectures

The conjecture of Chern can be considered a particular case of the following conjecture:

A closed
aspherical manifold In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups \pi_n(X) equal to 0 when n>1. If one works with CW complexes, one can reformulate this condition: an aspherical CW complex is a CW complex who ...
with nonzero Euler characteristic doesn't admit a flat structure

This conjecture was originally stated for general closed manifolds, not just for aspherical ones (but due to Smillie, there's a counterexample), and it itself can, in turn, also be considered a special case of even more general conjecture:

A closed aspherical manifold with nonzero
simplicial volume In the mathematical field of geometric topology, the simplicial volume (also called Gromov norm) is a certain measure of the topological complexity of a manifold. More generally, the simplicial norm measures the complexity of homology classes. Give ...
doesn't admit a flat structure

While generalizing the Chern's conjecture on affine manifolds in these ways, it's known as the generalized Chern conjecture for manifolds that are locally a product of surfaces.

Details

History

Related conjectures

References

Further reading