Chern–Gauss–Bonnet theorem

In mathematics, the Chern theorem (or sometimes Chern formula or Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss and Pierre Ossian Bonnet) states that the Euler characteristic of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial of its curvature. It is a direct generalization of the Gauss–Bonnet theorem (for 2d surfaces) to higher dimensions and was first published by Shiing-Shen Chern in 1945, connecting global topology with local geometry.[1]

Statement

One useful form of the Chern theorem is that[2][3]

${\displaystyle \chi (M)=\int _{M}e(\Omega )}$

where ${\displaystyle \chi (M)}$ denotes the Euler characteristic of M. The Euler class is defined as

${\displaystyle e(\Omega )={\frac {1}{(2\pi )^{n}}}\operatorname {Pf} (\Omega ).}$

where we have the Pfaffian ${\displaystyle \operatorname {Pf} (\Omega )}$. Here M is a compact orientable 2n-dimensional Riemannian manifold without boundary, and ${\displaystyle \Omega }$ is the associated curvature form of the Levi-Civita connection. In fact the statement holds with ${\displaystyle \Omega }$ the curvature form of any metric connection on the tangent bundle, as well as for other vector bundles over ${\displaystyle M}$.[4]

Since the dimension is 2n, we have that ${\displaystyle \Omega }$ is an ${\displaystyle {\mathfrak {s}}{\mathfrak {o}}(2n)}$-valued 2-differential form on M (see special orthogonal group). So ${\displaystyle \Omega }$ can be regarded as a skew-symmetric 2n × 2n matrix whose entries are 2-forms, so it is a matrix over the commutative ring ${\textstyle {\bigwedge }^{\text{even}}\,T^{*}M}$. Hence the Pfaffian is a 2n-form. It is also an invariant polynomial.

However, Chern's theorem in general is that for any closed ${\displaystyle C^{\infty }}$ orientable n-dimensional M,[2]

${\displaystyle \chi (M)=(e(TM),[M])}$

where the above denotes the cap product with the Euler class of the tangent bundle TM.

Applications

The Gauss–Bonnet theorem can be seen as a special instance in the theory of characteristic classes. The Gauss–Bonnet integrand is the Euler class. Since it is a top-dimensional differential form, it is closed. The naturality of the Euler class means that when you change the Riemannian metric, you stay in the same cohomology class. That means that the integral of the Euler class remains constant as you vary the metric and is thus a global invariant of the smooth structure.[3]

The theorem has also found numerous applications in physics, including:[3]

Special cases

Four-dimensional manifolds

In dimension ${\displaystyle 2n=4}$, for a compact oriented manifold, we get

${\displaystyle \chi (M)={\frac {1}{32\pi ^{2}}}\int _{M}\left(|{\text{Riem}}|^{2}-4|{\text{Ric}}|^{2}+R^{2}\right)\,d\mu }$

where ${\displaystyle {\text{Riem}}}$ is the full Riemann curvature tensor, ${\displaystyle {\text{Ric}}}$ is the Ricci curvature tensor, and ${\displaystyle R}$ is the scalar curvature. This is particularly important in general relativity, where spacetime is viewed as a 4-dimensional manifold.

Gauss–Bonnet theorem

The Gauss–Bonnet theorem is a special case when M is a 2d manifold. It arises as the special case where the topological index is defined in terms of Betti numbers and the analytical index is defined in terms of the Gauss–Bonnet integrand.

As with the two-dimensional Gauss–Bonnet theorem, there are generalizations when M is a manifold with boundary.

Further generalizations

Atiyah–Singer

A far-reaching generalization of the Gauss–Bonnet theorem is the Atiyah–Singer Index Theorem.[3]

Let ${\displaystyle D}$ be a weakly elliptic differential operator between vector bundles. That means that the principal symbol is an isomorphism. Strong ellipticity would furthermore require the symbol to be positive-definite.

Let ${\displaystyle D^{*}}$ be its adjoint operator. Then the analytical index is defined as

dim(ker(D)) − dim(ker(D*)),

By ellipticity this is always finite. The index theorem says that this is constant as you vary the elliptic operator smoothly. It is in fact equal to a topological index, which can be expressed in terms of characteristic classes like the Euler class.

The GB theorem is derived by considering the Dirac operator

${\displaystyle D=d+d^{*}}$

Odd dimensions

The Chen formula is defined for even dimensions because the Euler characteristic vanishes for odd dimension. There is some research being done on 'twisting' the index theorem in K-theory to give non-trivial results for odd dimension.[5][6]

There is also a version of Chen's formula for orbifolds.[7]

History

Chen Xingshen published his proof of the theorem in 1945 while at the Institute for Advanced Study. This was historically the first time that the formula was proven without assuming the manifold to be a hypersurface (manifolds embedded in Euclidean space). In 1940 Allendoerfer and Weil proved the special case for hypersurfaces, which Chern cited in his paper.[1]