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 sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an analytical invariant).
It is a highly non-trivial generalization of the classic Gauss–Bonnet theorem (for 2-dimensional manifolds / surfaces) to higher even-dimensional Riemannian manifolds. In 1943, Carl B. Allendoerfer and André Weil proved a special case for extrinsic manifolds. In a classic paper published in 1944, Shiing-Shen Chern proved the theorem in full generality connecting global topology with local geometry.
The Riemann–Roch theorem and the Atiyah–Singer index theorem are other generalizations of the Gauss–Bonnet theorem.
Statement
One useful form of the Chern theorem is that
Since the dimension is 2n, we have that <math>\Omega</math> is an <math>\mathfrak s\mathfrak o(2n)</math>-valued 2-differential form on <math> M </math> (see special orthogonal group). So <math>\Omega</math> can be regarded as a skew-symmetric 2n × 2n matrix whose entries are 2-forms, so it is a matrix over the commutative ring <math display="inline">{\bigwedge}^\text{even}\,T^*M</math>. Hence the Pfaffian is a 2n-form. It is also an invariant polynomial.
However, Chern's theorem in general is that for any closed <math>C^\infty</math> orientable n-dimensional <math> M </math>,
In 2013, a proof of the theorem via supersymmetric Euclidean field theories was also found.
The theorem has also found numerous applications in physics, including:
:<math>\int_M K\,dV = \frac{1}{2}\gamma_n\,\chi(M) </math>
where <math> dV </math> is the volume element of the hypersurface, <math>K</math> is the Jacobian determinant of the Gauss map, and <math>\gamma_n</math> is the surface area of the unit n-sphere.
Gauss–Bonnet theorem
The Gauss–Bonnet theorem is a special case when <math> M </math> is a 2-dimensional 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 <math> M </math> is a manifold with boundary.
Further generalizations
Atiyah–Singer
A far-reaching generalization of the Gauss–Bonnet theorem is the Atiyah–Singer Index Theorem.
There is also a version of Chern's formula for orbifolds.
History
Shiing-Shen Chern published his proof of the theorem in 1944 while at the Institute for Advanced Study. This was historically the first time that the formula was proven without assuming the manifold to be embedded in a Euclidean space, which is what it means by "intrinsic". The special case for a hypersurface (an (n-1)-dimensional submanifold in an n-dimensional Euclidean space) was proved by H. Hopf in which the integrand is the Gauss–Kronecker curvature (the product of all principal curvatures at a point of the hypersurface). This was generalized independently by Allendoerfer in 1939 and Fenchel in 1940 to a Riemannian submanifold of a Euclidean space of any codimension, for which they used the Lipschitz–Killing curvature (the average of the Gauss–Kronecker curvature along each unit normal vector over the unit sphere in the normal space; for an even dimensional submanifold, this is an invariant only depending on the Riemann metric of the submanifold). Their result would be valid for the general case if the Nash embedding theorem can be assumed. However, this theorem was not available then, as John Nash published his famous embedding theorem for Riemannian manifolds in 1956. In 1943 Allendoerfer and Weil published their proof for the general case, in which they first used an approximation theorem of H. Whitney to reduce the case to analytic Riemannian manifolds, then they embedded "small" neighborhoods of the manifold isometrically into a Euclidean space with the help of the Cartan–Janet local embedding theorem, so that they can patch these embedded neighborhoods together and apply the above theorem of Allendoerfer and Fenchel to establish the global result. This is, of course, unsatisfactory for the reason that the theorem only involves intrinsic invariants of the manifold, then the validity of the theorem should not rely on its embedding into a Euclidean space. Weil met Chern in Princeton after Chern arrived in August 1943. He told Chern that he believed there should be an intrinsic proof, which Chern was able to obtain within two weeks. The result is Chern's classic paper "A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds" published in the Annals of Mathematics the next year. The earlier work of Allendoerfer, Fenchel, Allendoerfer and Weil were cited by Chern in this paper. The work of Allendoerfer and Weil was also cited by Chern in his second paper related to the same topic.