Chern–Gauss–Bonnet theorem

A theorem equating the integral of a curvature-derived Euler form over a compact, oriented, even-dimensional Riemannian manifold to its Euler characteristic.

Euler form: A differential form constructed from curvature whose integral, with a fixed normalization, gives the Euler characteristic of an even-dimensional manifold. Its curvature-polynomial integral is the theorem’s geometric side.
Riemannian manifold: A smooth manifold equipped with a smoothly varying inner product on each tangent space. The metric supplies the curvature used in the theorem.
Carl Friedrich Gauss: A German mathematician whose work established foundational results in number theory, geometry, and mathematical physics. His theorema egregium and curvature formula underpin the surface version.
Gauss–Bonnet theorem with boundary: A Gauss–Bonnet formula that includes boundary curvature terms for a compact surface with boundary. Boundary contributions are absent from the stated closed-manifold setting.
Curvature tensor: A tensor measuring how a connection’s covariant derivatives fail to commute, encoding the intrinsic curvature of a Riemannian manifold. The Euler form is built from contractions of this tensor.
Orientability: The property of a manifold admitting a consistent choice of orientation across its coordinate charts. An orientation gives the Euler form an integral with a consistent sign.
Pierre Ossian Bonnet: A French mathematician who proved a global relation between total curvature and the topology of a surface. His work extended the curvature relation beyond Gauss’s local result.
Chern–Simons form: A differential form whose exterior derivative is a characteristic form, defined locally from a connection. It appears in boundary transgression formulas rather than the closed-manifold integral itself.
Pfaffian: A polynomial in the entries of a skew-symmetric matrix whose square is its determinant. In even dimensions, the Euler form is expressed using a Pfaffian of curvature.
Compact space: A topological space in which every open cover has a finite subcover. Compactness makes the manifold integral finite without boundary terms in the closed case.

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