GaussBonnet

GaussBonnet, typically written Gauss–Bonnet, refers to a fundamental result in differential geometry that connects curvature to topology, and to its higher-dimensional generalizations and applications in physics. The theorem was developed from the work of Carl Friedrich Gauss and Pierre Ossian Bonnet in the 19th century and was later generalized by Shiing-Shen Chern.

In its classical form, the Gauss-Bonnet theorem relates geometric quantities on a compact oriented surface. For a closed surface M without boundary, the integral of Gaussian curvature K over M equals 2π times the Euler characteristic χ(M): ∫M K dA = 2πχ(M). If the surface has a boundary ∂M, the formula includes a boundary term involving the geodesic curvature κg: ∫M K dA + ∮∂M κg ds = 2πχ(M). This result provides a bridge between local geometric data and global topological invariants.

The theorem extends to higher even dimensions via the Gauss-Bonnet-Chern theorem. In such cases, the Euler characteristic χ(M) is given by the integral of a curvature-derived differential form known as the Euler form. The statement holds for compact oriented Riemannian manifolds of even dimension; for odd dimensions, the Euler characteristic can vanish in many contexts, reflecting topological constraints.

In physics, the Gauss-Bonnet term appears as a curvature invariant in gravitational theories, notably in Lovelock gravity, where it contributes to the action in dimensions higher than four and yields second-order field equations. In four-dimensional spacetime, the Gauss-Bonnet term is topological for closed manifolds, influencing dynamics only under particular boundary or coupling conditions.

Historically, Gauss and Bonnet established the core identity, with later generalizations by Chern and others expanding its scope to broader geometric and physical contexts.