Knowra Gauss–Bonnet theorem Gauss–Bonnet theorem The Gauss–Bonnet theorem equates a surface’s total Gaussian curvature, with boundary curvature included when needed, to a quantity determined by its Euler characteristic.
Gaussian curvature : A measure of a surface’s intrinsic curvature at a point, defined as the product of its principal curvatures. Its surface integral is the curvature term in the theorem.
Differential geometry : The study of smooth spaces using calculus, geometric structures, and invariants. The theorem uses differential geometry to integrate curvature across surfaces.
Carl Friedrich Gauss : A German mathematician whose work shaped number theory, geometry, statistics, and mathematical physics. His 1827 surface-curvature theorem is a foundational precursor.
Chern–Gauss–Bonnet theorem : A higher-dimensional generalization relating curvature integrals on a compact even-dimensional Riemannian manifold to its Euler characteristic. It extends the surface theorem’s curvature-topology relation to even dimensions.
geodesic curvature : The curvature of a curve within a surface, measured by how it bends away from a surface geodesic. For surfaces with boundary, its boundary integral completes the curvature total.
Topology : The study of properties of spaces preserved under continuous deformation. The Euler characteristic on the theorem’s right side is topological.
Pierre Ossian Bonnet : A French mathematician known for contributions to differential geometry and the theory of surfaces. His work extended the curvature relation associated with Gauss.
Poincaré–Hopf theorem : A theorem equating the sum of indices of a vector field’s isolated zeros to a manifold’s Euler characteristic. It provides another way to compute the same topological invariant.
Euler characteristic : A topological invariant equal, for a finite cell decomposition, to vertices minus edges plus faces. The theorem’s curvature integral is fixed by this invariant.
Riemannian metric : A smoothly varying inner product on tangent spaces that defines lengths and angles on a manifold. A metric supplies the lengths and curvature integrated by the theorem.
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