High Quality Content by WIKIPEDIA articles! In mathematics, the generalized Gauss–Bonnet theorem (also called Chern–Gauss–Bonnet theorem) presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature. It is a direct generalization of the Gauss–Bonnet theorem to higher dimensions. Let M be a compact 2n-dimensional Riemannian manifold without boundary, and let ? be the curvature form of the Levi-Civita connection. As with the two-dimensional Gauss–Bonnet Theorem, there are generalizations when M is a manifold with boundary. 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 so is an invariant of smooth structure.

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