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 (named after Carl Friedrich Gauss and Pierre Ossian Bonnet) to higher dimensions. Let M be a compact orientable 2n-dimensional Riemannian manifold without boundary, and let be the curvature form of the Levi-Civita connection. This means that is an -valued 2-form on M. So can be regarded as a skew...

In mathematics, a diffeology on a set declares what the smooth parametrizations in the set are. In some sense a diffeology generalizes the concept of smooth charts in a differentiable manifold. The concept was first introduced by Jean-Marie Souriau in the 1980s and developed first by his students Paul Donato (homogeneous spaces and coverings) and Patrick Iglesias (diffeological fiber bundles, higher homotopy etc.), later by other people. A related idea was introduced by Kuo-Tsaï Chen (陳國才, Chen Guocai) in the 1970s, using convex sets instead of open sets for the domains of the plots. == D...