Euler Characteristic - Relations To Other Invariants

Relations To Other Invariants

The Euler characteristic of a closed orientable surface can be calculated from its genus g (the number of tori in a connected sum decomposition of the surface; intuitively, the number of "handles") as

The Euler characteristic of a closed non-orientable surface can be calculated from its non-orientable genus k (the number of real projective planes in a connected sum decomposition of the surface) as

For closed smooth manifolds, the Euler characteristic coincides with the Euler number, i.e., the Euler class of its tangent bundle evaluated on the fundamental class of a manifold. The Euler class, in turn, relates to all other characteristic classes of vector bundles.

For closed Riemannian manifolds, the Euler characteristic can also be found by integrating the curvature; see the Gauss–Bonnet theorem for the two-dimensional case and the generalized Gauss–Bonnet theorem for the general case.

A discrete analog of the Gauss–Bonnet theorem is Descartes' theorem that the "total defect" of a polyhedron, measured in full circles, is the Euler characteristic of the polyhedron; see defect (geometry).

Hadwiger's theorem characterizes the Euler characteristic as the unique (up to scalar multiplication) translation-invariant, finitely additive, not-necessarily-nonnegative set function defined on finite unions of compact convex sets in Rn that is "homogeneous of degree 0".