Editing Pi (section)

=== Topology ===
[[File:Order-7 triangular tiling.svg|thumb|right|[[Uniformization theorem|Uniformization]] of the [[Klein quartic]], a surface of [[genus (mathematics)|genus]] three and Euler characteristic −4, as a quotient of the [[Poincaré disk model|hyperbolic plane]] by the [[symmetry group]] [[PSL(2,7)]] of the [[Fano plane]].  The hyperbolic area of a fundamental domain is {{math|8π}}, by Gauss–Bonnet.]]
The constant {{pi}} appears in the [[Gauss–Bonnet formula]] which relates the [[differential geometry of surfaces]] to their [[topology]].  Specifically, if a [[compact space|compact]] surface {{math|Σ}} has [[Gauss curvature]] ''K'', then

<math display=block>\int_\Sigma K\,dA = 2\pi \chi(\Sigma)</math>

where {{math|''χ''(Σ)}} is the [[Euler characteristic]], which is an integer.<ref>{{cite book |title=A Comprehensive Introduction to Differential Geometry |volume=3 |first=Michael |last=Spivak |year=1999 |publisher=Publish or Perish Press |author-link=Michael Spivak}}; Chapter 6.</ref>  An example is the surface area of a sphere ''S'' of curvature 1 (so that its [[radius of curvature]], which coincides with its radius, is also 1.)  The Euler characteristic of a sphere can be computed from its [[homology group]]s and is found to be equal to two.  Thus we have

<math display=block>A(S) = \int_S 1\,dA = 2\pi\cdot 2 = 4\pi</math>

reproducing the formula for the surface area of a sphere of radius 1.

The constant appears in many other integral formulae in topology, in particular, those involving [[characteristic class]]es via the [[Chern–Weil homomorphism]].<ref>{{cite book |last1=Kobayashi |first1=Shoshichi |last2=Nomizu |first2=Katsumi |title=Foundations of Differential Geometry |volume=2 |publisher=[[Wiley Interscience]] |year=1996 |edition=New |page=293 |title-link=Foundations of Differential Geometry}}; Chapter XII ''Characteristic classes''</ref>