Euler Characteristic
χ = 2
| Property | Value |
|---|
Rotation
Classification Theorem
Every compact, connected, orientable surface without boundary is homeomorphic to a sphere with g handles.
The Euler characteristic χ = V − E + F = 2 − 2g is a topological invariant — unchanged by continuous deformation.
Non-orientable surfaces (Klein bottle, projective plane) are classified by their cross-cap number. The Klein bottle has χ = 0 but cannot be embedded in ℝ³ without self-intersection.
The Gauss-Bonnet theorem: ∮κ dA = 2πχ — total Gaussian curvature equals 2π times the Euler characteristic.