Euler Form of the Two-Sphere¶

The Euler class of an oriented rank-two tangent bundle is represented by a curvature two-form. For the round sphere,

$$\int_{S^2} e(TS^2)=\chi(S^2)=2.$$

We compute the metric from the embedding, transform the connection to an orthonormal frame, and apply Cartan's curvature equation.

Embedding and induced metric¶

The radius-$r$ sphere is parametrized by $(\theta,\phi)$. Dot products of its coordinate tangent vectors produce the metric rather than taking it as an input.

declare symbol r, θ, φ : MathValue

def x : Vector MathValue := [| θ, φ |]
def X : Vector MathValue :=
  [| r * sin θ * cos φ
   , r * sin θ * sin φ
   , r * cos θ |]

def e_i_j : Matrix MathValue := ∂/∂ X_j x~i
def g_i_j : Matrix MathValue :=
  generateTensor (\[a, b] -> V.* e_a_# e_b_#) [2, 2]
def g~i~j : Matrix MathValue := M.inverse g_#_#

g_#_#

$\begin{pmatrix} r^{2} & 0 \\ 0 & \sin(θ)^{2} r^{2} \\ \end{pmatrix}_{\#\#}^{\;\;}$

Levi-Civita connection and orthonormal frame¶

The diagonal vielbein rescales the coordinate basis by $r$ and $r\sin\theta$. Under a frame change $A$, the connection transforms as

$$\omega=A^{-1}\omega_0A+A^{-1}dA.$$

def Γ_i_j_k : Tensor MathValue :=
  (1 / 2) *
    (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i)

def Γ~i_j_k : Tensor MathValue := withSymbols [m]
  g~i~m . Γ_m_j_k

def A : Matrix MathValue :=
  [| [| 1 / r, 0 |], [| 0, 1 / (r * sin θ) |] |]

def d (t : Tensor MathValue) : Tensor MathValue :=
  !(flip ∂/∂) x t

def ω0~i_j : Matrix MathValue := Γ~i_j_#
def ω~i_j : Tensor MathValue := withSymbols [a, b]
  (M.inverse A)~i_a . ω0~a_b . A~b_j
  + (M.inverse A)~i_a . d A~a_j

Curvature and Euler form¶

Cartan's second equation gives $\Omega$. In two dimensions the Pfaffian reduces to the difference of the two off-diagonal curvature components.

def Ω~i_j : Tensor MathValue := withSymbols [k]
  antisymmetrize (d ω~i_j + ω~i_k ∧ ω~k_j)

def eulerForm : Tensor MathValue :=
  (1 / (4 * π)) * withSymbols [t1, t2]
    (Ω~1_2_t1_t2 - Ω~2_1_t1_t2)

eulerForm

$\begin{pmatrix} 0 & \frac{1}{4} \sin(θ) π^{-1} \\ \frac{-1}{4} \sin(θ) π^{-1} & 0 \\ \end{pmatrix}$

The upper tensor component is

$$e_{12}=\frac{\sin\theta}{4\pi}.$$

A full antisymmetric tensor stores both $e_{12}$ and $e_{21}$, so the corresponding oriented differential-form density is

$$e(TS^2)=\frac{\sin\theta}{2\pi}\,d\theta\wedge d\phi.$$

Therefore

$$\int_0^{2\pi}\!\int_0^\pi \frac{\sin\theta}{2\pi}\,d\theta\,d\phi=2,$$

recovering the Euler characteristic of the sphere. The radius cancels, illustrating that the Euler number is topological rather than metric-size dependent.

Euler Form of the Two-Sphere¶

Embedding and induced metric¶

Levi-Civita connection and orthonormal frame¶

Curvature and Euler form¶

Links