Riemann Curvature Tensor of the Embedded Torus $T^2$¶

A ring torus with tube radius $a$ and major radius $b$ is embedded by

$$ X(\theta,\phi)=\bigl((b+a\cos\theta)\cos\phi, (b+a\cos\theta)\sin\phi,a\sin\theta\bigr),\qquad b>a>0. $$

Unlike the intrinsically flat quotient torus, this embedded torus has curvature that changes sign. We compute its connection, Riemann tensor, Ricci tensor, and scalar curvature.

Coordinates and metric¶

The coordinate vectors are orthogonal and have squared lengths $a^2$ and $(b+a\cos\theta)^2$. Therefore

$$ g=\begin{pmatrix}a^2&0\\0&(b+a\cos\theta)^2\end{pmatrix}. $$

declare symbol a, b, θ, φ: MathValue

def x : Vector MathValue := [| θ, φ |]
def X : Vector MathValue :=
  [| `(a * cos θ + b) * cos φ
   , `(a * cos θ + b) * sin φ
   , a * sin θ
   |]

def e_i_j : Matrix MathValue := ∂/∂ X_j x~i

def g_i_j : Matrix MathValue :=
  generateTensor (\[u, v] -> V.* e_u_# e_v_#) [2, 2]

def g~i~j : Matrix MathValue := M.inverse g_#_#

$\begin{pmatrix} \cos(φ) (\cos(θ) a + b) \\ \sin(φ) (\cos(θ) a + b) \\ \sin(θ) a\\ \end{pmatrix}$

g_#_#

$\begin{pmatrix} a^{2} & 0 \\ 0 & (\cos(θ) a + b)^{2} \\ \end{pmatrix}_{\#\#}^{\;\;}$

g~#~#

$\begin{pmatrix} a^{-2} & 0 \\ 0 & (\cos(θ) a + b)^{-2} \\ \end{pmatrix}_{\;\;}^{\#\#}$

Levi-Civita connection¶

We use $\Gamma_{ijk}=\tfrac12(\partial_jg_{ik}+\partial_kg_{ij}-\partial_ig_{jk})$ and raise the first index with $g^{ij}$.

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

Γ~1_#_#

$\begin{pmatrix} 0 & 0 \\ 0 & \sin(θ) (\cos(θ) a + b) a^{-1} \\ \end{pmatrix}_{\#\#}^{\;\;}$

Γ~2_#_#

$\begin{pmatrix} 0 & -\sin(θ) (\cos(θ) a + b)^{-1} a \\ -\sin(θ) (\cos(θ) a + b)^{-1} a & 0 \\ \end{pmatrix}_{\#\#}^{\;\;}$

Riemann tensor¶

The last two indices are antisymmetric. Lowering the first index also exposes the familiar pair symmetries of $R_{ijkl}$.

def R~i_j_k_l : Tensor MathValue := withSymbols [m]
  ∂/∂ Γ~i_j_l x~k - ∂/∂ Γ~i_j_k x~l
    + Γ~m_j_l . Γ~i_m_k - Γ~m_j_k . Γ~i_m_l

def R_i_j_k_l : Tensor MathValue := withSymbols [m]
  g_i_m . R~m_j_k_l

R~#_#_1_2

$\begin{pmatrix} 0 & \cos(θ) (\cos(θ) a + b) a^{-1} \\ -\cos(θ) (\cos(θ) a + b)^{-1} a & 0 \\ \end{pmatrix}_{\;\#}^{\#\;}$

R_1_2_1_2

$\cos(θ) (\cos(θ) a + b) a$

Ricci, scalar, and Gaussian curvature¶

In two dimensions the scalar curvature is twice the Gaussian curvature: $\mathcal R=2K$. For this torus,

$$ K(\theta)=\frac{\cos\theta}{a(b+a\cos\theta)}. $$

def Ric_i_j : Matrix MathValue := withSymbols [m]
  sum (contract R~m_i_m_j)

def scalarCurvature : MathValue := withSymbols [i, j]
  g~i~j . Ric_i_j

def gaussianCurvature : MathValue := scalarCurvature / 2

Ric_#_#

$\begin{pmatrix} \cos(θ) (\cos(θ) a + b)^{-1} a & 0 \\ 0 & \cos(θ) (\cos(θ) a + b) a^{-1} \\ \end{pmatrix}_{\#\#}^{\;\;}$

scalarCurvature

$2 \cos(θ) (\cos(θ) a + b)^{-1} a^{-1}$

gaussianCurvature

$\cos(θ) (\cos(θ) a + b)^{-1} a^{-1}$

Interpretation¶

The outside of the torus ($\cos\theta>0$) is positively curved, the inside ($\cos\theta<0$) is negatively curved, and the top and bottom circles are parabolic. The positive and negative contributions cancel in the Gauss--Bonnet integral, consistently with $\chi(T^2)=0$.