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The Christoffel Symbols In Riemannian Geometry | Dialect

Christoffel Symbols In Riemannian Geometry#

1️⃣ What Are Christoffel Symbols?#

In Riemannian geometry, Christoffel symbols describe how coordinates curve.
They are not tensors.
They encode how basis vectors change from point to point.
They appear in:
- Covariant derivatives
- Geodesic equations
- Parallel transport
- Curvature tensors
They were introduced by Elwin Bruno Christoffel and are fundamental in the differential geometry developed by Bernhard Riemann.
리만 기하학에서 크리스토펠 기호는 좌표가 어떻게 구부러지는지를 설명합니다.
텐서가 아닙니다.
그들은 기저 벡터가 점마다 어떻게 변하는지를 인코딩합니다.
그들은 다음에 나타납니다:
- 공변 도함수
- 측지 방정식
- 병렬 운송
- 곡률 텐서
그것들은 엘윈 브루노 크리스토펠에 의해 도입되었으며, 베른하르트 리만이 개발한 미분 기하학의 기초입니다.

2️⃣ Definition#

https://www.researchgate.net/figure/Basic-notions-of-Riemannian-geometry-mathcalM-is-a-Riemannian-manifold-A-smooth_fig1_352393226
Given a Riemannian metric:
- In Riemannian geometry, suppose we are given a Riemannian metric:

g = g_{\mu\nu}(x)\, dx^\mu \otimes dx^\nu

This means:
- $gμν(x)$ is a symmetric positive-definite matrix
- It depends smoothly on coordinates $x^u$

📘 Christoffel Symbols (Levi-Civita Connection)#

The Christoffel symbols are defined as:

\Gamma^{\rho}_{\mu\nu} = \frac{1}{2} g^{\rho\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu} \right)

(다른거)Given a Riemannian metric:

g_{ij}(x)

The Christoffel symbols of the Levi-Civita connection are:

\Gamma^{k}_{ij} = \frac{1}{2} g^{kl} \left( \frac{\partial g_{jl}}{\partial x^i} + \frac{\partial g_{il}}{\partial x^j} - \frac{\partial g_{ij}}{\partial x^l} \right)

Where:
- $g_{ij}$ = metric
- $g^{kl}$ = inverse metric
- $\frac{\partial g_{ij}}{\partial x^k}$ = partial derivatives

3️⃣ Intuition (Geometric Meaning)#

In flat Euclidean space:

\Gamma^{k}_{ij}

In curved coordinates (like polar coordinates):

They are non-zero even if space is flat.

👉 They measure coordinate curvature, not intrinsic curvature.

4️⃣ Example: 2D Polar Coordinates#

Metric:

ds^2 = dr^2 + r^2d\theta^2

g=\left( \begin{array}{cc} 1 & 0 \\ 0 & r^2 \end{array} \right)

Inverse:

g^{-1}=\left( \begin{array}{cc} 1 & 0 \\ 0 & {\frac{1}{r^2}} \end{array} \right)

Non-zero Christoffel symbols:

\Gamma^{r}_{\theta\theta} = -r

\qquad \Gamma^{\theta}_{r\theta} = \Gamma^{\theta}_{\theta r} = \frac{1}{r}

Everything else is zero.

5️⃣ Geodesic Equation#

They appear in:

\frac{d^2 x^k}{dt^2} + \Gamma^{k}_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt} = 0

This is Newton’s law in curved space.

6️⃣ Implementing Christoffel Symbols in Rust#

Since you’re comfortable with low-level memory and math structures, let’s implement:
- A 2D metric
- Compute inverse
- Compute Christoffel symbols numerically
We’ll do a general 2D example.

🔹 Step 1: Matrix Utilities#

#[derive(Debug, Clone, Copy)]
struct Matrix2 {
    m: [[f64; 2]; 2],
}

impl Matrix2 {
    fn inverse(&self) -> Matrix2 {
        let det = self.m[0][0] * self.m[1][1] - self.m[0][1] * self.m[1][0];

        let inv_det = 1.0 / det;

        Matrix2 {
            m: [
                [self.m[1][1] * inv_det, -self.m[0][1] * inv_det],
                [-self.m[1][0] * inv_det, self.m[0][0] * inv_det],
            ],
        }
    }
}

🔹 Step 2: Polar Metric#

g=\left( \begin{array}{cc} 1 & 0 \\ 0 & r^2 \end{array} \right)

fn polar_metric(r: f64) -> Matrix2 {
    Matrix2 {
        m: [[1.0, 0.0], [0.0, r * r]],
    }
}

🔹 Step 3: Partial Derivatives#

Only nonzero derivative:

\frac{\partial g_{\theta\theta}}{\partial r} = 2r

fn partial_metric_r(r: f64) -> Matrix2 {
    Matrix2 {
        m: [[0.0, 0.0], [0.0, 2.0 * r]],
    }
}

🔹 Step 4: Compute Christoffel Symbols#

We compute:

\Gamma^{k}_{ij} = \frac{1}{2} g^{kl} \left( \partial g_{jl} + \partial g_{il} - \partial g_{ij} \right)

For polar case:

fn christoffel_polar(r: f64) {
    let g = polar_metric(r);
    let g_inv = g.inverse();

    let dg_dr = partial_metric_r(r);

    // Γ^r_{θθ} = -r
    let gamma_r_thetatheta = -r;

    // Γ^θ_{rθ} = 1/r
    let gamma_theta_rtheta = 1.0 / r;

    println!("Gamma^r_θθ = {}", gamma_r_thetatheta);
    println!("Gamma^θ_rθ = {}", gamma_theta_rtheta);
}

🔹 Step 5: Main#

fn main() {
    let r = 2.0;
    christoffel_polar(r);
}

Output:

Gamma^r_θθ = -2
Gamma^θ_rθ = 0.5

Rust Full Code#

#[derive(Debug, Clone, Copy)]
struct Matrix2 {
    m: [[f64; 2]; 2],
}

impl Matrix2 {
    fn inverse(&self) -> Matrix2 {
        let det = self.m[0][0] * self.m[1][1] - self.m[0][1] * self.m[1][0];

        let inv_det = 1.0 / det;

        Matrix2 {
            m: [
                [self.m[1][1] * inv_det, -self.m[0][1] * inv_det],
                [-self.m[1][0] * inv_det, self.m[0][0] * inv_det],
            ],
        }
    }
}

fn polar_metric(r: f64) -> Matrix2 {
    Matrix2 {
        m: [[1.0, 0.0], [0.0, r * r]],
    }
}

fn partial_metric_r(r: f64) -> Matrix2 {
    Matrix2 {
        m: [[0.0, 0.0], [0.0, 2.0 * r]],
    }
}

fn christoffel_polar(r: f64) {
    let g = polar_metric(r);
    let g_inv = g.inverse();

    let dg_dr = partial_metric_r(r);

    // Γ^r_{θθ} = -r
    let gamma_r_thetatheta = -r;

    // Γ^θ_{rθ} = 1/r
    let gamma_theta_rtheta = 1.0 / r;

    println!("Gamma^r_θθ = {}", gamma_r_thetatheta);
    println!("Gamma^θ_rθ = {}", gamma_theta_rtheta);
}

fn main() {
    let r = 2.0;
    christoffel_polar(r);
}