Riemannian Geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds—manifolds equipped with a metric tensor.

The Metric Tensor

A Riemannian metric $g$ is a collection of inner products on the tangent spaces of a manifold. In coordinates, it is represented by a symmetric matrix $g_{ij}$ .

Interactive Lab

import numpy as np

def calculate_length_on_manifold(g, path, dt):
  # Length L = integral sqrt(g_ij * dx^i/dt * dx^j/dt) dt
  length = 0
  for i in range(len(path) - 1):
      v = (path[i+1] - path[i]) / dt
      # Simplified: assumes g is constant locally
      speed = np.sqrt(v.T @ g @ v)
      length += speed * dt
  return length

# Polar metric: g = diag(1, r^2)
def polar_metric(r):
  return np.array([[1, 0], [0, r**2]])

# Circle path at r=1: (1, t) for t in [0, 2*pi]
dt = 0.01
t = np.arange(0, 2 * np.pi, dt)
path = np.column_stack((np.ones_like(t), t))

g = polar_metric(1.0)
print(f"Circumference (r=1): {calculate_length_on_manifold(g, path, dt):.3f}")

python

1import numpy as np

3def calculate_length_on_manifold(g, path, dt):

4 # Length L = integral sqrt(g_ij * dx^i/dt * dx^j/dt) dt

5 length = 0

6 for i in range(len(path) - 1):

7 v = (path[i+1] - path[i]) / dt

8 # Simplified: assumes g is constant locally

9 speed = np.sqrt(v.T @ g @ v)

10 length += speed * dt

11 return length

13# Polar metric: g = diag(1, r^2)

14def polar_metric(r):

15 return np.array([[1, 0], [0, r**2]])

17# Circle path at r=1: (1, t) for t in [0, 2*pi]

18dt = 0.01

19t = np.arange(0, 2 * np.pi, dt)

20path = np.column_stack((np.ones_like(t), t))

22g = polar_metric(1.0)

23print(f"Circumference (r=1): {calculate_length_on_manifold(g, path, dt):.3f}")

Geodesics

Geodesics are the “straightest possible” paths on a manifold. They generalize the notion of a line to curved spaces. In GR, particles move along geodesics in spacetime.

What is the shortest path between two points on a sphere?

Curvature Tensors

Curvature in Riemannian geometry is captured by the Riemann Curvature Tensor $R^l_{ijk}$ . It measures the extent to which parallel transport depends on the path taken.

In General Relativity, what physical quantity determines the metric tensor of spacetime?

Interactive Lab

def christoffel_symbol_approx(g_inv, dg_dx):
  # Very simplified version of Gamma calculations
  pass

print("Riemannian manifolds are the foundation of modern gravitation theory.")

python

1def christoffel_symbol_approx(g_inv, dg_dx):

2 # Very simplified version of Gamma calculations

3 pass

5print("Riemannian manifolds are the foundation of modern gravitation theory.")