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← Back to Part I: Differential Geometry

Chapter 1: Manifolds and Topology

A manifold is the mathematical arena for General Relativity—a space that looks locally like ℝⁿ but may have global curvature and topology. Spacetime itself is a 4-dimensional Lorentzian manifold.

What is a Manifold?

A differentiable manifold M is a topological space that:

1. Locally Euclidean

Every point has a neighborhood homeomorphic to an open set in ℝⁿ. This means locally, the manifold "looks flat"—like ordinary Euclidean space.

2. Hausdorff and Second Countable

Technical conditions ensuring the space is "well-behaved": distinct points can be separated, and the topology has a countable basis.

3. Smooth Structure

Coordinate charts overlap smoothly (C∞ transition functions), allowing calculus on the manifold.

Key Examples:

• ℝⁿ (Euclidean space) — the simplest manifold
• S² (2-sphere) — curved but compact
• T² (2-torus) — has different topology than sphere
• Spacetime M⁴ — our 4D universe!

Coordinate Charts and Atlases

A coordinate chart (U, φ) is an open set U ⊂ M with a homeomorphism φ: U → ℝⁿ assigning coordinates to each point.

$ \phi: U \rightarrow \mathbb{R}^n, \quad p \mapsto (x^1(p), x^2(p), \ldots, x^n(p)) $

An atlas is a collection of charts covering the entire manifold. For the 2-sphere, we need at least two charts (e.g., stereographic projections from north and south poles).

Transition Functions

Where charts overlap, the transition function $ \phi_\beta \circ \phi_\alpha^{-1} $ must be smooth (C∞). This is what makes M a differentiable manifold.

Tangent Spaces

At each point p ∈ M, the tangent space T_pM is the vector space of all tangent vectors at p. These are the "velocities" of curves passing through p.

Definition via Derivations

A tangent vector v at p is a linear map v: C∞(M) → ℝ satisfying the Leibniz rule:

$ v(fg) = f(p)v(g) + g(p)v(f) $

Coordinate Basis

In coordinates (x¹, ..., xⁿ), the tangent space has basis vectors:

$ \left\{ \frac{\partial}{\partial x^1}, \ldots, \frac{\partial}{\partial x^n} \right\} $

Any tangent vector: $ v = v^\mu \frac{\partial}{\partial x^\mu} $ (Einstein summation)

Python Example: 2-Sphere Manifold

This Python code demonstrates coordinate charts, the metric tensor, and curvature calculations on the 2-sphere—a simple curved manifold with interactive visualization.

2-Sphere Manifold Analysis

Python

Coordinate charts, metric tensor, and curvature on the 2-sphere

manifold_sphere.py128 lines

#!/usr/bin/env python3
"""
manifold_sphere.py - Manifolds and Coordinate Charts in General Relativity
Demonstrates coordinate transformations on a 2-sphere
"""
import numpy as np
import matplotlib.pyplot as plt
import base64
from io import BytesIO

# Define the 2-sphere embedding in R^3
def sphere_embedding(theta, phi, R=1):
    """Map spherical coordinates to Cartesian"""
    x = R * np.sin(theta) * np.cos(phi)
    y = R * np.sin(theta) * np.sin(phi)
    z = R * np.cos(theta)
    return x, y, z

# Create coordinate grid
theta = np.linspace(0.01, np.pi-0.01, 50)
phi = np.linspace(0, 2*np.pi, 50)
THETA, PHI = np.meshgrid(theta, phi)

# Compute embedding
X, Y, Z = sphere_embedding(THETA, PHI)

# Calculate metric components g_ab for 2-sphere
# ds² = R²(dθ² + sin²θ dφ²)
R = 1.0
g_theta_theta = R**2 * np.ones_like(THETA)
g_phi_phi = R**2 * np.sin(THETA)**2

# Compute Gaussian curvature K = 1/R² (constant for sphere)
K = 1.0 / R**2

print("╔═══════════════════════════════════════════════════════════════════════╗")
print("║            2-Sphere Manifold Analysis                                 ║")
print("╚═══════════════════════════════════════════════════════════════════════╝")
print()
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  METRIC TENSOR COMPONENTS")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print("  The 2-sphere metric in spherical coordinates:")
print()
print("  $ ds^2 = R^2(d\\theta^2 + \\sin^2\\theta \\, d\\phi^2) $")
print()
print(f"  At θ = π/4:")
print(f"    $ g_{{\\theta\\theta}} = {R**2:.4f} $")
print(f"    $ g_{{\\phi\\phi}} = {R**2 * np.sin(np.pi/4)**2:.4f} $")
print(f"    $ \\det(g) = {R**4 * np.sin(np.pi/4)**2:.4f} $")
print()
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  GAUSSIAN CURVATURE")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print(f"  $ K = \\frac{{1}}{{R^2}} = {K:.4f} $")
print()
print("  This demonstrates the intrinsic curvature of the sphere.")
print("  The sphere cannot be flattened without distortion!")
print()
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  COORDINATE SINGULARITIES")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print("  At θ → 0:    $ g_{{\\phi\\phi}} \\to 0 $  (coordinate artifact at poles)")
print("  At θ = π/2:  $ g_{{\\phi\\phi}} = R^2 $  (maximum at equator)")
print()

# Stereographic projection (alternative chart)
def stereographic_north(theta, phi):
    """Project from north pole"""
    r = 2 * R * np.tan(theta / 2)
    x = r * np.cos(phi)
    y = r * np.sin(phi)
    return x, y

print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  COORDINATE CHARTS AND ATLASES")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print("  Stereographic projection covers all points except one.")
print("  ✓ Two charts needed for complete atlas!")
print()
print("  A single coordinate chart cannot cover the entire sphere.")
print("  This is a fundamental topological property.")
print()

# Plot the sphere
fig = plt.figure(figsize=(12, 5))

ax1 = fig.add_subplot(121, projection='3d')
ax1.plot_surface(X, Y, Z, cmap='viridis', alpha=0.8, edgecolor='none')
ax1.set_xlabel('X', fontsize=10)
ax1.set_ylabel('Y', fontsize=10)
ax1.set_zlabel('Z', fontsize=10)
ax1.set_title('2-Sphere Embedding in R³', fontsize=12, fontweight='bold')
ax1.view_init(elev=20, azim=45)

# Plot coordinate lines in stereographic projection
ax2 = fig.add_subplot(122)
for t in np.linspace(0.1, np.pi-0.1, 8):
    x_stereo, y_stereo = stereographic_north(t, phi)
    ax2.plot(x_stereo, y_stereo, 'b-', alpha=0.5, linewidth=1.5)
for p in np.linspace(0, 2*np.pi, 12):
    x_s, y_s = stereographic_north(theta, p*np.ones_like(theta))
    ax2.plot(x_s, y_s, 'r-', alpha=0.5, linewidth=1.5)
ax2.set_xlim(-5, 5)
ax2.set_ylim(-5, 5)
ax2.set_xlabel('x (stereographic)', fontsize=10)
ax2.set_ylabel('y (stereographic)', fontsize=10)
ax2.set_title('Stereographic Projection Chart', fontsize=12, fontweight='bold')
ax2.set_aspect('equal')
ax2.grid(True, alpha=0.3)

plt.tight_layout()

# Save plot to base64 string for display
buffer = BytesIO()
plt.savefig(buffer, format='png', dpi=150, bbox_inches='tight')
buffer.seek(0)
image_base64 = base64.b64encode(buffer.read()).decode()
print(f'<img src="data:image/png;base64,{image_base64}" alt="Manifold Sphere Visualization" />')
buffer.close()

print()
print("✓ Visualization complete!")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Example: Metric Calculations

This Fortran code computes metric components and Christoffel symbols for the 2-sphere—essential calculations for understanding geodesic motion on curved manifolds.

Fortran + Python: 2-Sphere Metric Visualization

Python

Compiles and runs Fortran code, then plots metric components and Christoffel symbols vs theta

manifold_sphere_plot.py116 lines

import subprocess, sys
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fortran_source = r"""
program manifold_sphere
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp), parameter :: PI = 4.0_dp * atan(1.0_dp)
    real(dp) :: R, theta, phi, g11, g22, det_g
    real(dp) :: Gamma_theta_phi_phi, Gamma_phi_theta_phi, K_gaussian
    real(dp) :: th, dth
    integer :: i, npts

R = 1.0_dp
    theta = PI / 4.0_dp
    phi = PI / 6.0_dp

g11 = R**2
    g22 = R**2 * sin(theta)**2
    det_g = g11 * g22

write(*,'(A)') '2-Sphere Metric Calculation (Fortran)'
    write(*,'(A)') '======================================'
    write(*,'(A,F10.6)') 'R = ', R
    write(*,'(A,F10.6)') 'theta = ', theta
    write(*,'(A,F10.6)') 'phi = ', phi
    write(*,'(A)') ''
    write(*,'(A,F10.6)') 'g_theta_theta = ', g11
    write(*,'(A,F10.6)') 'g_phi_phi     = ', g22
    write(*,'(A,F10.6)') 'det(g)        = ', det_g
    write(*,'(A,F10.6)') 'sqrt(det(g))  = ', sqrt(det_g)

Gamma_theta_phi_phi = -sin(theta) * cos(theta)
    Gamma_phi_theta_phi = cos(theta) / sin(theta)
    K_gaussian = 1.0_dp / R**2

write(*,'(A)') ''
    write(*,'(A,F12.6)') 'Gamma^theta_phi_phi = ', Gamma_theta_phi_phi
    write(*,'(A,F12.6)') 'Gamma^phi_theta_phi = ', Gamma_phi_theta_phi
    write(*,'(A,F10.6)') 'Gaussian curvature K = ', K_gaussian
    write(*,'(A)') ''

npts = 100
    dth = (PI - 0.02_dp) / real(npts - 1, dp)
    write(*,'(A)') 'DATA_START'
    write(*,'(A)') 'theta,sqrt_det_g,Gamma_tpp,g_phi_phi'
    do i = 1, npts
        th = 0.01_dp + real(i-1, dp) * dth
        write(*,'(F12.6,A,F12.6,A,F12.6,A,F12.6)') &
            th, ',', R*abs(sin(th)), ',', -sin(th)*cos(th), ',', R**2*sin(th)**2
    end do
    write(*,'(A)') 'DATA_END'
end program manifold_sphere
"""

with open('manifold_sphere.f90', 'w') as f:
    f.write(fortran_source)

comp = subprocess.run(['gfortran', '-o', './manifold_sphere', 'manifold_sphere.f90'],
                      capture_output=True, text=True)
if comp.returncode != 0:
    print("Compilation error:", comp.stderr)
    sys.exit(1)

result = subprocess.run(['./manifold_sphere'], capture_output=True, text=True, timeout=30)

lines = result.stdout.strip().split('\n')
data_lines = []
in_data = False
for line in lines:
    if 'DATA_START' in line:
        in_data = True
        continue
    if 'DATA_END' in line:
        in_data = False
        continue
    if in_data:
        data_lines.append(line.strip())
    else:
        print(line)

data = np.array([[float(x) for x in line.split(',')] for line in data_lines[1:]])
theta = data[:, 0]
sqrt_det = data[:, 1]
gamma_tpp = data[:, 2]
g_pp = data[:, 3]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
fig.patch.set_facecolor('#0f172a')
for ax in [ax1, ax2]:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

ax1.plot(theta, sqrt_det, color='#22d3ee', linewidth=2.5, label=r'$\sqrt{\det g}=|\sin\theta|$')
ax1.plot(theta, g_pp, color='#a78bfa', linewidth=2, linestyle='--', label=r'$g_{\phi\phi}=\sin^2\theta$')
ax1.set_title('Metric on 2-Sphere (R=1)', color='#e2e8f0', fontsize=14, fontweight='bold')
ax1.set_xlabel('theta (rad)', color='#94a3b8')
ax1.set_ylabel('Value', color='#94a3b8')
ax1.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

ax2.plot(theta, gamma_tpp, color='#fb923c', linewidth=2.5)
ax2.axhline(y=0, color='#475569', linestyle='--', linewidth=1)
ax2.set_title('Christoffel Symbol vs theta', color='#e2e8f0', fontsize=14, fontweight='bold')
ax2.set_xlabel('theta (rad)', color='#94a3b8')
ax2.set_ylabel(r'$\Gamma^\theta_{\phi\phi}$', color='#94a3b8')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0f172a')
print("\nVisualization saved to output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Topology and Global Structure

Topology captures the "global shape" of a manifold—properties unchanged by continuous deformations.

Compact Manifolds

Closed and bounded (like S²). Every sequence has a convergent subsequence. The universe might be spatially compact!

Simply Connected

Every loop can be continuously shrunk to a point. S² is simply connected; T² (torus) is not.

Orientable

A consistent notion of "clockwise" exists globally. The Möbius strip is non-orientable.

Euler Characteristic

Topological invariant: χ = V - E + F for polyhedra. For S²: χ = 2. For T²: χ = 0.

← Part I Overview Next: Tensors on Manifolds →