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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.

2-Sphere Metric and Christoffel Symbols

Fortran

Compute metric determinant and connection coefficients

manifold_sphere.f9080 lines

! manifold_sphere.f90 - Manifold calculations in Fortran
! Computes metric determinant and Christoffel symbols for 2-sphere

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
    real(dp) :: g11, g22, det_g
    real(dp) :: Gamma_theta_phi_phi, Gamma_phi_theta_phi
    real(dp) :: K_gaussian

R = 1.0_dp
    theta = PI / 4.0_dp  ! 45 degrees
    phi = PI / 6.0_dp    ! 30 degrees

print '(A)', '╔═══════════════════════════════════════════════════════════════════════╗'
    print '(A)', '║            2-Sphere Metric Calculation                                ║'
    print '(A)', '╚═══════════════════════════════════════════════════════════════════════╝'
    print '(A)', ''

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  COORDINATE POINT'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A,F10.6)', '  Radius:    \( R = ', R, ' \)'
    print '(A,F10.6)', '  Theta:     \( \theta = ', theta, ' \) rad'
    print '(A,F10.6)', '  Phi:       \( \phi = ', phi, ' \) rad'
    print '(A)', ''

! Metric components for 2-sphere: ds² = R²(dθ² + sin²θ dφ²)
    g11 = R**2              ! g_θθ
    g22 = R**2 * sin(theta)**2  ! g_φφ

det_g = g11 * g22  ! Determinant

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  METRIC TENSOR'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', '  Line element: \( ds^2 = R^2(d\theta^2 + \sin^2\theta \, d\phi^2) \)'
    print '(A)', ''
    print '(A,F10.6)', '  \( g_{\theta\theta} = ', g11, ' \)'
    print '(A,F10.6)', '  \( g_{\phi\phi} = ', g22, ' \)'
    print '(A)', ''
    print '(A,F10.6)', '  \( \det(g) = ', det_g, ' \)'
    print '(A,F10.6)', '  \( \sqrt{\det(g)} = ', sqrt(det_g), ' \)'
    print '(A)', ''

! Non-zero Christoffel symbols
    ! Γ^θ_φφ = -sin(θ)cos(θ)
    Gamma_theta_phi_phi = -sin(theta) * cos(theta)

! Γ^φ_θφ = Γ^φ_φθ = cot(θ)
    Gamma_phi_theta_phi = cos(theta) / sin(theta)

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  CHRISTOFFEL SYMBOLS'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', '  Non-zero components at this point:'
    print '(A)', ''
    print '(A,F12.6)', '  \( \Gamma^\theta_{\phi\phi} = ', Gamma_theta_phi_phi, ' \)'
    print '(A,F12.6)', '  \( \Gamma^\phi_{\theta\phi} = ', Gamma_phi_theta_phi, ' \)'
    print '(A)', ''

! Gaussian curvature K = 1/R²
    K_gaussian = 1.0_dp / R**2

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  GAUSSIAN CURVATURE'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A,F10.6)', '  \( K = 1/R^2 = ', K_gaussian, ' \)'
    print '(A)', ''
    print '(A)', '  The sphere has constant positive curvature!'
    print '(A)', ''

end program manifold_sphere

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed 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 →