Tangent Spaces & Cotangent Bundles

The tangent space at a point encodes the set of all possible "velocities" or "directional derivatives" at that point. Together, tangent spaces at all points form the tangent bundle, the arena for vector fields and differential equations on manifolds.

Historical Context

The modern algebraic definition of tangent vectors as derivations on the algebra of smooth functions was developed in the 1930s–1950s by Elie Cartan, Charles Ehresmann, and the Bourbaki school. This replaced the older geometric intuition of "infinitesimal arrows" with a rigorous algebraic framework. The cotangent bundle and its role in Hamiltonian mechanics was clarified by the symplectic geometry revolution of the 1960s–1970s.

Derivation 1: Tangent Vectors as Derivations

Let $M$ be a smooth manifold and $p \in M$. A tangent vector at $p$ is a linear map $v: C^\infty(M) \to \mathbb{R}$ satisfying the Leibniz rule:

$v(fg) = f(p) \cdot v(g) + g(p) \cdot v(f) \quad \forall f, g \in C^\infty(M)$

The set of all such derivations at $p$ forms the tangent space $T_pM$, which is a real vector space. In local coordinates $(x^1, \ldots, x^n)$, a basis is given by the coordinate partial derivatives:

$\left\{\frac{\partial}{\partial x^1}\bigg|_p, \frac{\partial}{\partial x^2}\bigg|_p, \ldots, \frac{\partial}{\partial x^n}\bigg|_p\right\}$

Thus $\dim T_pM = \dim M = n$. Any tangent vector can be written as:

$\boxed{v = v^i \frac{\partial}{\partial x^i}\bigg|_p, \quad v(f) = v^i \frac{\partial f}{\partial x^i}(p)}$

Proof that Derivations Span $T_pM$

Step 1: For any $f \in C^\infty(M)$, using Hadamard's lemma in local coordinates around $p = 0$:

$f(x) = f(0) + x^i g_i(x), \quad g_i(0) = \frac{\partial f}{\partial x^i}(0)$

Step 2: Apply the derivation $v$ and use Leibniz rule:

$v(f) = v(f(0)) + v(x^i g_i) = 0 + v(x^i) g_i(0) + x^i(0) v(g_i) = v(x^i) \frac{\partial f}{\partial x^i}(0)$

Setting $v^i = v(x^i)$, we obtain $v = v^i \partial_i$, proving the coordinate partials span. Linear independence follows since $\partial_i(x^j) = \delta_i^j$.

Derivation 2: The Pushforward (Differential)

Given a smooth map $F: M \to N$ and a point $p \in M$, the pushforward (or differential) is the linear map:

$F_{*,p}: T_pM \to T_{F(p)}N, \quad (F_{*,p}v)(g) = v(g \circ F)$

In local coordinates $x^i$ on $M$ and $y^a$ on $N$, with $y^a = F^a(x^1, \ldots, x^m)$:

The pushforward of the coordinate basis:

$F_*\left(\frac{\partial}{\partial x^i}\right) = \frac{\partial F^a}{\partial x^i} \frac{\partial}{\partial y^a}$

The matrix representation is the Jacobian:

$\boxed{(F_*)^a_{\ i} = \frac{\partial F^a}{\partial x^i}}$

Chain Rule as Functoriality

For composable maps $M \xrightarrow{F} N \xrightarrow{G} P$, the chain rule becomes:

$(G \circ F)_* = G_* \circ F_*$

This says the tangent functor $T$ preserves composition—a key property making differential geometry a natural framework for category theory.

Derivation 3: The Cotangent Space and Pullback

The cotangent space $T_p^*M$ is the dual vector space of $T_pM$: the space of linear functionals on tangent vectors.

$T_p^*M = \{\omega: T_pM \to \mathbb{R} \mid \omega \text{ is linear}\}$

For a smooth function $f: M \to \mathbb{R}$, its differential at $p$ is the covector:

$df_p: T_pM \to \mathbb{R}, \quad df_p(v) = v(f)$

The dual basis to $\{\partial/\partial x^i\}$ is $\{dx^i\}$ with:

$\boxed{dx^i\left(\frac{\partial}{\partial x^j}\right) = \delta^i_j}$

The Pullback

For $F: M \to N$, the pullback goes in the opposite direction from the pushforward:

$F^*: T^*_{F(p)}N \to T^*_pM, \quad (F^*\omega)(v) = \omega(F_*v)$

The pullback is contravariant: it reverses the direction of maps. In coordinates, if$\omega = \omega_a \, dy^a$, then $F^*\omega = \omega_a \frac{\partial F^a}{\partial x^i} dx^i$.

Derivation 4: Musical Isomorphisms

A Riemannian metric $g$ provides a natural isomorphism between $T_pM$ and $T_p^*M$. These are called the musical isomorphisms (raising and lowering indices).

Flat Map (Index Lowering)

$\flat: T_pM \to T_p^*M, \quad v^\flat(w) = g(v, w)$

In coordinates: $v = v^i \partial_i \mapsto v^\flat = v_j \, dx^j$ where $v_j = g_{ji} v^i$.

Sharp Map (Index Raising)

$\sharp: T_p^*M \to T_pM, \quad \omega^\sharp = g^{ij} \omega_j \partial_i$

Key identity:

$\boxed{(\sharp) \circ (\flat) = \text{id}_{T_pM}, \quad g^{ik} g_{kj} = \delta^i_j}$

Physics notation: In general relativity, lowering an index on $v^\mu$ gives $v_\nu = g_{\mu\nu} v^\mu$, and raising gives $v^\mu = g^{\mu\nu} v_\nu$. The metric's role as the bridge between vectors and covectors is fundamental to writing covariant equations.

Derivation 5: The Tangent Bundle

The tangent bundle is the disjoint union of all tangent spaces:

$TM = \bigsqcup_{p \in M} T_pM = \{(p, v) : p \in M, v \in T_pM\}$

The projection $\pi: TM \to M$ with $\pi(p,v) = p$ makes $TM$ a smooth manifold of dimension $2n$. A chart $(U, x^i)$ on $M$ induces a chart on $TM$:

$(p, v) \mapsto (x^1(p), \ldots, x^n(p), v^1, \ldots, v^n)$

Vector Fields as Sections

A vector field $X$ on $M$ is a smooth section of $TM$: a smooth map $X: M \to TM$ with $\pi \circ X = \text{id}_M$. In coordinates:

$X = X^i(x) \frac{\partial}{\partial x^i}$

The space of all smooth vector fields $\mathfrak{X}(M)$ forms an infinite-dimensional Lie algebra under the Lie bracket:

$[X, Y]f = X(Yf) - Y(Xf)$

In components:

$[X, Y]^k = X^i \frac{\partial Y^k}{\partial x^i} - Y^i \frac{\partial X^k}{\partial x^i}$

Hairy Ball Theorem: There is no nonvanishing continuous vector field on $S^{2n}$. This is a deep topological obstruction: the Euler characteristic$\chi(S^{2n}) = 2 \neq 0$. Physically, this means you cannot comb a hairy ball flat without creating a cowlick.

Applications to Physics

General Relativity

The 4-velocity of a particle is a tangent vector $u^\mu = dx^\mu/d\tau \in T_p\mathcal{M}$. The momentum covector $p_\mu = g_{\mu\nu} u^\nu \in T_p^*\mathcal{M}$ lives in the cotangent space. The geodesic equation describes how tangent vectors parallel-transport themselves along curves in spacetime.

Hamiltonian Mechanics

The phase space of classical mechanics is the cotangent bundle $T^*Q$ of configuration space $Q$. Position coordinates $q^i$ parametrize $Q$, while conjugate momenta $p_i$ are cotangent vectors. The symplectic form $\omega = dp_i \wedge dq^i$ is canonical on $T^*Q$.

Condensed Matter

Crystal momentum $\mathbf{k}$ lives in the cotangent space of position space. The Berry connection on the Brillouin zone torus involves tangent vectors in $k$-space, and the Berry curvature is a 2-form on this tangent bundle.

Interactive Simulation

This simulation visualizes the tangent plane on $S^2$ with basis vectors $e_\theta, e_\phi$, a gradient vector field, the pushforward under stereographic projection, and the metric components as functions of latitude.

Tangent Spaces: Vectors, Pushforward & Cotangent Bundle

Python

script.py212 lines

#!/usr/bin/env python3
"""
tangent_spaces.py - Tangent Vectors, Pushforward, and Cotangent Spaces
Visualizes tangent planes, vector fields, and the pushforward map
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import base64
from io import BytesIO

# ============================================================
# 1. Tangent plane to the 2-sphere at various points
# ============================================================
theta_grid = np.linspace(0, np.pi, 40)
phi_grid = np.linspace(0, 2*np.pi, 40)
TH, PH = np.meshgrid(theta_grid, phi_grid)

X = np.sin(TH) * np.cos(PH)
Y = np.sin(TH) * np.sin(PH)
Z = np.cos(TH)

# Tangent vectors at a point (theta0, phi0)
theta0, phi0 = np.pi/3, np.pi/4

# Position vector
p = np.array([np.sin(theta0)*np.cos(phi0), np.sin(theta0)*np.sin(phi0), np.cos(theta0)])

# Basis tangent vectors: e_theta and e_phi
e_theta = np.array([np.cos(theta0)*np.cos(phi0), np.cos(theta0)*np.sin(phi0), -np.sin(theta0)])
e_phi = np.array([-np.sin(theta0)*np.sin(phi0), np.sin(theta0)*np.cos(phi0), 0])

# Metric components at this point
g_tt = np.dot(e_theta, e_theta)  # R^2 = 1
g_pp = np.dot(e_phi, e_phi)      # R^2 sin^2(theta)
g_tp = np.dot(e_theta, e_phi)    # 0

print("=" * 72)
print("  TANGENT SPACES AND COTANGENT BUNDLES")
print("=" * 72)
print()
print("  Point: theta = pi/3, phi = pi/4")
print("  Position: ({:.4f}, {:.4f}, {:.4f})".format(p[0], p[1], p[2]))
print()
print("  Tangent basis vectors (embedding coordinates):")
print("    e_theta = ({:.4f}, {:.4f}, {:.4f})".format(*e_theta))
print("    e_phi   = ({:.4f}, {:.4f}, {:.4f})".format(*e_phi))
print()
print("  Metric tensor components:")
print("    g_theta,theta = {:.6f}".format(g_tt))
print("    g_phi,phi     = {:.6f}".format(g_pp))
print("    g_theta,phi   = {:.6f}".format(g_tp))
print()

# Pushforward of a map f: S^2 -> R^2 (stereographic projection)
# f(theta, phi) = (sin(theta)cos(phi)/(1-cos(theta)), sin(theta)sin(phi)/(1-cos(theta)))
denom = 1 - np.cos(theta0)
fx = np.sin(theta0)*np.cos(phi0)/denom
fy = np.sin(theta0)*np.sin(phi0)/denom

# Jacobian of stereographic projection
# df/d(theta) and df/d(phi) at (theta0, phi0)
J11 = (np.cos(theta0)*np.cos(phi0)*(1-np.cos(theta0)) - np.sin(theta0)**2*np.cos(phi0)) / denom**2
J12 = -np.sin(theta0)*np.sin(phi0) / denom
J21 = (np.cos(theta0)*np.sin(phi0)*(1-np.cos(theta0)) - np.sin(theta0)**2*np.sin(phi0)) / denom**2
J22 = np.sin(theta0)*np.cos(phi0) / denom

print("  PUSHFORWARD (Stereographic Projection)")
print("  " + "-" * 40)
print("  Image point: ({:.4f}, {:.4f})".format(fx, fy))
print()
print("  Jacobian matrix (pushforward):")
print("    | {:.4f}  {:.4f} |".format(J11, J12))
print("    | {:.4f}  {:.4f} |".format(J21, J22))
print("  det(J) = {:.6f}".format(J11*J22 - J12*J21))
print()

# Cotangent vectors (dual basis)
# For orthogonal coordinates: dx^i = g^{ii} e_i (where e_i are tangent)
print("  COTANGENT SPACE (Dual Basis)")
print("  " + "-" * 40)
print("  Dual basis: dtheta, dphi")
print("  Evaluated on tangent vectors:")
print("    dtheta(e_theta) = 1,  dtheta(e_phi) = 0")
print("    dphi(e_theta)   = 0,  dphi(e_phi)   = 1")
print()
print("  Musical isomorphism (index lowering):")
print("    v = a e_theta + b e_phi")
print("    v_flat = a g_tt dtheta + b g_pp dphi")
print("    = {:.4f} a dtheta + {:.4f} b dphi".format(g_tt, g_pp))
print()

# ============================================================
# Visualization
# ============================================================
fig = plt.figure(figsize=(15, 10))
fig.patch.set_facecolor('#0a0a0a')

# Plot 1: Sphere with tangent plane
ax1 = fig.add_subplot(221, projection='3d')
ax1.set_facecolor('#0a0a0a')
ax1.plot_surface(X, Y, Z, alpha=0.3, color='#0d9488', edgecolor='none')

# Draw tangent plane
s = np.linspace(-0.4, 0.4, 10)
t = np.linspace(-0.4, 0.4, 10)
S, T = np.meshgrid(s, t)
Xp = p[0] + S*e_theta[0] + T*e_phi[0]
Yp = p[1] + S*e_theta[1] + T*e_phi[1]
Zp = p[2] + S*e_theta[2] + T*e_phi[2]
ax1.plot_surface(Xp, Yp, Zp, alpha=0.5, color='#f59e0b', edgecolor='none')

# Draw tangent vectors
scale = 0.5
ax1.quiver(p[0], p[1], p[2], scale*e_theta[0], scale*e_theta[1], scale*e_theta[2],
           color='#ef4444', linewidth=2, arrow_length_ratio=0.15)
ax1.quiver(p[0], p[1], p[2], scale*e_phi[0], scale*e_phi[1], scale*e_phi[2],
           color='#3b82f6', linewidth=2, arrow_length_ratio=0.15)
ax1.quiver(0, 0, 0, p[0], p[1], p[2],
           color='#a78bfa', linewidth=1.5, arrow_length_ratio=0.1, linestyle='--')

ax1.set_title('Tangent Plane on S^2', color='#99f6e4', fontsize=12, fontweight='bold')
ax1.set_xlim(-1.2, 1.2)
ax1.set_ylim(-1.2, 1.2)
ax1.set_zlim(-1.2, 1.2)
ax1.view_init(elev=25, azim=35)

# Plot 2: Vector field on sphere (gradient of height function)
ax2 = fig.add_subplot(222, projection='3d')
ax2.set_facecolor('#0a0a0a')
ax2.plot_surface(X, Y, Z, alpha=0.2, color='#0d9488', edgecolor='none')

# Sample points for vector field
th_pts = np.linspace(0.3, 2.8, 8)
ph_pts = np.linspace(0, 2*np.pi, 12, endpoint=False)
for th in th_pts:
    for ph in ph_pts:
        pos = np.array([np.sin(th)*np.cos(ph), np.sin(th)*np.sin(ph), np.cos(th)])
        # Gradient of z = cos(theta): grad z = -e_theta/R^2
        vec = -np.array([np.cos(th)*np.cos(ph), np.cos(th)*np.sin(ph), -np.sin(th)])
        sc = 0.12
        ax2.quiver(pos[0], pos[1], pos[2], sc*vec[0], sc*vec[1], sc*vec[2],
                   color='#f59e0b', linewidth=0.8, arrow_length_ratio=0.2)

ax2.set_title('Gradient Vector Field on S^2', color='#99f6e4', fontsize=12, fontweight='bold')
ax2.set_xlim(-1.2, 1.2)
ax2.set_ylim(-1.2, 1.2)
ax2.set_zlim(-1.2, 1.2)
ax2.view_init(elev=25, azim=35)

# Plot 3: Pushforward visualization
ax3 = fig.add_subplot(223)
ax3.set_facecolor('#111111')
for spine in ax3.spines.values():
    spine.set_color('#2dd4bf')
ax3.tick_params(colors='#99f6e4')
ax3.grid(True, color='#134e4a', alpha=0.4)

# Draw some curves on sphere mapped to stereographic plane
for t0 in np.linspace(0.3, 2.5, 8):
    ph_curve = np.linspace(0, 2*np.pi, 100)
    d = 1 - np.cos(t0)
    sx = np.sin(t0)*np.cos(ph_curve)/d
    sy = np.sin(t0)*np.sin(ph_curve)/d
    ax3.plot(sx, sy, color='#2dd4bf', alpha=0.4, linewidth=0.8)

# Mark the tangent vectors in stereographic coords
ax3.plot(fx, fy, 'o', color='#f59e0b', markersize=8)
push_eth = np.array([J11, J21])
push_eph = np.array([J12, J22])
ax3.quiver(fx, fy, push_eth[0]*0.5, push_eth[1]*0.5, color='#ef4444', linewidth=2, scale=1, scale_units='xy')
ax3.quiver(fx, fy, push_eph[0]*0.5, push_eph[1]*0.5, color='#3b82f6', linewidth=2, scale=1, scale_units='xy')

ax3.set_xlim(-4, 4)
ax3.set_ylim(-4, 4)
ax3.set_aspect('equal')
ax3.set_title('Pushforward in Stereographic Chart', color='#99f6e4', fontsize=12, fontweight='bold')
ax3.set_xlabel('x', color='#5eead4')
ax3.set_ylabel('y', color='#5eead4')

# Plot 4: Musical isomorphism (metric components vs theta)
ax4 = fig.add_subplot(224)
ax4.set_facecolor('#111111')
for spine in ax4.spines.values():
    spine.set_color('#2dd4bf')
ax4.tick_params(colors='#99f6e4')
ax4.grid(True, color='#134e4a', alpha=0.4)

th_range = np.linspace(0.01, np.pi-0.01, 300)
g_tt_vals = np.ones_like(th_range)
g_pp_vals = np.sin(th_range)**2

ax4.plot(th_range, g_tt_vals, color='#ef4444', linewidth=2.5, label='g_theta,theta = 1')
ax4.plot(th_range, g_pp_vals, color='#3b82f6', linewidth=2.5, label='g_phi,phi = sin^2(theta)')
ax4.fill_between(th_range, g_pp_vals, alpha=0.1, color='#3b82f6')
ax4.set_xlabel('theta', color='#5eead4')
ax4.set_ylabel('Metric component', color='#5eead4')
ax4.set_title('S^2 Metric Components', color='#99f6e4', fontsize=12, fontweight='bold')
ax4.legend(facecolor='#111111', edgecolor='#134e4a', labelcolor='#99f6e4', fontsize=9)

plt.tight_layout(pad=2.0)
buf = BytesIO()
plt.savefig(buf, format='png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
buf.seek(0)
img_b64 = base64.b64encode(buf.read()).decode()
print('<img src="data:image/png;base64,' + img_b64 + '" alt="Tangent Spaces Visualization" />')
buf.close()
print()
print("Visualization complete.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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