Mathematics · Advanced Topics

Tensor Calculus & Differential Geometry

Tensors generalize vectors and matrices to arbitrary rank and coordinate system. Differential geometry is their natural home — curved spaces, covariant derivatives, and the Riemann curvature tensor are the mathematical language of general relativity and gauge theories.

Prerequisites

Linear algebra (Ch. LA) \cdot Vectors & calculus (Ch. 21-22) \cdot General relativity (Ch. GR) \cdot G

roup theory (Ch. GT)

Learning Goals

Define a tensor of type (p, q) by its transformation law and explain why tensor equations are valid in all coordinate systems.
Compute Christoffel symbols from the metric and use them to write the covariant derivative and the geodesic equation.
Calculate the Riemann tensor from Christoffel symbols, and obtain the Ricci tensor, Ricci scalar, and Einstein tensor by contraction.
State Stokes' theorem in differential-form language and show how it unifies the fundamental theorem of calculus, Green's theorem, and the divergence theorem.
$Write Maxwell's equations as F = dA, dF = 0, d(*F) = *J and identify gauge invariance as A$ $\to A + d\lambda$

TC.1 Tensors

Definition TC.1 — Tensor

A tensor of type (p, q) is a multilinear map from p copies of a dual space V* and q copies of a vector space V to the reals. In component form, a (1,1) tensor T transforms as:

T'^\mu_\nu = (\partialx'^\mu/\partialx^\alpha)(\partialx^\beta/\partialx'^\nu) T^\alpha_\beta

A tensor equation valid in one coordinate system is valid in all — this is the power of tensor notation for expressing physical laws (general covariance).

Contraction: sum over a repeated upper and lower index. Contracts a (p,q) tensor to (p−1, q−1). The trace: T = T^μ_μ is a scalar.

Raising and lowering indices: with the metric g_μν: T_μ = g_μν T^ν. In flat Minkowski space: g_μν = diag(−1, +1, +1, +1).

The Levi-Civita tensor ε^μνρσ (totally antisymmetric, ε^0123 = 1/√(−g)) is used to construct dual tensors and volume forms.

TC.2 Covariant Derivative and Christoffel Symbols

On a curved manifold, ordinary partial derivatives do not transform as tensors. The covariant derivative corrects for the change of basis:

\nabla_\mu V^\nu = \partial_\mu V^\nu + Γ^\nu_\mu\lambda V^\lambda \qquad (covariant derivative of a vector)

(TC.1)

Γ^\lambda_\mu\nu = \frac{1}{2} g^\lambda\sigma (\partial_\mu g_\nu\sigma + \partial_\nu g_\mu\sigma - \partial_\sigma g_\mu\nu) \qquad (Christoffel symbols)

(TC.2)

The Christoffel symbols are NOT tensors (they vanish in local inertial frames). Key properties: Γ^λ_μν = Γ^λ_νμ (symmetric in lower indices), metric compatibility ∇_λ g_μν = 0 (parallel transport preserves lengths).

Geodesic equation: the straightest path on a curved manifold (generalizes straight lines). Free-falling particles follow geodesics:

d^{2}x^\mu/d\tau^{2} + Γ^\mu_\nu\rho (dx^\nu/d\tau)(dx^\rho/d\tau) = 0 \qquad (geodesic equation)

(TC.3)

TC.3 Curvature

The Riemann curvature tensor measures the failure of parallel transport around a loop. The commutator of covariant derivatives:

[\nabla_\mu, \nabla_\nu] V^\rho = R^\rho_\sigma\mu\nu V^\sigma \qquad (defines the Riemann tensor)

(TC.4)

R^\rho_\sigma\mu\nu = \partial_\mu Γ^\rho_\nu\sigma - \partial_\nu Γ^\rho_\mu\sigma + Γ^\rho_\mu\lambda Γ^\lambda_\nu\sigma - Γ^\rho_\nu\lambda Γ^\lambda_\mu\sigma

(TC.5)

Contractions give lower-rank tensors:Ricci tensor: R_μν = R^λ_μλν (trace of Riemann).Ricci scalar: R = g^μν R_μν (full trace).Einstein tensor: G_μν = R_μν − ½ g_μν R (divergence-free: ∇^μ G_μν = 0).

Symmetries of Riemann: R_μνρσ = −R_νμρσ, R_μνρσ = −R_μνσρ, R_μνρσ = R_ρσμν, R_μ[νρσ] = 0 (first Bianchi). Number of independent components: n²(n²−1)/12 (= 20 in 4D spacetime).

Example TC.1 — Christoffel Symbols for the 2-Sphere

$Calculate the Christoffel symbols for the 2-sphere ds^{2} = R^{2}(d\theta^{2} + sin^{2}\theta d\phi^{2}$ .

Metric:

g_\theta\theta = R^{2}, g_\phi\phi = R^{2} sin^{2}\theta, g_\theta\phi = 0. Inverse: g^\theta\theta = 1/R^{2}, g^\phi\phi = 1/(R^{2}sin^{2}\theta

Non-zero symbols:

Γ^\theta_\phi\phi = -\frac{1}{2} g^\theta\theta \partial_\theta g_\phi\phi = -\frac{1}{2}(1/R^{2}) \times 2R^{2} sin\theta cos\theta = -sin\theta cos\theta

More symbols:

Γ^\phi_\theta\phi = Γ^\phi_\phi\theta = \frac{1}{2} g^\phi\phi \partial_\theta g_\phi\phi = \frac{1}{2}(1/R^{2}sin^{2}\theta) \times 2R^{2} sin\theta cos\theta = cos\theta/sin\theta = cot \theta

Geodesics:

Great circles! The geodesic equation gives d^{2}\theta/ds^{2} - sin\theta cos\theta (d\phi/ds)^{2} = 0. For equatoria

l orbit \theta=\pi/2: dΓ = const (great circle). Deviation from great circle: parallel transport

of a vector around the equator rotates it by 2\pi steradians solid angle = 2\pi (1-cos\theta) for a

loop at colatitude \theta. This geometric phase is holonomy — related to Berry phase

TC.4 Differential Forms

A p-form is a totally antisymmetric (0,p) tensor. They integrate naturally over p-dimensional surfaces:

\omega = \omega_{\mu_{1}\cdots\mu_{p}} dx^\mu_{1} ∧ \cdots ∧ dx^\mu_{p} \qquad (p-form)

(TC.6)

The exterior derivative d maps p-forms to (p+1)-forms: d² = 0 (nilpotent). The Hodge star * maps p-forms to (n−p)-forms (n = dimension).

Theorem TC.1 — Stokes' Theorem (General)

For any (n-1)-form \omega on an n-manifold M with boundary \partialM

\int_M d\omega = \int_{\partialM} \omega

This unifies: the fundamental theorem of calculus (n=1), Green's theorem (n=2), Stokes' th

eorem (n=3), and the divergence theorem (n=3). Maxwell's equations in form language: dF =

0 (Bianchi), d(*F) = *J (equation of motion) — elegant and coordinate-free

Gauge theory in form language: the electromagnetic potential is a 1-form A = A_μ dx^μ. Field strength: F = dA = (∂_μ A_ν − ∂_ν A_μ) dx^μ ∧ dx^ν. Gauge transformation: A → A + dλ (for scalar λ). F is gauge-invariant: d(dλ) = 0. Non-Abelian gauge fields (Yang-Mills): F = dA + A ∧ A — the A∧A term encodes self-interaction (gluons interact with gluons).

TC.5 Fiber Bundles and Topology

A fiber bundle is a manifold E (total space) that locally looks like a product B × F (base × fiber) but may be twisted globally. Physical examples:

Tangent bundle TM: at each point of spacetime, attach the tangent space. Tensors are sections of tensor products of TM and T*M.

Principal bundle: a Lie group G acts on the fiber. Gauge fields are connections on principal bundles. Curvature of the connection = field strength F. The Chern-Weil homomorphism maps topological invariants (Chern classes) to integrals of F ∧ F — this is the mathematical basis of the TKNN formula in topological insulators.

Characteristic classes: Chern class c₁(E) = [F/(2π)] ∈ H²(M, ℤ) — the integral of F/(2π) over a closed surface is an integer (first Chern number). This is the mathematical origin of quantization in physics: charge quantization, magnetic monopole charge, quantum Hall conductance.

Definition TC.2 — Common Traps

Indices are bookkeeping, not decoration: upper and lower positions encode transformation behavior.
Repeated indices imply summation: free indices must match on both sides of an equation.
Partial derivatives are not tensorial in curved coordinates: covariant derivatives correct for connection terms.
The metric raises and lowers indices: it also defines lengths, angles, and contractions.

Exercises — TC.1–TC.5 Tensor Calculus

Calculate the Riemann tensor and Ricci scalar for a 2-sphere of radius R. Verify that R =

2/R^{2}

R⁻²

Straightforward

arallel transport a vector around a spherical triangle with vertices at the North Pole, (\theta

\pi/2, \phi=0), and (\theta=\pi/2, \phi=\pi/2). By what angle (degrees) is the vector rotated

degrees

Intermediate

Write Maxwell's equations in differential form language: F = dA, dF = 0, d(*F) = *J. Show

how these reduce to the standard vector equations.

Intermediate

4.Describe Chern-Simons theory and its role in the fractional quantum Hall effect. What is the connection between the Chern-Simons level k, anyon statistics, and the filling fraction?

Challenging

Key Takeaways

Tensors transform as products of Jacobians. Tensor equations are coordinate-independent.
$Covariant derivative: \nabla_\mu V^\nu = \partial_\mu V^\nu + Γ^\nu_\mu\lambda V^\lambda. Metric compatible: \nablag = 0$ .
$Riemann tensor R^\rho_\sigma\mu\nu: measures curvature via [\nabla_\mu, \nabla_\nu] V^\rho. Ricci scalar R = g^\mu\nu R_\mu\nu$ .
$Differential forms: d^{2} = 0. Stokes: \int_M d\omega = \int_{\partialM} \omega — unifies all integration theorems$ .
$Maxwell in forms: F = dA, dF = 0, d*F = *J. Gauge: A \to A + d\lambda leaves F invariant$ .
$Fiber bundles: connections = gauge fields, curvature = field strength. Chern numbers are t$ opological integers.