Differential Geometry Primer

This document covers the mathematics underlying XAct.jl. It's aimed at contributors with a programming background who want a working understanding of the relevant concepts — enough to read tensor algebra code and xAct's notation.

1. Manifolds

A manifold is a space that locally looks like ordinary Euclidean space, but may have global structure that's more complex. Think of the surface of the Earth: any small patch looks flat (like ℝ²), but the whole thing is a sphere.

More precisely, an n-dimensional manifold M is a topological space where every point has a neighborhood that is homeomorphic to ℝⁿ.

Charts, atlases, and coordinate systems

A chart (or coordinate patch) is a pair (U, φ) where:

U ⊆ M is an open subset
φ: U → ℝⁿ is a homeomorphism (a smooth invertible map to Euclidean space)

The map φ assigns coordinates (x¹, x², …, xⁿ) to each point in U.

An atlas is a collection of charts that covers all of M. Where two charts overlap, the transition maps φ₂ ∘ φ₁⁻¹ must be smooth.

Why this matters for xAct: xAct's DefManifold creates a manifold object and DefChart introduces coordinates. Coordinate expressions live in specific charts; chart-free (abstract) expressions don't.

2. Vectors and Tensors

Tangent spaces

At each point p ∈ M there is a tangent space TₚM — the vector space of all "directions you can move from p." It has dimension n.

Concretely, a tangent vector at p is an equivalence class of curves through p, or equivalently, a directional derivative operator:

v = vⁱ ∂/∂xⁱ

where ∂/∂xⁱ are the coordinate basis vectors and vⁱ are the components.

Cotangent spaces

The cotangent space Tₚ*M is the dual of TₚM: the space of linear maps TₚM → ℝ. Elements are called covectors or one-forms.

The coordinate basis for covectors is dxⁱ, defined by dxⁱ(∂/∂xʲ) = δⁱⱼ. A covector writes as ω = ωᵢ dxⁱ.

Tensors as multilinear maps

A tensor of type (r, s) at a point is a multilinear map:

T: Tₚ*M × … × Tₚ*M × TₚM × … × TₚM → ℝ
       r covector slots          s vector slots

In components: T^{a₁…aᵣ}_{b₁…bₛ}

Upper indices (contravariant) = vector slots
Lower indices (covariant) = covector slots

A scalar is (0,0), a vector is (1,0), a covector is (0,1), the metric is (0,2).

3. Tensor Algebra

Tensor product

Given tensors T of type (r,s) and S of type (p,q), their tensor product T⊗S is a tensor of type (r+p, s+q):

(T⊗S)^{ab}_{cd} = T^a_c · S^b_d

In xAct: tensors written next to each other are multiplied.

Contraction

Contraction sums over a matching upper and lower index pair, reducing type (r,s) to (r-1, s-1):

Tᵃ_ₐ = Σₐ Tᵃ_ₐ    (trace)

In abstract index notation (see below), repeated indices with one up and one down signal a contraction: T[a, -a] in xAct.

Abstract index notation

xAct uses abstract index notation (Penrose notation):

Indices are labels, not component numbers. v[a] means the vector v, with label a marking its single slot.
A repeated index, one up and one down, means contraction: T[a, -a] = trace.
Concrete components are written with coordinate indices (integers or chart labels).

This notation lets you write tensorial equations that hold in any coordinate system, without choosing coordinates.

4. Metric Tensor

The metric tensor g is a (0,2) tensor that provides an inner product on each tangent space:

g: TₚM × TₚM → ℝ
g(v, w) = gₐb vᵃ wᵇ

In GR the metric has signature (-,+,+,+) (one time, three space dimensions).

Raising and lowering indices

The metric and its inverse g^{ab} (defined by g^{ac} g_{cb} = δᵃb) convert between upper and lower indices:

vₐ = gₐb vᵇ      (lower)
vᵃ = gᵃᵇ vᵦ      (raise)

In xAct: g[-a, -b] is the metric; delta[-a, b] is the identity tensor.

Lengths, angles, volumes

Length of a vector: |v|² = g_{ab} vᵃ vᵇ
Angle between vectors: cos θ = g(v,w)/(|v||w|)
Volume element: √|det g| dx¹∧…∧dxⁿ

5. Differential Forms

Differential forms are totally antisymmetric covariant tensors.

A 0-form is a function f.
A 1-form is a covector ω = ωₐ dxᵃ.
A 2-form is F = F{ab} dxᵃ∧dxᵇ with F{ab} = -F_{ba}.
A p-form is a rank-(0,p) antisymmetric tensor.

Wedge product

The wedge product of a p-form α and q-form β is a (p+q)-form:

(α∧β)_{a₁…aₚb₁…bq} = (p+q)!/(p!q!) α_{[a₁…aₚ} β_{b₁…bq]}

where [·] denotes antisymmetrization. The wedge product is graded-commutative: α∧β = (-1)^{pq} β∧α.

Exterior derivative

The exterior derivative d maps p-forms to (p+1)-forms:

(dω)_{a₀a₁…aₚ} = (p+1) ∂_{[a₀} ω_{a₁…aₚ]}

Key properties:

d(df) = 0 for any function f (= d² = 0)
Leibniz rule: d(α∧β) = dα∧β + (-1)ᵖ α∧dβ

6. Covariant Derivative

An ordinary partial derivative ∂a of a tensor does not transform as a tensor. The covariant derivative ∇a fixes this by accounting for how basis vectors change across the manifold.

Connection and parallel transport

A connection (or covariant derivative) ∇ specifies how to "transport" vectors along curves while keeping them "parallel." This requires additional structure beyond the manifold itself — it's not determined by the topology alone.

The Levi-Civita connection is the unique connection that is:

Compatible with the metric: ∇a g{bc} = 0
Torsion-free: ∇a ∇b f = ∇b ∇a f for scalars

Covariant derivative of a tensor

For a vector field v:

∇_a v^b = ∂_a v^b + Γ^b_{ac} v^c

For a covector ω:

∇_a ω_b = ∂_a ω_b - Γ^c_{ab} ω_c

The sign flips and the free index goes in the Γ position for each lower index.

Christoffel symbols

The Christoffel symbols Γ^c_{ab} encode the connection in coordinates:

Γ^c_{ab} = ½ g^{cd} (∂_a g_{bd} + ∂_b g_{ad} - ∂_d g_{ab})

They're symmetric in the lower indices: Γ^c{ab} = Γ^c{ba}.

In xAct: CD[-a] is ∇_a using the CD connection. ChristoffelCD[a, -b, -c] gives the Christoffel symbols.

7. Curvature

The Riemann curvature tensor measures how much parallel transport around a closed loop rotates vectors:

R^d_{cab} v^c = (∇_a ∇_b - ∇_b ∇_a) v^d

In components:

R^d_{cab} = ∂_a Γ^d_{bc} - ∂_b Γ^d_{ac} + Γ^d_{ae}Γ^e_{bc} - Γ^d_{be}Γ^e_{ac}

Ricci tensor

The Ricci tensor is the trace of the Riemann tensor:

R_{ab} = R^c_{acb}

It's symmetric: R{ab} = R{ba}.

Ricci scalar

The Ricci scalar (scalar curvature) is the trace of the Ricci tensor:

R = g^{ab} R_{ab} = Rᵃ_a

Einstein tensor

The Einstein tensor appears in the Einstein field equations of GR:

G_{ab} = R_{ab} - ½ g_{ab} R

It's divergence-free: ∇^a G_{ab} = 0 (contracted Bianchi identity).

In xAct: RiemannCD[-a,-b,-c,-d], RicciCD[-a,-b], RicciScalarCD[], EinsteinCD[-a,-b].

8. Symmetries of Tensors

Symmetric and antisymmetric parts

For a rank-(0,2) tensor T:

T_{(ab)} = ½(T_{ab} + T_{ba})    symmetric part
T_{[ab]} = ½(T_{ab} - T_{ba})    antisymmetric part

In xAct: (...) denotes symmetrization, [...] antisymmetrization in index lists.

Symmetries of the Riemann tensor

The Riemann tensor R_{abcd} (fully covariant) has these symmetries:

Symmetry	Equation
Antisymmetric in first pair	R{abcd} = -R{bacd}
Antisymmetric in second pair	R{abcd} = -R{abdc}
Symmetric under pair swap	R{abcd} = R{cdab}
First Bianchi identity	R_{a[bcd]} = 0
Second Bianchi identity	∇{[e}R{ab]cd} = 0

These symmetries reduce the number of independent components of R from n⁴ to n²(n²-1)/12 (in 4D: 20 components out of 256).

Bianchi identities

The second (differential) Bianchi identity ∇{[e}R{ab]cd} = 0 implies the contracted form ∇^a G_{ab} = 0, which is the conservation law for the Einstein tensor.

xAct can verify and apply these symmetries automatically via the canonicalization routines in XPerm.jl (Butler-Portugal algorithm).