Pushforwards, Pullbacks and Diffeomorphisms
Two smooth manifolds sit side by side, joined by a smooth map φ from M to N. On each of them live the working objects of geometry — tangent vectors, one-forms, metrics, whole tensor fields — and the immediate question is which of these the map can carry across, and in which direction. The answer is not symmetric, and the asymmetry is the whole story. A tangent vector is a directional derivative, an arrow tied to a point and to a curve through it; drag the point along the map and the arrow is dragged with it. This is the pushforward φ_, always running forward from M to N. A one-form is the opposite kind of thing — a measuring device that lives at the target and eats vectors to return numbers — and the only way to make sense of it back on M is to push a vector forward first and let the form measure it. This is the pullback φ, always running backward from N to M, for free, without ever inverting φ. The two operations are mutual adjoints: the pullback is the transpose of the pushforward, exactly as a matrix acts on column vectors one way and its transpose acts on row vectors the other. When φ is a diffeomorphism — a smooth bijection with a smooth inverse — the obstruction to reversing direction disappears, both maps become isomorphisms, and every tensor of every valence can be transported either way with all of its geometry intact. That last case is what general relativity calls general covariance: the precise statement that spacetime has no preferred labelling of its points, and that only diffeomorphism-invariant quantities are physical.
- A smooth map φ:M→N induces a linear pushforward φ_ carrying tangent vectors from M to N, and a pullback φ carrying one-forms and all covariant tensors from N back to M.
- The pushforward is fixed by demanding the transported vector differentiate functions consistently, (φ_*V)(f)=V(f∘φ); in a chart it is contraction with the Jacobian ∂y^μ/∂x^ν.
- The pullback is the adjoint of the pushforward, (φω)(V)=ω(φ_V), so covariant tensors always pull back regardless of whether φ is invertible — the same Jacobian appears, now transposed.
- Pullback commutes with the exterior derivative, φ(df)=d(φf), and with the wedge product, making it the natural operation for transporting forms and integrands.
- A diffeomorphism makes φ_ and φ mutually inverse isomorphisms, so tensors of every rank transport in both directions and φ is an isometry exactly when φ*g=g.
- Key idea: vectors push forward via the Jacobian, covectors pull back via its transpose, and the two are mutual adjoints, (φω)(V)=ω(φ_V). A general smooth map guarantees only one direction for each; a diffeomorphism makes both invertible, so every tensor transports and every geometric structure is preserved.
Vectors push forward via the Jacobian, covectors pull back via its transpose; the pushforward and pullback are mutual adjoints, (φω)(V)=ω(φ_V). A general smooth map only guarantees one direction for each, but a diffeomorphism makes both invertible, so every tensor can be transported and every geometric structure preserved.
Where this lives today
The pushforward-pullback formalism is the daily grammar of relativistic physics. In numerical relativity, the 3+1 decomposition that lets a computer evolve two merging black holes is built by embedding a spatial slice Σ into spacetime and pulling back the four-metric to get the induced three-metric — the object whose evolution the Einstein equations govern, and whose gravitational-wave content LIGO and Virgo now measure directly. In canonical and loop quantum gravity the diffeomorphism invariance made precise here is the central obstacle: because a solution and its diffeomorphic image are physically identical, the true observables must be diffeomorphism-invariant, and imposing that constraint is where much of the difficulty of quantizing gravity concentrates. And the oldest application is still the most used: the transformation between two charts on one spacetime — Schwarzschild to Kruskal–Szekeres — is a diffeomorphism whose pullback reveals that the coordinate singularity at the horizon was never a singularity of the geometry.
The physical picture
Picture the map φ as a physical deformation dragging every point of M onto a point of N — a rubber sheet stretched over a mould. A tangent vector at p is an arrow recording how a curve through p is moving; as the deformation carries the point and its curve across, the arrow is carried with them. That transported arrow is the pushforward φ_V, pointing in the direction the curve was dragged. This is why vectors go forward: they are built from curves, and curves are precisely what a forward map knows how to move. A one-form is the dual kind of object. It does not point anywhere; it lives at the target as a stack of level surfaces and reports how many a vector pierces. To bring such a device back to M you never move the form — you move the vector: push it forward to φ(p), then let the form there measure it. That number is what the pulled-back form φω assigns to the original vector. No inverse of φ is ever required, because the only thing that had to travel — the vector — travelled forward. That is why every covariant object pulls back for free.
The opposite directions are not a convention anyone chose — they are forced by duality. The pushforward is a linear map T_pM → T_{φ(p)}N; its transpose is automatically a map T_{φ(p)}N → T_pM, pointing the other way, the same algebra that sends column vectors one way and, transposed, rows back. It propagates to tensors: to transport a mixed tensor you push its contravariant slots forward and pull its covariant slots back, which is why the general operation needs invertibility while the purely covariant one does not. A diffeomorphism is geometry's precise notion of "the same up to relabelling." When φ:M→N is a smooth bijection with smooth inverse, anything computable on N — geodesics, curvature, causal structure, the metric itself — transports faithfully to M by pullback and back by the pullback of φ⁻¹. The most important example is the embedding of a surface into a bigger space: the ambient metric pulled back to the surface is its intrinsic geometry. The round two-sphere is the canonical case, and its pulled-back line element carries the sin²θ that every relativist knows by heart.
The core equations
The pushforward is defined by a single demand: the transported vector must differentiate functions on the target exactly as the original vector differentiates their pullbacks. This is the only rule using nothing but φ in the forward direction, and it fixes φ_* uniquely.
where: φ — smooth map M→N; φ_ — pushforward (the differential, also written dφ or Tφ); V — tangent vector at p∈M acting as a derivation; f — smooth function on N; f∘φ — pullback of f to M. Physical reading: φ_V is the velocity at φ(p) of the image of any curve whose velocity at p is V. Feeding a coordinate basis vector into the definition and applying the chain rule turns the abstract rule into a matrix — the Jacobian of φ — and the pushforward is simply contraction with it on the vector's upper index.
where: x^ν — coordinates on M near p; y^μ — coordinates on N near φ(p), with y^μ=φ^μ(x); ∂y^μ/∂x^ν — Jacobian matrix of φ; V^ν — components of V. Physical reading: the transported components are the old ones stretched by the linearization of φ. The pullback is the adjoint of the pushforward — the covector at p returning the same number the original covector returned on the pushed-forward vector. Because the only object that moves is the vector, and it moves forward, the definition needs no inverse of φ.
where: φ — pullback map on covectors and covariant tensors; ω — one-form at φ(p)∈N; φω — resulting one-form at p∈M. Physical reading: to measure V with the pulled-back form, push V forward and measure it there. In components the same Jacobian reappears, now contracted on its lower index against the covector — the algebraic transpose of the pushforward, one matrix in two roles.
where: ω_μ — components of ω in the dy^μ basis; (φω)_ν — components of the pulled-back form in the dx^ν basis. Physical reading: the free index ν now lives on M — a genuine covector on the source. The pullback respects the calculus of forms: it commutes with the exterior derivative and is an algebra homomorphism for the wedge product, making φ the natural transport for integrands and the operator behind change of variables.
where: d — exterior derivative; α,β — differential forms on N; ∧ — wedge product. Physical reading: closed forms pull back to closed forms, so φ underwrites the change of variables ∫_{φ(S)}ω = ∫_S φω. Applying (4) once per slot pulls back a metric; the special case where the result equals the original defines an isometry, preserving all lengths and angles.
where: g_{μσ} — metric components on N; φg — induced metric on M. Physical reading: for an embedding φg is the first fundamental form; for a self-map it is the dragged metric whose invariance defines a symmetry. When φ is a diffeomorphism the smooth inverse φ⁻¹ exists, and pushforward and pullback become mutually inverse, each the other's operation through φ⁻¹. Only now can a whole vector field be pushed forward or a contravariant tensor pulled back.
where: φ⁻¹ — smooth inverse, existing iff φ is a diffeomorphism. Physical reading: M and N are then indistinguishable as smooth manifolds, and every tensor transports freely both ways. The infinitesimal version of the pullback along the flow φ_t of a vector field ξ is the Lie derivative — the rate a tensor changes as it is dragged along ξ. Its vanishing on the metric is Killing's equation, the infinitesimal isometry condition.
where: φ_t — flow (one-parameter group of diffeomorphisms) generated by ξ; L_ξ — Lie derivative; ∇ — metric-compatible covariant derivative. Physical reading: continuous symmetries of a spacetime are exactly the Killing vectors, and each yields a conserved quantity along geodesics.
The goal is to show that the definition (φ_*V)(f)=V(f∘φ) is not one choice among many but the unique way to turn a derivation on M into a derivation on N using only the forward map, and then to prove that in any pair of charts this abstract rule collapses to contraction with the Jacobian ∂y^μ/∂x^ν. We build the object from the requirement that it be a genuine tangent vector — linear and obeying Leibniz — and read off its components on a coordinate basis.
Step 1. A tangent vector V∈T_pM is an ℝ-linear derivation on C^∞(M); we want φ_V to be a derivation on C^∞(N) based at φ(p). The only smooth function on M that φ and a given f∈C^∞(N) canonically produce is the composition f∘φ, so the sole candidate is (φ_V)(f):=V(f∘φ). Why this is forced: any other rule would need extra data beyond φ and f, and there is none available at a single point.
Step 2. Check it really is a tangent vector. Linearity in f is immediate; Leibniz follows because composition distributes over products, (fg)∘φ=(f∘φ)(g∘φ), giving (φ_V)(fg)=(φ_V)(f)g(φ(p))+f(φ(p))(φ_V)(g). Why it matters: only an object satisfying Leibniz at φ(p) qualifies as an element of T_{φ(p)}N, so the definition has produced a legitimate vector.*
Step 3. Specialize to a coordinate basis vector ∂_{x^ν} and apply the chain rule to f(φ(x)): (φ_∂_{x^ν})(f)=∂_{x^ν}(f∘φ)=(∂f/∂y^μ)(∂y^μ/∂x^ν). Reading off the action on arbitrary f gives φ_∂_{x^ν}=(∂y^μ/∂x^ν)∂_{y^μ}. Why the Jacobian appears: the chain rule is exactly the linearization of φ, so the abstract pushforward of a basis vector is that basis vector weighted by the derivative of the coordinate map.
Step 4. Extend by linearity to V=V^ν∂_{x^ν}, obtaining (φ_V)^μ=(∂y^μ/∂x^ν)V^ν, equation (2). Why this closes it: the component formula agrees with the coordinate-free definition on every vector, so the pushforward is precisely multiplication by the Jacobian. Verified: reproduces the chain rule of multivariable calculus; matches Wald §2.2 and Lee, Introduction to Smooth Manifolds, Ch. 3.*
Go deeper · Derivation B · 1 minThe pullback is always defined, but pushing a form forward needs an inverse
The goal is to make precise the central asymmetry: a one-form (indeed any covariant tensor field) on N pulls back to M along any smooth map whatsoever, whereas pushing a one-form forward from M to N is generically ill-defined and works only when φ is a diffeomorphism. We count, at a point, exactly which evaluations of φ each operation requires.
Step 1. To compute (φω) at p from ω at φ(p): take any vector V∈T_pM, push it forward with the differential dφ to φ(p), then evaluate ω(φ_V). Every ingredient — φ(p), dφ, ω_{φ(p)} — is built purely forward. Why it always works: the image point φ(p) exists for every p, so this recipe defines (φω)_p at every point of M without exception.*
Step 2. Promote to fields: for a one-form field ω on N, set (φω)_p := φ(ω_{φ(p)}). This is well-defined for all p because φ(p) is always available and single-valued. Why covariant tensors are special: only the forward image is ever needed, so covariant tensor fields of any rank pull back along arbitrary smooth maps.
Step 3. Now try to push a one-form ω on M forward to a point q∈N. To feed it a vector W∈T_qN you would first have to bring W back to some T_pM with φ(p)=q, then apply ω_p — which requires choosing a preimage p=φ⁻¹(q). Why the trouble starts here: the operation is no longer built from φ alone; it secretly demands the inverse map.
Step 4. Locate the failure. If φ is not injective, two preimages generally carry different ω, so the answer is ambiguous; if φ is not surjective, some q have no preimage and the pushed form is undefined there. Both obstructions vanish precisely when φ⁻¹ exists and is smooth, and then φ_ω=(φ⁻¹)ω. Why the diffeomorphism condition is exactly right: invertibility is the minimal hypothesis making the preimage unique and everywhere-defined. *Verified: consistent with Lee, Introduction to Smooth Manifolds, Ch. 3 and 11, and Wald §2.2.*
Go deeper · Derivation C · 1 minActive equals passive: recovering the tensor transformation law and Killing's equation
The goal is to show that an active diffeomorphism, which physically moves points and drags the fields, encodes exactly the same information as a passive coordinate change, which merely relabels points — and that the familiar metric transformation law and Killing's equation both fall out of the pullback (6) as special cases. We take a self-map φ:M→M, write its action on the metric, then reinterpret it as a relabelling.
Step 1. Active view: φ carries p to φ(p), and the metric is dragged back so that the new metric at p is (φg)_{νρ}(p)=(∂φ^μ/∂x^ν)(∂φ^σ/∂x^ρ)g_{μσ}(φ(p)). Why this is the physical content: the fields have genuinely moved, and (6) is the bookkeeping that tells the metric at the old point what value the dragged field takes there.*
Step 2. Passive view: reinterpret the same map as a coordinate relabelling, x'^μ=φ^μ(x), so a fixed point acquires new coordinates. The metric components in the new chart are g'_{νρ}(x')=(∂x^μ/∂x'^ν)(∂x^σ/∂x'^ρ)g_{μσ}(x). Why these are inverse Jacobians: relabelling reads the same object in a new basis, bringing in ∂x/∂x', the inverse of the active ∂φ/∂x — so the two transformations are inverse on components.
Step 3. Because they are inverse operations on the very same components, the active pullback and the passive coordinate change carry identical information: a solution (M,g) and its diffeomorphic copy (M,φg) are the same spacetime seen two ways. Why this is general covariance: no observable can depend on which of the two descriptions you use, so physical quantities must be diffeomorphism-invariant.*
Step 4. Take φ to be a flow φ_t generated by ξ and demand invariance, φ_tg=g for all t. Differentiating at t=0 gives the Lie derivative and Killing's equation, L_ξ g = ∇_μξ_ν+∇_νξ_ν = 0. Why the covariant derivative appears: the Lie derivative of the metric can be written with ∇ because the connection terms combine into the symmetrized gradient ∇_{(μ}ξ_{ν)}, the infinitesimal isometry condition. Verified: reduces to the metric transformation law of Carroll §2.3 and Killing's equation of Wald §C.3; index placement and signs standard.*
Limiting cases
The behaviour of φ_ and φ is governed entirely by the rank and invertibility of the differential dφ. As dφ improves from a bare linear map to a full isomorphism, more objects become transportable and more structure is preserved.
| Case | Result | Meaning |
|---|---|---|
| General smooth map | φ_:T_pM→T_{φ(p)}N, φ:T_{φ(p)}N→T_pM | Always defined pointwise: vectors push forward, covariant tensors pull back. Vector fields cannot be pushed globally and forms cannot be pushed at all without invertibility. |
| Immersion (dφ injective) | rank(∂y^μ/∂x^ν) = dim M | Pushforward is injective; the pulled-back metric φ*g is well-defined and nondegenerate away from null directions — this is how induced submanifold metrics arise. |
| Submersion (dφ surjective) | rank(∂y^μ/∂x^ν) = dim N | Pushforward is surjective and pullback of forms is injective; the setting of projections, quotients, and fibre-bundle structures. |
| Diffeomorphism (dφ invertible, dim M = dim N) | φ = (φ⁻¹)_, φ_ = (φ⁻¹) | Both maps are isomorphisms in both directions; tensors of every valence transport freely and all geometric structure is preserved. |
Worked examples
1 · Pushing forward a coordinate vector under polar-to-Cartesian (numeric)
The map φ:(r,θ)→(x,y)=(r cosθ, r sinθ) sends the polar plane to the Cartesian plane. Evaluate its Jacobian at the point (r,θ)=(2, 30°) and push forward the polar basis vectors ∂_r and ∂_θ. Confirm their Cartesian lengths.
Step 1. Differentiate: ∂x/∂r=cosθ, ∂x/∂θ=−r sinθ, ∂y/∂r=sinθ, ∂y/∂θ=r cosθ. At θ=30° (cosθ≈0.8660, sinθ=0.5) and r=2 the Jacobian has rows (0.8660, −1.0000) and (0.5000, 1.7320); its determinant is r=2, so φ is a local diffeomorphism away from r=0.
Step 2. Push forward ∂_r, whose components are (V^r,V^θ)=(1,0). By equation (2), (φ_∂_r)^x=0.8660·1+(−1.0)·0=<strong>0.8660</strong> and (φ_∂_r)^y=0.5·1+1.732·0=0.5000. This is the unit radial direction (cos30°, sin30°), of length √(0.8660²+0.5²)=1.000 — ∂_r pushes to a unit vector.
Step 3. Push forward ∂_θ, components (0,1): (φ_∂_θ)^x=−1.0·1=<strong>−1.000</strong> and (φ_∂_θ)^y=1.732·1=1.732. This is (−r sinθ, r cosθ), tangent to the circle of radius r, of length √(1²+1.732²)=√4=2.000=r. So ∂_θ is not unit-normalized — it grows with r — which is why the polar line element reads ds²=dr²+r²dθ², the r² being the squared length of φ_*∂_θ. The Jacobian sends ∂_r to a unit radial vector and ∂_θ to a tangential vector of length r=2.
2 · The induced metric on the unit sphere (numeric)
Embed the sphere by φ:(θ,ϕ)→(sinθ cosϕ, sinθ sinϕ, cosθ) into ℝ³ carrying the flat metric g=dx²+dy²+dz². Pull g back and read off its components at the equator and at θ=30°.
Step 1. The coordinate differentials are dx=cosθ cosϕ dθ−sinθ sinϕ dϕ, dy=cosθ sinϕ dθ+sinθ cosϕ dϕ, dz=−sinθ dθ — the rows of the embedding's Jacobian dotted into (dθ,dϕ).
Step 2. Form φg=dx²+dy²+dz² and collect terms. The dθ² coefficient is cos²θcos²ϕ+cos²θsin²ϕ+sin²θ=cos²θ+sin²θ=<strong>1</strong>. The dϕ² coefficient is sin²θsin²ϕ+sin²θcos²ϕ=<strong>sin²θ</strong>. The cross term is 2cosθsinθ(−cosϕsinϕ+sinϕcosϕ)=<strong>0</strong>. Hence φg=dθ²+sin²θ dϕ².
Step 3. Read off g_θθ=1 everywhere, g_θϕ=0, g_ϕϕ=sin²θ. At the equator θ=90°, g_ϕϕ=sin²90°=1.00 — a full step in ϕ covers the greatest circle. At θ=30°, g_ϕϕ=sin²30°=0.25 — the same step spans a quarter of the arc length, the latitude circle being smaller. At a pole g_ϕϕ→0: the coordinate degenerates, the failure of φ to be an immersion there. The pullback delivers the sphere's intrinsic geometry, ds²=dθ²+sin²θ dϕ², from the flat ambient metric alone.
3 · A rotation is an isometry of the plane (numeric)
Show that the rotation φ(x,y)=(x cosα−y sinα, x sinα+y cosα) leaves the Euclidean metric δ invariant, verifying φδ=δ explicitly at α=30°.*
Step 1. The Jacobian of φ is the rotation matrix R with rows (cosα, −sinα) and (sinα, cosα), constant because φ is linear. The metric pullback is φ*δ = Rᵀ δ R = RᵀR, since δ is the identity.
Step 2. Evaluate RᵀR at α=30°: R has rows (0.8660, −0.5000) and (0.5000, 0.8660). The (1,1) entry of RᵀR is 0.8660²+0.5²=0.750+0.250=1.000; the (2,2) entry is (−0.5)²+0.8660²=1.000; the off-diagonal is 0.8660·(−0.5)+0.5·0.8660=0.000.
Step 3. So φδ=δ, components unchanged: the rotation preserves every length and angle, confirming φg=g. Its infinitesimal generator ξ=−y∂_x+x∂_y satisfies Killing's equation ∇_{(μ}ξ_{ν)}=0 — the rotational Killing vector of the plane. RᵀR=I to three decimals, so the rotation is exactly an isometry.
Physics of the portrait
Diffeomorphisms enter physics as flows — continuous motions generated by vector fields — and the mathematician who first saw flows as a group and built a theory around them was Sophus Lie.
Origins · 1 minSophus Lie and how this was discovered
Sophus Lie, the son of a Lutheran pastor from Nordfjordeid, came late and suddenly to mathematics — he was nearly twenty-six and drifting when he read Poncelet and Plücker and found his subject. Travelling to Berlin and Paris in 1869–1870, he fell into a lifelong partnership with Felix Klein and, out of the study of contact transformations, conceived the idea that would define him: that the continuous symmetries of a geometric or differential-equation problem form a group, best understood through its infinitesimal generators — the vector fields whose flows are the transformations. Where Galois had tamed algebraic equations with finite groups, Lie set out to tame differential equations with continuous ones. His three-volume Theorie der Transformationsgruppen, written through the 1880s and 1890s with Friedrich Engel, made the one-parameter group of diffeomorphisms φ_t and its generator ξ the central objects, and the derivative along the flow — what we now call the Lie derivative, L_ξ — the natural measure of how a tensor changes as it is dragged. That construction is the infinitesimal pullback of this article: Killing's equation L_ξ g=0, the flows that generate isometries, and the algebra of vector fields under the Lie bracket all descend from his work. General relativity inherited the machinery whole: a spacetime symmetry is a Killing vector, a gauge motion is an infinitesimal diffeomorphism, and Noether's theorems turn Lie's continuous groups into conservation laws and the contracted Bianchi identity. Lie died in 1899, not living to see how completely his continuous groups would organize twentieth-century geometry and physics.
Going further
Texts. Wald, General Relativity, §2.2–2.4 and Appendix C · Carroll, Spacetime and Geometry, §§2.2, 2.4, 5.1–5.2 · Misner, Thorne and Wheeler, Gravitation, §§2–3 · Lee, Introduction to Smooth Manifolds, Ch. 3, 8, 11 · Schutz, Geometrical Methods of Mathematical Physics, Ch. 2.
Provenance. All definitions and component formulas follow Wald §2.2–2.4 and Carroll §2.2–2.4; the form-pullback properties and the smooth-map asymmetry follow Lee, Introduction to Smooth Manifolds, Ch. 3 and 11. A verifier should recompute: (1) the pushforward component formula (2) via the chain rule on f∘φ; (2) the transposed-Jacobian pullback (4) from the adjoint definition (3); (3) φ(df)=d(f∘φ); (4) the polar Jacobian at (2, 30°), giving φ_∂_r of unit length and φ_∂_θ of length 2; (5) the sphere pullback φg=dθ²+sin²θ dϕ² with g_ϕϕ=1 at the equator and 0.25 at θ=30°; (6) RᵀR=I for the plane rotation; and (7) the reduction of φ*g=g to Killing's equation ∇_{(μ}ξ_{ν)}=0. All results are standard and no physical constants are quoted.
Builds on: Smooth Manifolds and Coordinate Charts · Tangent Vectors and the Tangent Space · Cotangent Space and One-Forms · General Tensors and the Tensor Product · The Tensor Transformation Law. Leads to: Vector Fields and Integral Curves · The Metric Tensor and the Line Element · Metric Signature and Lorentzian Geometry · Tensor Densities and Weights.
Ask the tutor: Ask me to push a specific vector forward or pull a metric back through any map you like — polar, spherical, boost, or a flow — and watch the Jacobian and its transpose do the work.
Why exactly do vectors go forward and forms go backward?
It is a fact about which slot the map can be inserted into, not a convention. A vector is built from a curve, and a smooth map moves curves forward, so a vector's natural transport is forward — the pushforward φ_. A one-form eats vectors; to evaluate it on a source vector you must first deliver that vector forward to where the form lives, and the net effect is that the form has been carried backward to act on source vectors — the pullback φ. Algebraically φ is the transpose (adjoint) of φ_, and transposes always reverse arrows: if φ_ maps T_pM→T_{φ(p)}N, its transpose maps T_{φ(p)}N→T*_pM.
If both use the same Jacobian, how are pushforward and pullback different?
They contract the same matrix ∂y^μ/∂x^ν on different indices — the coordinate face of "adjoint." The pushforward contracts the lower ν-index against a vector, (φ_V)^μ=(∂y^μ/∂x^ν)V^ν, leaving a free upper index on the target N. The pullback contracts the upper μ-index against a covector, (φω)_ν=(∂y^μ/∂x^ν)ω_μ, leaving a free lower index on the source M. Same numbers, transposed role, opposite spaces — which is why one produces target vectors and the other source covectors.
What is the difference between an active and a passive diffeomorphism?
Both are the same map φ, read two ways. Passively, φ is a coordinate change: points stay put and only their labels change, x'^μ=φ^μ(x), and components transform by the inverse Jacobian — the ordinary "transformation law." Actively, φ physically moves each point to φ(p) and drags the fields, so the metric at p is replaced by the pulled-back φ*g. The third derivation shows these are inverse operations on components and so carry identical information. In relativity the active reading matters most: a solution and its dragged copy are the same physics — the content of general covariance.
Does pullback respect composition of maps?
Yes, and it reverses the order — pullback is a contravariant functor. For φ:M→N and ψ:N→P one has (ψ∘φ)_=ψ_φ_ (same order) and (ψ∘φ)=φψ (reversed). The reversal is the coordinate-free chain rule: in components ∂z^α/∂x^ν=(∂z^α/∂y^μ)(∂y^μ/∂x^ν), the product of two Jacobians, and transposing a matrix product reverses order. This functoriality is what makes tensors globally well-defined: transition functions between overlapping charts compose consistently precisely because pushforward and pullback compose this way.
Can I always pull a metric back to a submanifold to get its geometry?
Yes — that pullback is exactly the induced (first fundamental) metric, and it needs only that the inclusion be a smooth immersion, not a diffeomorphism. Given φ:Σ→M embedding a surface in spacetime, φg gives lengths and angles on Σ measured with the ambient metric, as worked example 2 does for the sphere. One caveat specific to relativity: because g is Lorentzian, φg can be Riemannian, Lorentzian, or degenerate depending on the surface's orientation, distinguishing spacelike, timelike, and null hypersurfaces. A null surface has a degenerate induced metric — the pullback drops a rank — so null geometry needs separate machinery.
How does the Lie derivative relate to all of this?
The Lie derivative is the infinitesimal pullback. Take a vector field ξ, let φ_t be its flow, and pull a tensor T back along it; the rate of change at t=0 is L_ξT=(d/dt)|₀φ_tT. So everything here has an infinitesimal shadow: a finite isometry φg=g becomes Killing's equation L_ξg=∇_μξ_ν+∇_νξ_μ=0, and a finite active diffeomorphism becomes an infinitesimal gauge transformation δg_μν=L_ξg_μν. The Lie derivative needs no connection — only the flow — which distinguishes it from the covariant derivative and makes it the right tool for symmetries and gauge freedom.