Differential geometry
through exercises

Problem 1 · Contraction of a symmetrized pair with an antisymmetrized pair

H. S. Reall, Part III General Relativity (2022), §1.7 “Tensors”, exercise right after eq. (1.42), p. 20 — damtp.cam.ac.uk.

Show that T^(ab) X_[a|cd|b] = 0.

In Sucuri: all of it: the whole identity (section I).

Problem 15 · Conserved current associated with a Killing vector

H. S. Reall, Part III General Relativity (2022), §5.3 “Lie derivative”, last exercise of the section, p. 57 — damtp.cam.ac.uk; and D. Tong, General Relativity: Example Sheet 3 (2019), question 5, p. 2 — davidtong.org.

Reall: Let J^a = T^a_bX^b where T_ab is symmetric and satisfies ∇_aT^ab = 0 (e.g. the energy-momentum tensor: see later) and X^b is a Killing vector field. Show that ∇_aJ^a = 0, i.e., J^a is a conserved current.

Tong: Let K^μ be a Killing vector field and T_μν the energy momentum tensor. Let J^μ = T^μ_νK^ν. Show that J^μ is a conserved current, meaning ∇_μJ^μ = 0.

In Sucuri: all of it: both versions: conservation from the two hypotheses.

Problem 16 · Contraction of two Levi-Civita tensors

H. S. Reall, Part III General Relativity (2022), §8.5 “Volume form”, lemma with eq. (8.54) (“Proof. Optional exercise.”), p. 107 — damtp.cam.ac.uk.

Lemma.

ε^{a₁…a_p c_p+1…c_n} ε_{b₁…bp cp+1…cn} = ± p!(n − p)! δ^[a₁_b₁ … δ^a_p]_bp   (8.54)

where the upper (lower) sign holds for Riemannian (Lorentzian) signature.

Proof. Optional exercise.

[Context: (8.52) ε_12…n = √|g| and (8.53) ε^12…n = ±1/√|g| in a right-handed chart. The special case ε^abcdε_abcd = −24 (p = 0, n = 4, Lorentzian signature) is not written out in the text.]

In Sucuri, partly: the cases p = 0 and p = 2 in 4D, with both signatures; the lemma for general n and p, no.

Problem 2 · Covariant derivative of a (1,1) tensor

H. S. Reall, Part III General Relativity (2022), §3.1 “Introduction”, exercise right after eq. (3.17), p. 34 — damtp.cam.ac.uk.

Now the Leibniz rule can be used to obtain the formula for the coordinate basis components of ∇T where T is a (r, s) tensor:

Tμ₁…μrν₁…νs;ρ = Tμ₁…μrν₁…νs,ρ + Γμ₁σρ Tσμ₂…μrν₁…νs + … + Γμrσρ Tμ₁…μr−1σν₁…νs − Γσν₁ρ Tμ₁…μrσν₂…νs − … − Γσνsρ Tμ₁…μrν₁…νs−1σ   (3.17)

Exercise. Prove this result for a (1, 1) tensor.

[Context: eqs. (3.9) and (3.16) of the same text give, in the coordinate basis, Y^μ_;ν = Y^μ_,ν + Γ^μ_ρνY^ρ and η_μ;ν = η_μ,ν − Γ^ρ_μνη_ρ. The result asked for is therefore T^μ_ν;ρ = T^μ_ν,ρ + Γ^μ_σρT^σ_ν − Γ^σ_νρT^μ_σ. This is a specialization of (3.17) and does not appear literally in the text.]

In Sucuri: all of it: the (1,1) tensor, in Reall's convention, read from the written definition.

Problem 3 · The transformation of the Christoffel symbols

MIT OpenCourseWare, 8.962 General Relativity (Spring 2020), Problem Set 3, question 4, parts (a)–(b), p. 2 — ocw.mit.edu.

(a) Show that, under a coordinate transformation, the components of the Christoffel symbol transform as follows:

Γ^α′_β′γ′ = (∂x^α′/∂x^α)(∂x^β/∂x^β′)(∂x^γ/∂x^γ′) Γ^α_βγ − (∂²x^α′/∂x^β∂x^γ)(∂x^β/∂x^β′)(∂x^γ/∂x^γ′).

Do this by considering the form of the Christoffel symbol in terms of derivatives of the metric.

(b) Show that, using this rule, the components of the covariant derivative of a vector transform as tensors should: ∇_α′A^β′ = (∂x^α/∂x^α′)(∂x^β′/∂x^β) ∇_αA^β.

In Sucuri: all of it: (a) and (b), via the form of the Christoffel symbols in terms of derivatives of the metric, as the problem asks. The ∂′ of a product — Leibniz with the chain rule, ∂′_λ = J^σ_λ∂_σ — is written as a hypothesis: Sucuri has a single set of coordinates, and the rest is index algebra, which it does.

Problem 4 · Connection identities: derivatives of the metric

MIT OpenCourseWare, 8.962 General Relativity (Spring 2020), Problem Set 3, question 7, parts (a)–(c), p. 3 — ocw.mit.edu.

Prove the following connection identities:

(a) ∂_λg_μν = Γ_μνλ + Γ_νμλ.

(b) g_μκ∂_λg^κν = −g^κν∂_λg_μκ.

(c) ∂_λg^μν = −Γ^μ_λκg^κν − Γ^ν_λκg^κμ.

In Sucuri: all of it: (a), (b) and (c).

Problem 17 · Lie derivative of a covector and of the metric

H. S. Reall, Part III General Relativity (2022), §5.3 “Lie derivative”, “Exercises (examples sheet 2)” 1 and 2, eqs. (5.18)–(5.21), pp. 55–56 (the same content is in D. Tong, Example Sheet 1, Q5, and Example Sheet 2, Q7) — damtp.cam.ac.uk.

(1) Derive the formula for the Lie derivative of a covector field ω_a in a coordinate basis:

(L_Xω)_μ = X^ν∂_νω_μ + ω_ν∂_μX^ν   (5.18)

Show that this can be written in the basis-independent form

(L_Xω)_a = X^b∇_bω_a + ω_b∇_aX^b   (5.19)

where ∇ is any torsion-free connection.

(2) Show that the Lie derivative of a (0, 2) tensor field g_ab in a coordinate basis is

(L_Xg)_μν = X^ρ∂_ρg_μν + g_μρ∂_νX^ρ + g_ρν∂_μX^ρ   (5.20)

and that this can be written in the basis-independent form

(L_Xg)_ab = ∇_aX_b + ∇_bX_a   (5.21)

where ∇ is the Levi-Civita connection.

Note that we cannot use abstract indices in (5.18) and (5.20) because they hold only in a coordinate basis.

In Sucuri: all of it: (1) and (2): the forms with ∇ in place of ∂.

Problem 65 · Conformal metrics and Nordstrøm's theory

J. M. Evans, Part II General Relativity, Example Sheet 1 (Cambridge, 2026), question 8, p. 2 — damtp.cam.ac.uk.

Two metrics g_αβ and ĝ_αβ are conformally related if ĝ_αβ = Ω²g_αβ for some scalar function Ω. Show that their Christoffel symbols Γ^α_βγ and Γ̂^α_βγ are related by

Γ̂^α_βγ = Γ^α_βγ + Ω⁻¹(δ^α_βΩ_,γ + δ^α_γΩ_,β − g^αδg_βγΩ_,δ).

In Nordstrøm's theory of gravity the metric is given by g_αβ = e^2φη_αβ, where φ is a scalar function of position and η_αβ is the Minkowski metric. Compute the equation of a geodesic in Nordstrøm's theory and use your result to show that T^αT^βg_αβ is constant on a geodesic, where T^α is the tangent vector corresponding to an affine parameter.

Using the results of question 5, show that a null geodesic is also a null geodesic in Minkowski spacetime, and hence deduce that light rays are not subject to gravitational deflection.

Show that for any time-like geodesic, with suitably chosen parameter μ,

d²x^α/dμ² = −η^αγψ_,γ

for some function ψ.

[A comma denotes differentiation with respect to a coordinate, e.g. φ_,α = ∂_αφ, and the Christoffel symbols are defined by Γ^α_βγ = ½ g^αμ(g_βμ,γ + g_μγ,β − g_βγ,μ).]

In Sucuri, partly: the relation between the Christoffel symbols; Nordstrøm's theory, no.

Problem 5 · Covariant divergences and the d'Alembertian in a coordinate basis

MIT OpenCourseWare, 8.962 General Relativity (Spring 2020), Problem Set 3, question 7, parts (d)–(f), p. 3 — ocw.mit.edu.

(Continuation of question 7: “Prove the following identities of the connection”.) The next three parts rely on an identity I will prove on either Thursday March 2nd or on Tuesday March 7th. The quantity g is the determinant of the metric g_μν.

(d) ∇_νA_μ^ν = |g|^−1/2∂_ν(|g|^1/2A_μ^ν) − Γ^λ_νμA_λ^ν in a coordinate basis.

(e) ∇_νF^μν = |g|^−1/2∂_ν(|g|^1/2F^μν) in a coordinate basis, if F^μν is antisymmetric.

(f) □S ≡ g^μν∇_μ∇_νS = |g|^−1/2∂_μ(|g|^1/2g^μν∂_νS) in a coordinate basis. (S is a scalar function.)

In Sucuri: all of it: (d), (e) and (f); and the divergence of a vector, which is the special case.

Problem 64 · Variation of the determinant and contraction of the Christoffel symbols

J. M. Evans, Part II General Relativity, Example Sheet 3 (Cambridge, 2026), question 10, parts (i)–(iv), p. 3 — damtp.cam.ac.uk.

(i) Let M be an invertible matrix. Show that under a small change δM, the corresponding change in the determinant is, to first order, δ(det M) = (det M) tr(M⁻¹δM). [Hint: if the entries of a matrix A are small then, to first order, det(I + A) = 1 + tr A, where I is the identity matrix.]

(ii) Let g_αβ be a metric with Lorentzian signature and let g = det(g_αβ). Use the result in (i) to show that

Γ_α^β_β = (1/2g) ∂g/∂x^α = (1/√−g) ∂(√−g)/∂x^α,

where Γ_α^γ_β is the Levi-Civita connection. (Note that g < 0 for a metric with Lorentzian signature.)

(iii) A tensor density of weight q is defined to be a quantity that transforms as a tensor under a change in coordinates from {x^μ} to {x̃^α} but with an additional factor of Δ^q, where Δ = det(∂x^μ/∂x̃^α), the Jacobian. Show that g transforms as a scalar density of weight 2.

(iv) For ψ a scalar density of weight q, the covariant derivative is defined by

∇_αψ = ∂ψ/∂x^α − q Γ_α^β_β ψ.

Show that ∇_αψ is a covector density of weight q.

In Sucuri, partly: part (ii); (i) is Jacobi's formula, which in Sucuri is the declaration of det g; (iii) and (iv), densities, no.

Problem 66 · Maxwell's equations with √−g

G. 't Hooft, Introduction to General Relativity (Utrecht, 2013), ch. 10 “Electromagnetism”, exercise after eq. (10.12), eqs. (10.13)–(10.14), p. 45 — webspace.science.uu.nl.

[Context: varying A_μ, we find that the inhomogeneous equation becomes D_μF_μν = g^αβD_αF_βν = −J_ν (10.12), and hence receives a contribution from the gravitational field Γ^λ_μν and the potential g^αβ.]

Exercise: show that Eq. (10.12) can also be written as

∂_μ(√−g F^μν) = −√−g J^ν,   (10.13)

and that

∂_μ(√−g J^μ) = 0.   (10.14)

Thus √−g J^μ is the real conserved current, and Eq. (10.13) implies that √−g acts as the dielectric constant of the vacuum.

In Sucuri: all of it: (10.13), via the identity ∇_μF^μν = (−g)^−1/2∂_μ(√−g F^μν), and (10.14): ∂_ν of the left-hand side of (10.13) is identically zero.

Problem 10 · Contracted Bianchi identity

D. Tong, General Relativity: Example Sheet 2 (2019), question 4, p. 1 — davidtong.org.

The Riemann tensor constructed from the Levi-Civita connection obeys the Bianchi identity R^μ_ν[ρσ;λ] = 0. Use this fact to derive the contracted Bianchi identity G^μ_ν;μ = 0, where G_μν = R_μν − ½ R g_μν is the Einstein tensor.

In Sucuri: all of it: the contracted identity, from the second Bianchi identity.

Problem 14 · Second derivative of a Killing vector and the Killing vectors of Minkowski

D. Tong, General Relativity: Example Sheet 3 (2019), question 6, p. 2 — davidtong.org.

Show that a Killing vector field K^μ satisfies the equation

∇_μ∇_νK^ρ = R^ρ_νμσ K^σ

[Hint: use the identity R^ρ_[μνσ] = 0.]

Deduce that in Minkowski spacetime the components of Killing covectors are linear functions of the coordinates.

In Sucuri: all of it: the identity, and, in Minkowski, ∂∂K = 0: K linear in the coordinates.

Problem 68 · Second-order equation for Killing covectors; the 10 Killing vectors of Minkowski

J. M. Evans, Part II General Relativity, Example Sheet 3 (Cambridge, 2026), question 6, p. 2 — damtp.cam.ac.uk.

Let ξ_α be a Killing covector field, satisfying ξ_α;β + ξ_β;α = 0 (see question 5 on Example Sheet 2). Use the Ricci identity and R^α_[βγδ] = 0 to show that

ξ_α;βγ = −R^δ_γαβ ξ_δ .

In the case of Minkowski space, integrate this equation twice and deduce that there are 10 independent Killing vectors.

In Sucuri: all of it: the identity, in Evans's notation, and the 10 Killing fields of Minkowski.

Problem 69 · Scalar field with ∇∇φ = Ricci

J. M. Evans, Part II General Relativity, Example Sheet 3 (Cambridge, 2026), question 4, p. 1 — damtp.cam.ac.uk.

Let φ be a scalar field in curved spacetime such that

∇_α∇_βφ = R_αβ,

where R_αβ is the Ricci tensor. Show that

∇_α(∇_β∇^βφ) = −2R_αβ∇^βφ

and hence deduce that ∇_αφ∇^αφ + R is constant.

[Hint: use the Ricci identity and the contracted Bianchi identity, ∇^βR_αβ = ½∇_αR.]

In Sucuri: all of it: the conclusion, with the contracted Bianchi identity as a hypothesis, as in the hint.

Problem 6 · Torsion and the commutator of covariant derivatives of a function

D. Tong, General Relativity: Example Sheet 2 (2019), question 2, p. 1 — davidtong.org.

Let ∇ be a connection that is not torsion-free. Let T(X, Y) = ∇_XY − ∇_YX − [X, Y], where X and Y are vector fields. Show that this defines a (1, 2) tensor field T. This is called the torsion tensor. Show that, for any function f,

2∇_[μ∇_ν] f = −T^ρ_μν ∇_ρ f.

In Sucuri: all of it: the identity, with the torsion written in a chart, T^ρ_μν = Γ^ρ_μν − Γ^ρ_νμ; that T is a tensor comes out index-free (problem 21).

Problem 7 · Ricci identity

H. S. Reall, Part 3 General Relativity (2022), §4.2 “The Riemann tensor”, exercise with eq. (4.8), p. 42 — damtp.cam.ac.uk.

Context: we saw earlier that, with vanishing torsion, the second covariant derivatives of a function commute. The same is not true of covariant derivatives of tensor fields. The failure to commute arises from the Riemann tensor.

Exercise. Let ∇ be a torsion-free connection. Prove the Ricci identity:

∇_c∇_dZ^a − ∇_d∇_cZ^a = R^a_bcd Z^b (4.8)

Hint. Show that the equation is true when multiplied by arbitrary vector fields X^c and Y^d.

In Sucuri: all of it: the identity (4.8), from the components (4.6) of the Riemann tensor, with the torsion-free connection.

Problem 8 · Ricci identity for a 1-form

D. Tong, General Relativity: Example Sheet 2 (2019), question 3, p. 1 — davidtong.org.

Let ∇ be a torsion-free connection. Derive the analogue of the Ricci identity for a 1-form ω,

2∇_[μ∇_ν] ω_ρ = −R^σ_ρμν ω_σ.

In Sucuri: all of it: the identity on the 1-form, from the definition of the Riemann tensor on a vector.

Problem 9 · Pair-exchange symmetry of the Riemann tensor

S. M. Carroll, Lecture Notes on General Relativity (1997), ch. 3 “Curvature”, paragraph right after eq. (3.81), p. 79 — arxiv.org.

This last property is equivalent to the vanishing of the antisymmetric part of the last three indices:

R_ρ[σμν] = 0.   (3.81)

All of these properties have been derived in a special coordinate system, but they are all tensor equations; therefore they will be true in any coordinates. Not all of them are independent; with some effort, you can show that (3.64), (3.78) and (3.81) together imply (3.79). The logical interdependence of the equations is usually less important than the simple fact that they are true.

[References in the text: (3.64) R^ρ_σμν = −R^ρ_σνμ; (3.78) R_ρσμν = −R_σρμν; (3.79) R_ρσμν = R_μνρσ; (3.80) R_ρσμν + R_ρμνσ + R_ρνσμ = 0.]

In Sucuri: all of it: the pair exchange, rewritten index-free, with the three properties as hypotheses.

Problem 11 · Riemann from its symmetrized part

H. S. Reall, Part III General Relativity (2022), §4.5 “Geodesic deviation”, exercise with eq. (4.25), p. 47 (the symmetries used are eqs. (4.16)–(4.17) of §4.4, p. 45) — damtp.cam.ac.uk.

[…] Hence by measuring the LHS above [of the geodesic deviation equation (4.24)] we can determine R^a_(bc)d. From this we can determine R^a_bcd:

Exercise. Show that, for a torsion-free connection,

R^a_bcd = ⅔ (R^a_(bc)d − R^a_(bd)c).   (4.25)

[Symmetries available in the text: (4.16) R^a_b(cd) = 0 and, for a torsion-free connection, (4.17) R^a_[bcd] = 0.]

In Sucuri: all of it: (4.25), rewritten index-free.

Problem 59 · Vector identities in index notation

R. E. Hunt, Vector Calculus: Example Sheet 1 (Part IA, Cambridge, Lent 2026), question 3, p. 1 — damtp.cam.ac.uk.

Use suffix notation to show that for vector fields u(x) and v(x),

∇ × (u × v) = (∇ · v)u − (∇ · u)v + (v · ∇)u − (u · ∇)v

and

(u · ∇)u = ∇(½|u|²) − u × (∇ × u).

In Sucuri: all of it: both identities.

Problem 60 · Decomposition of a second-rank tensor

R. E. Hunt, Vector Calculus: Example Sheet 4 (Part IA, Cambridge, Lent 2026), section “Symmetries of Tensors”, question 6, p. 2 — damtp.cam.ac.uk.

(i) By first decomposing into symmetric and antisymmetric parts, show that an arbitrary second rank tensor T_ij can be written in the form

T_ij = α δ_ij + ε_ijk ω_k + D_ij

for some scalar α, vector ω and symmetric traceless tensor D_ij.

(ii) Let u(x) be a vector field and let T_ij = ∂u_i/∂x_j. Show that α = ⅓ ∇ · u and ω = −½ ∇ × u. Find D_ij (the “deviatoric strain tensor”) for the case u(x) = (xy², yz², zx²).

* Verify that (0, 0, 1) is a principal axis of D_ij at the point x = (2, 3, 0), and find the others.

In Sucuri, partly: part (i), through the antisymmetric part; (ii), no.

Problem 51 · d² = 0 in ℝ³: curl grad = 0 and div curl = 0

C. Wendl, Differential Geometry I, Problem Set 6 (Humboldt-Universität zu Berlin, 2016–17), problem 6(c) — mathematik.hu-berlin.de.

Context (preamble of problem 6 and part (b)): given a volume form μ ∈ Ωⁿ(M) on an n-manifold M, the divergence of X ∈ Vec(M) is the unique real-valued function div(X) : M → ℝ such that L_Xμ = div(X) μ. On M = ℝ³, with μ = dx ∧ dy ∧ dz and X = X^x∂_x + X^y∂_y + X^z∂_z, div(X) = ∂_xX^x + ∂_yX^y + ∂_zX^z, an expression sometimes also denoted by ∇ · X.

(c) Recall that on ℝ³, the gradient of a function f : ℝ³ → ℝ is the vector field

grad(f) = ∇f := (∂_xf)∂_x + (∂_yf)∂_y + (∂_zf)∂_z,

and the curl of a vector field X = X^x∂_x + X^y∂_y + X^z∂_z is the vector field

curl(X) = ∇ × X := (∂_yX^z − ∂_zX^y)∂_x + (∂_zX^x − ∂_xX^z)∂_y + (∂_xX^y − ∂_yX^x)∂_z.

Using the relations of these operations to differential forms and the exterior derivative, deduce from d² = 0 the formulas

∇ × (∇f) = 0   and   ∇ · (∇ × X) = 0

for all f ∈ C^∞(ℝ³) and X ∈ Vec(ℝ³).

In Sucuri: all of it: curl grad = 0 and div curl = 0, with forms and in index notation.

Problem 63 · Ricci identity for rank-2 tensors

J. M. Evans, Part II General Relativity, Example Sheet 3 (Cambridge, 2026), question 1, parts (i)–(ii), p. 1 — damtp.cam.ac.uk.

The Ricci identity for a vector field V^μ is ∇_α∇_βV^μ − ∇_β∇_αV^μ = R^μ_ναβV^ν.

(i) Deduce the corresponding result for a covector field W_μ by considering ∇_α∇_β(V^μW_μ) − ∇_β∇_α(V^μW_μ). Is there an easier way to find this result?

(ii) [part corresponding to exercise 63] Given two vector fields U^μ and V^μ, evaluate ∇_α∇_β(U^μV^ν) − ∇_β∇_α(U^μV^ν). Deduce that, for an arbitrary tensor field T^μν,

∇_α∇_βT^μν − ∇_β∇_αT^μν = R^μ_σαβT^σν + R^ν_σαβT^μσ.

Hence show that ∇_α∇_βT^αβ = ∇_β∇_αT^αβ, for any tensor field T^αβ.

In Sucuri: all of it: (i) (problem 8) and (ii).

Problem 13 · The Weyl tensor

D. Tong, General Relativity: Example Sheet 2 (2019), question 9, p. 3 — davidtong.org.

In a d-dimensional spacetime, define a tensor

C_μνρσ = R_μνρσ + α(R_μρg_νσ + R_νσg_μρ − R_μσg_νρ − R_νρg_μσ) + βR(g_μρg_νσ − g_μσg_νρ),

where α and β are constants. Show that C_μνρσ has the same symmetries as R_μνρσ.

What values of α and β give C^μ_νμσ = 0? Determine them. With this extra condition C_μνρσ is called the Weyl tensor. Show that it vanishes if d = 2, 3.

Setting d = 4, how many independent components do R_μν and C_μνρσ have? Show that in vacuum

∇^μC_μνρσ = 0.

What does the Weyl tensor represent physically?

In Sucuri: all of it: the symmetries, α and β, the vanishing Weyl in d = 2, 3, the counts in d = 4 and ∇^μC_μνρσ = 0 in vacuum; the physics question is left to the reader.

Problem 12 · Independent components of the Riemann; the two-dimensional case

D. Tong, General Relativity: Example Sheet 2 (2019), question 8, p. 2 — davidtong.org.

How many independent components does the Riemann tensor (of the Levi-Civita connection) have in two, three and four dimensions? Show that in two dimensions

R_μνρσ = ½ R (g_μρg_νσ − g_μσg_νρ).

Discuss the implications for general relativity in two spacetime dimensions.

In Sucuri: all of it: the counts (1, 6 and 20), the 2D formula and, for the discussion, that G_μν ≡ 0 in 2D.

Problem 61 · Counting components of antisymmetric tensors

A. Guth, MIT 8.962 General Relativity (Spring 2018), Problem Set 4, Problem 1 (“Counting Tensors”), p. 1 — web.mit.edu.

In lecture we derived the fact that if S_μ1···μR is fully symmetric in its indices, and each index can take on D different values, then the number of independent components is

N = (R + D − 1)! / [R! (D − 1)!].

Suppose that A_μ1···μR is fully antisymmetric in its indices, with each index again taking on D different values. How many independent components does A_μ1···μR have?

In Sucuri, partly: the count dimension by dimension; the general formula in R and D, no.

Problem 62 · Rotational Killing fields in 3D Euclidean space

Y. Ali-Haïmoud, General Relativity (NYU, Fall 2019), Homework 6, Exercise 1, parts (v) and (vi) (with the preamble that precedes them), p. 1 — cosmo.nyu.edu.

Consider flat 3-D space, with cartesian coordinates x¹, x², x³ and associated coordinate basis {∂_(i)}, i = 1, 2, 3, and line element dℓ² = δ_ijdxⁱdx^j. Define the three vector fields V_(i) ≡ ε_ijkx^j∂_(k), where ε_ijk is the Levi-Civita symbol (fully antisymmetric, such that ε₁₂₃ = 1), and we sum over repeated indices, even if they are not up and down.

(v) Show that these three vector fields are Killing vector fields.

(vi) Show that they satisfy the commutation relations [V_(i), V_(j)] = −ε_ijkV_(k).

In Sucuri: all of it: (v) and (vi).

Problem 71 · Linearized curvature

J. M. Evans, Part II General Relativity, Example Sheet 4 (Cambridge, 2026), question 10, part (i), p. 4 — damtp.cam.ac.uk.

(i) A weak gravitational field has the spacetime metric g_αβ = η_αβ + εh_αβ + O(ε²), where η_αβ is the Minkowski metric and ε is small. Show that

R_αβγδ = ½ε(h_αδ,βγ + h_βγ,αδ − h_αγ,βδ − h_βδ,αγ) + O(ε²).

Let h = h^γ_γ and define h̄_αβ = h_αβ − ½h η_αβ. Check that h_αβ = h̄_αβ − ½h̄ η_αβ where h̄ = h̄^γ_γ, and show that

R_αβ = ½ε(−□h̄_αβ + h̄_α^γ_,βγ + h̄_β^γ_,αγ + ½η_αβ□h̄) + O(ε²),

where □ = η^αβ∂_α∂_β. What is the linearised vacuum Einstein equation for h̄_αβ?

In Sucuri: all of it: the linearized Riemann and Ricci; the vacuum equation is read off the result.

Problem 67 · Bianchi identity for F = dA and conservation of the Maxwell stress tensor

J. M. Evans, Part II General Relativity, Example Sheet 3 (Cambridge, 2026), question 5, p. 2 — damtp.cam.ac.uk.

The Maxwell tensor F_αβ for the electromagnetic field in curved spacetime is given in terms of a vector potential A_α by F_αβ = ∇_αA_β − ∇_βA_α. Show that this implies ∇_[γF_αβ] = 0. Show further that if ∇_βF^αβ = 0, then the energy momentum tensor

T^αβ = F^α_γF^βγ − ¼ g^αβF_γδF^γδ,

is conserved, i.e. ∇_βT^αβ = 0.

In Sucuri: all of it: ∇_[γF_αβ] = 0 for F = dA written with ∇, via the first Bianchi identity, and the conservation of T^αβ.

Problem 70 · Exterior derivative in coordinates: the case of 1-forms and 2-forms

C. Wendl, Differential Geometry I (Humboldt-Universität zu Berlin, 2016–17), Problem Set 6, Problem 4(b) (pp. 2–3) — mathematik.hu-berlin.de.

4. Recall that if (x¹, …, xⁿ) : U → ℝⁿ is a chart defined on an open subset in some n-manifold M, any k-form ω ∈ Ω^k(M) can be written on U as

ω = ω_i1…ik dx^i₁ ⊗ … ⊗ dx^i_k = (1/k!) ω_i1…ik dx^i₁ ∧ … ∧ dx^i_k = Σ_i1<…<ik ω_i1…ik dx^i₁ ∧ … ∧ dx^i_k,

where the first two expressions use the Einstein summation convention and the third one does not. Here the component functions ω_i1…ik : U → ℝ can be written in terms of the coordinate vector fields ∂₁, …, ∂_n as ω_i1…ik = ω(∂_i1, …, ∂_ik). In order to write down a coordinate formula for the exterior derivative, we introduce the following notation: given any collection of functions T_i1…ik on U labeled by the indices i₁, …, i_k, define

T_[i1…ik] := (1/k!) Σ_σ∈Sk (−1)^|σ| T_{iσ(1)…iσ(k)},

so for instance if T_i1…ik are the components of a tensor field T, then Alt(T)_i1,…,ik = T_[i1…ik], and the wedge product of α ∈ Ω^k(M) and β ∈ Ω^ℓ(M) can now be written in coordinates as

(α ∧ β)_{i1…ikj1…jℓ} = α_[i1…ikβ_j1…jℓ].

(a) Prove that the exterior derivative d : Ω^k(M) → Ω^k+1(M) satisfies (dω)_i1…ik+1 = (k + 1) ∂_[i1ω_i2…ik+1].

(b) Show that for λ ∈ Ω¹(M) and ω ∈ Ω²(M), the above formula reduces to

(dλ)_ij = ∂_iλ_j − ∂_jλ_i,  and  (dω)_ijk = ∂_iω_jk + ∂_jω_ki + ∂_kω_ij.

In Sucuri, partly: the cyclic sum, by explicit antisymmetrization; ∂_[iω_jk], with the antisymmetrization reaching across the derivative's index, cannot be written yet.

Problem 35 · Unit sphere: Christoffel, Ricci and Ricci scalar

J. M. Cline, A Short Course in General Relativity (2026), ch. 7, §7.5 Problems, problem 7.2, p. 43 — arxiv.org.

(a) Compute the nonvanishing Christoffel symbols for the unit sphere metric, with ds² = dθ² + sin²θ dϕ².

(b) Compute the Ricci tensor.

(c) Compute the Ricci scalar.

In Sucuri: all of it: (a), (b) and (c).

Problem 36 · Curvature of a sphere

MIT OpenCourseWare, 8.962 General Relativity (Spring 2020), Problem Set 5, question 4 (“Curvature of a sphere”), parts (a)–(c), p. 3 — ocw.mit.edu.

(a) Compute all the nonvanishing components of the Riemann tensor R_ijkl [(i, j, k, l) ∈ (θ, φ)] for the surface of a 2-sphere.

(b) Consider the parallel transport of a tangent vector A = A^θ e_θ + A^φ e_φ on the sphere around an infinitesimal parallelogram of sides e_θ dθ and e_φ dφ. Using the results of part (a), show that to first order in dΩ ≡ sin θ dθ dφ, the length of A is unchanged, but its direction rotates through an angle equal to dΩ.

(c) Show that, if A is parallel transported around the boundary of any simply connected solid angle Ω, its direction rotates through an angle Ω. (“Simply connected” is a topological term meaning that the boundary of the region could be shrunk to a point; it tells us that there are no holes in the manifold or other pathologies.) Compare with the result of Problem 3 (parallel transport around the parallel θ = θ₀ on the 2-sphere of radius a, ds² = a²(dθ² + sin²θ dφ²)).

In Sucuri, partly: (a) the Riemann tensor, and (b): δA = R(·)A dθ dφ is orthogonal to A and has |δA| = sin θ |A| dθ dφ — a rotation by dΩ; (c), for an arbitrary region, is summing the infinitesimal rotations, an argument, and does not count.

Problem 37 · Covariant derivative and vector Laplacian in polar coordinates

J. M. Cline, A Short Course in General Relativity (2026), Problem 7.6, §7.5 “Problems”, pp. 43–44 — arxiv.org.

7.6. (a) Compute the components A_r;r, A_r;θ, A_θ;r, A_θ;θ of a vector in 2D polar coordinates. Recall that Γ^r_θθ = −r and Γ^θ_rθ = 1/r.

(b) Using the results of (a), find A_r;rr, A_r;θθ, A_θ;rr and A_θ;θθ.

(c) Put these together to obtain the covariant Laplacian g^ij∇_i∇_jA_k for the covariant components of A⃗. It is known as the vector Laplacian. Hint: you can consider A_θ to be dimensionless, and A_r to have dimensions of 1/r, as a way of checking dimensions. I get

∇²A_r = A_r,rr + r⁻²(A_r,θθ − 2r⁻¹A_θ,θ + rA_r,r − A_r),
∇²A_θ = A_θ,rr + r⁻²A_θ,θθ − r⁻¹A_θ,r + 2r⁻¹A_r,θ.   (7.44)

(This does not agree with Wolfram MathWorld, because the latter assumes a basis where the θ unit vector is normalized to 1, whereas x̂^θ · x̂^θ = g^θθ = 1/r² here. In other words, A_θ is divided by r in Wolfram to give it the same dimensions as A_r. See the box on coordinate versus orthonormal bases below.)

In Sucuri: all of it: (a) and (b), the first and second covariant derivatives, and (c) the vector Laplacian, which agrees with (7.44).

Problem 38 · Connection in Rindler spacetime

MIT OpenCourseWare, 8.962 General Relativity (Spring 2020), Problem Set 4, question 1, p. 1 — ocw.mit.edu.

The spacetime for an accelerated observer that we derived on Pset 2,

ds² = −(1 + g x̄)² dt̄² + dx̄² + dȳ² + dz̄²   (1)

is known as “Rindler spacetime”. Compute all non-zero Christoffel symbols for this spacetime. (Carroll problem 3.3 will help you quite a bit here.)

In Sucuri: all of it: the Christoffel symbols.

Problem 39 · Riemann tensor for 1+1 static spacetimes

MIT OpenCourseWare, 8.962 General Relativity (Spring 2020), Problem Set 5, question 6, parts (a)–(b), p. 3 — ocw.mit.edu.

(a) Compute all the nonvanishing components of the Riemann tensor for the spacetime with line element

ds² = −e^2φ(x) dt² + e^−2ψ(x) dx².

(b) For the case φ = ψ = ½ ln|g(x − x₀)| where g and x₀ are constants, show that the spacetime is flat and find a coordinate transformation to globally flat coordinates (t̄, x̄) such that ds² = −dt̄² + dx̄².

In Sucuri, partly: part (a); in (b), the change to flat coordinates, no.

Problem 40 · Christoffel symbols and curvature of a static spherically symmetric metric

D. Tong, General Relativity: Example Sheet 2 (2019), question 11*, p. 3 — davidtong.org.

Consider metrics of the form

ds² = −f(r)² dt² + f(r)⁻² dr² + r²(dθ² + sin²θ dφ²).

Use the action for a test particle to write down the geodesic equations in this metric, and hence extract the Christoffel symbols in coordinates (t, r, θ, φ).

Use a basis of vierbeins to determine the curvature 2-form, and hence the components of the Riemann tensor in coordinates (t, r, θ, φ).

In Sucuri: all of it: the geodesic equations — from the Christoffel symbols and checked against the test-particle action —, the Christoffel symbols, the tetrad, the curvature 2-forms and the Riemann tensor, in the frame and in coordinates.

Problem 41 · Schwarzschild curvature 2-forms in an orthonormal basis

H. S. Reall, Part 3 General Relativity (Cambridge, 2022), §8.4 “Curvature 2-forms”, Exercise (examples sheet 4) after the Example, eqs. (8.45)–(8.51), pp. 105–106 — damtp.cam.ac.uk.

Context (§8.2, eq. (8.27), p. 102): the Schwarzschild spacetime admits the obvious tetrad

e⁰ = f dt,   e¹ = f⁻¹ dr,   e² = r dθ,   e³ = r sin θ dφ,   where f = √(1 − 2M/r),

with connection 1-forms ω⁰₁ = f′e⁰, ω²₁ = (f/r)e², ω³₁ = (f/r)e³, ω³₂ = (1/r) cot θ e³ (and those related by ω_μν = −ω_νμ).

Context (§8.4): in an orthonormal basis the curvature 2-forms are Θ^μ_ν = ½ R^μ_νρσ e^ρ ∧ e^σ (8.45); the antisymmetry of the Riemann tensor implies Θ_μν = −Θ_νμ (8.46); and Θ^μ_ν = dω^μ_ν + ω^μ_ρ ∧ ω^ρ_ν (8.47). In the Example, from ω⁰₁ = f′e⁰ one obtains

Θ₀₁ = −Θ⁰₁ = (ff″ + f′²) e⁰ ∧ e¹  (8.50),   R₀₁₀₁ = −R₀₁₁₀ = ff″ + f′² = ½(f²)″ = −2M/r³  (8.51).

Exercise (examples sheet 4). Determine the remaining curvature 2-forms Θ₀₂, Θ₀₃, Θ₁₂, Θ₁₃, Θ₂₃ (all others are related to these by (8.46)). Hence determine the Riemann tensor components. Check that the Ricci tensor vanishes.

In Sucuri: all of it: the five requested 2-forms, the frame components of the Riemann tensor, and the vanishing Ricci tensor.

Problem 42 · Christoffel symbols and geodesics of the hyperbolic plane

J. Ross, Part III 2016: Differential Geometry — Example Sheets (Cambridge, 2016), Example Sheet 4, Question 9 (p. 13 of the compiled PDF) — dec41.user.srcf.net.

9. Define hyperbolic space H to be the upper half plane {(x, y) : y > 0} and set

g = (dx ⊗ dx + dy ⊗ dy) / y².

Verify that g defines a Riemannian metric on H, and compute the Christoffel symbols of the Levi-Civita connecion [sic]. Show that γ = (u, v) is a geodesic if and only if

u″v = 2u′v′  and  v″v = v′² − u′²

and use this to find the geodesics.

In Sucuri: all of it: g is Riemannian (the diagonal is positive for y > 0), the Christoffel symbols, the geodesic equations and the geodesics: semicircles (x − x₀)² + y² = κ/L² and, for L = 0, vertical lines.

Problem 43 · The unit 3-sphere

J. M. Cline, A Short Course in General Relativity (2026), ch. 7, §7.5 Problems, problem 7.10, pp. 44–45 — arxiv.org.

The unit 3-sphere can be embedded into 4D Euclidean space with coordinates (w, x, y, z) by the constraint x² + y² + z² + w² = 1.

(a) Show that the constraint is satisfied by parametrizing the Euclidean coordinates by

w = c_ψ,   x = s_ψs_θc_ϕ,   y = s_ψs_θs_ϕ,   z = s_ψc_θ   (7.46)

where c and s stand for cosine and sine of the subscripted angles 1 ≤ θ, ψ ≤ π and 0 ≤ ϕ ≤ 2π. [sic: read 0 ≤ θ, ψ ≤ π]

(b) By computing the line element in Euclidean space, show that the induced metric is given by diag(1, s_ψ², s_ψ²s_θ²) for the coordinates (ψ, θ, ϕ). Hint: start with dx² + dy² to isolate the dϕ² term, then add dz² to get the dθ² term. Add dw² last. Find the volume element and compute the volume of the 3-sphere.

(c) Plug the metric into a symbolic manipulation package and show that the Ricci curvature is R = 6. Also find R_ij.

(d) Compute the volume V of a small region ψ < ϵ to O(ϵ⁵) and compare it to the corresponding volume V_E of a sphere of radius ϵ in flat (Euclidean) space. Thereby check Eq. (7.20), V/V_E = 1 − r²R/[6(d + 2)] + O(r⁴), for the 3-sphere.

In Sucuri: all of it: (a) the constraint, (b) the induced metric, the volume element and the volume 2π², (c) R = 6 and R_ij = 2g_ij, (d) V to O(ϵ⁵) and the ratio V/V_E = 1 − ϵ²/5, which is (7.20) with R = 6 and d = 3.

Problem 44 · Isometries and the so(3) algebra

D. Tong, General Relativity: Example Sheet 3 (2019), question 4, p. 1 — davidtong.org.

(i) Let X and Y be two vector fields. Show that ℒ_X(ℒ_YQ) − ℒ_Y(ℒ_XQ) = ℒ_[X,Y]Q, when Q is either a function or a vector field. Use the Leibniz property of the Lie derivative to show that this also holds when Q is a one-form.

(ii) Demonstrate that if a Riemannian or Lorentzian manifold has two “independent” isometries then it has a third, and define what is meant by independent here.

(iii) Consider the unit sphere with metric ds² = dθ² + sin²θ dφ². Show that X = ∂/∂φ and Y = sin φ ∂/∂θ + cot θ cos φ ∂/∂φ are Killing vectors. Find a third, and show that they obey the Lie algebra of so(3).

In Sucuri: all of it: (i) on functions (the definition of the bracket), on vector fields (Jacobi) and on 1-forms (Leibniz, problem 46); (ii) the bracket of two Killing fields is Killing — “independent” means it is not a combination of the two; (iii) all of it.

Problem 45 · Curvature of the Robertson–Walker metric

S. M. Carroll, Lecture Notes on General Relativity (1997), ch. 8 “Cosmology”, metric (8.7) on p. 219 and eqs. (8.12)–(8.14) on p. 220 — arxiv.org.

[Metric (8.7): ds² = −dt² + a²(t)[dr²/(1 − kr²) + r²(dθ² + sin²θ dφ²)].]

With the metric in hand, we can set about computing the connection coefficients and curvature tensor. Setting ȧ ≡ da/dt, the Christoffel symbols are given by

Γ⁰₁₁ = aȧ/(1 − kr²),   Γ⁰₂₂ = aȧr²,   Γ⁰₃₃ = aȧr² sin²θ,
Γ¹₀₁ = Γ¹₁₀ = Γ²₀₂ = Γ²₂₀ = Γ³₀₃ = Γ³₃₀ = ȧ/a,
Γ¹₂₂ = −r(1 − kr²),   Γ¹₃₃ = −r(1 − kr²) sin²θ,
Γ²₁₂ = Γ²₂₁ = Γ³₁₃ = Γ³₃₁ = 1/r,
Γ²₃₃ = −sin θ cos θ,   Γ³₂₃ = Γ³₃₂ = cot θ.   (8.12)

The nonzero components of the Ricci tensor are

R₀₀ = −3ä/a,   R₁₁ = (aä + 2ȧ² + 2k)/(1 − kr²),   R₂₂ = r²(aä + 2ȧ² + 2k),   R₃₃ = r²(aä + 2ȧ² + 2k) sin²θ,   (8.13)

and the Ricci scalar is then

R = (6/a²)(aä + ȧ² + k).   (8.14)

In Sucuri: all of it: the Christoffel symbols, the Ricci tensor and the scalar.

Problem 72 · Negative curvature: 3D hyperbolic space

A. Guth, MIT 8.962 General Relativity (Spring 2018), Problem Set 6, Problem 4 (“Negative Curvature”), p. 4 — web.mit.edu.

Consider a 3-dimensional space given by the set of points {(x, y, z), x ∈ ℝ, y ∈ ℝ, z > 0} with the metric

ds² = (a/z²)(dx² + dy² + dz²).   (4.1)

Compute the metric, connection, Riemann curvature tensor, Ricci tensor, and Ricci scalar.

In Sucuri: all of it: the connection, the Riemann tensor, the Ricci tensor and the scalar.

Problem 73 · Surfaces of revolution

C. Hirata, Ph 236 – Homework 4 (Caltech, 2011), problem 1 “Surfaces of revolution”, p. 1 — tapir.caltech.edu.

Consider a 2-dimensional surface M embedded in 3-dimensional Euclidean space ℝ³. Suppose further that it is a surface of revolution, i.e. it can be described in cylindrical coordinates (z, ϖ, φ) by an equation of the form

z = f(ϖ).

The two-dimensional surface can be written locally with coordinates (ϖ, φ).

(a) Show that the line element for the surface is ds² = F(ϖ) dϖ² + ϖ² dφ², where F(ϖ) = 1 + [f′(ϖ)]².

(b) Compute the Christoffel symbols for this surface.

(c) Compute the Riemann tensor R_ijkl for this surface (hint: use symmetries of the Riemann tensor to avoid having to do 16 tedious calculations).

(d) Show that the Ricci scalar is R = F′(ϖ) / (ϖ[F(ϖ)]²).

(e) Prove that a 2-dimensional surface has zero Riemann tensor if and only if its Ricci scalar vanishes.

(f) Use (e) to completely classify the surfaces of revolution with zero Riemann tensor.

In Sucuri: all of it: (a) the induced metric, (b)–(c) Christoffel and Riemann, (d) the scalar, (e) via the 2-dimensional identity of problem 12, (f) by solving F′ = 0: z = C₁ + C₂ϖ, planes and cones.

Problem 74 · Geodesics, connection and curvature on the sphere and the cone

P. van Baal, Problem Set – Theory of General Relativity (Leiden), problem 5 “Geodesics, connections, curvature”, pp. 1–2 — lorentz.leidenuniv.nl.

To get more familiar to curved spaces and geodesics we now consider the two dimensional sphere and cone embedded in ℝ³ with ds² = dx² + dy² + dz²:

sphere: x = a sin θ cos φ, y = a sin θ sin φ, z = a cos θ;
cone: x = r cos φ, y = r sin φ, z = r.

The following questions are to be answered for both the sphere and the cone. (Hint: see example FN.2.2.1 and problem FN.2.3.)

a) What is the metric tensor?

b) Calculate the Christoffel symbols Γ^a_bc.

c) Calculate the curvature tensor R^a_bcd ≡ ∂_cΓ^a_bd − ∂_dΓ^a_bc + Γ^e_bdΓ^a_ec − Γ^e_bcΓ^a_ed. Furthermore, calculate R_bd ≡ R^a_bad and R ≡ R^a_a. Give an interpretation of the results.

d) What are the geodesic equations? Determine the most general solutions. Hint for the sphere: Show that the angular momentum is conserved, and use this to determine the geodesics. Hint for the cone: If you cut open the cone, what are the geodesics? Does this agree with what you found for the curvature?

In Sucuri: all of it: for the sphere and the cone: the induced metric, Christoffel, Riemann, Ricci and R, the geodesic equations and their solutions — great circles, cot θ = M sin(φ − φ₀); and, on the cone, straight lines of the unrolled cone. The interpretation is left to the reader.

Problem 75 · Christoffel symbols in the Newtonian limit

A. Guth, MIT 8.962 General Relativity (Spring 2018), Problem Set 3, Problem 2 (“An Inertial Coordinate System in the Newtonian Limit”), part (a), p. 2 — web.mit.edu.

Consider the spacetime metric with components

g₀₀ = −(1 + az),   g_0i = 0,   g_ij = δ_ij

in a region near the origin O = (0, 0, 0, 0) where az ≪ 1, and a is a constant.

(a) Compute the nonzero Christoffel coefficients of this metric.

In Sucuri: all of it: the Christoffel symbols.

Problem 76 · de Sitter spacetime in flat coordinates

C. Hirata, Ph 236 – Homework 5 (Caltech, 2011), problem 1 “Cosmological constant”, p. 1 — tapir.caltech.edu.

Consider the highly symmetrical spacetime with line element given by

ds² = (1/(H²η²)) (−dη² + dx² + dy² + dz²),

where H > 0 is a constant, and in the domain η < 0. (This is known as de Sitter spacetime, and H is the Hubble constant.)

(a) Find the Einstein tensor for this spacetime.

(b) Prove that de Sitter spacetime is a vacuum solution of Einstein's equations in the presence of a positive cosmological constant for a particular choice of H. What is the relation between H and Λ?

(c) Consider an observer whose world line is given by fixed spatial coordinates (x, y, z). Explain why this trajectory is a geodesic, and show that an infinite amount of proper time elapses before the observer reaches η = 0.

In Sucuri: all of it: (a) and (b), via the Ricci tensor and the scalar; (c) the geodesic equations, which constant x, y, z solve, and the proper time, which diverges.

Problem 77 · Two-dimensional de Sitter spacetime

J. M. Evans, Part II General Relativity, Example Sheet 1 (Cambridge, 2026), question 9, p. 3 — damtp.cam.ac.uk.

2-dimensional de Sitter space-time has the line element

ds² = −du² + cosh²u dφ²,

where −∞ < u < ∞ and 0 ≤ φ < 2π. Compute the Christoffel symbols and hence the geodesic equations. Verify that the equations of an affinely parametrized geodesic x^α = x^α(λ) can be derived from the variational principle

δ ∫ (u̇² − cosh²u φ̇²) dλ = 0,

where u̇ = du/dλ etc. Verify from the variational principle that there are two first integrals

cosh²u φ̇ = K,   cosh²u u̇² = K² + L cosh²u,

along the geodesic, where K and L are constants.

Show that if K = 0 the geodesics are φ = const. If K ≠ 0, show that λ may be eliminated in favour of φ as a parameter along the geodesic and obtain the equation

v′² = M² − v²,

where v = tanh u, v′ = dv/dφ and M is a constant depending on L and K. Hence show that the K ≠ 0 geodesics are given by

tanh u = M sin(φ − φ₀),

where φ₀ is a constant. Show also that M² > 1 for timelike geodesics, M² = 1 for null geodesics and M² < 1 for spacelike geodesics. Regarding u and φ as cartesian coordinates, sketch the set of geodesics starting from (0, 0).

Show from your diagram that no two such timelike geodesics will meet again, but that spacelike geodesics may recross each other. Demonstrate also that there are pairs of points which cannot be joined by a geodesic. Which, if any, of these statements would be valid in Minkowski spacetime?

In Sucuri, partly: the Christoffel symbols, the geodesic equations and the variational principle (checked), the two first integrals, and the geodesics tanh u = M sin(φ − φ₀) with M² = 1 − κ/L² — M² > 1, = 1, < 1 for timelike, null, spacelike; and R = 2. The sketch, and the arguments drawn from it — which geodesics meet again, which pairs of points cannot be joined — no.

Problem 78 · Spherically symmetric Einstein–Maxwell: Reissner–Nordström and Λ

J. M. Evans, Part II General Relativity, Example Sheet 4 (Cambridge, 2026), question 6, p. 2 — damtp.cam.ac.uk.

Show that the Einstein–Maxwell equations (i.e. the Einstein equations with energy momentum tensor for an electromagnetic field T^αβ = F^αγF^β_γ − ¼ F^γδF_γδg^αβ) can be written

R_αβ = κ(F_αγF_β^γ − ¼ g_αβF_γδF^γδ).

For a line element of the form

ds² = −f(r)c²dt² + (1/f(r)) dr² + r²(dθ² + sin²θ dφ²) ,

the only non-zero components of the Ricci tensor are given by

R_tt/(c²f) = −f R_rr = ½ f″ + f′/r , R_θθ = R_φφ/sin²θ = 1 − r f′ − f .

In the case

F_tr = −F_rt = Q/r² and F_αβ = 0 otherwise,

show that a solution can be found that reduces to the Schwarzschild solution when Q = 0.

Find an analogous solution in the case R_αβ = Λg_αβ.

In Sucuri: all of it: the form R_αβ = κ(…), via the trace in 4D; the Ricci components the problem statement gives, checked; f for Einstein–Maxwell, which with Q = 0 is Schwarzschild; and f for R_αβ = Λg_αβ. (With c = 1.)

Problem 79 · Ricci curvature of Reissner–Nordström

T. Dray, MTH 437/537 – HW #6 (Oregon State, Spring 2024), problem 1, p. 1 — sites.science.oregonstate.edu.

Consider the Reissner–Nordström geometry, with line element

−f(r) dt² + dr²/f(r) + r²(dθ² + sin²θ dφ²),

where f(r) = 1 − 2m/r + q²/r². Determine the components R_ij of the Ricci curvature.

You may use any formalism you wish, and any coordinate system you wish. Tedious computation by hand is discouraged, as is trying to fully typeset your results. Other options include using computer algebra and/or using or adapting a computation published elsewhere. Such strategies must of course be documented. However, make sure you understand — and clearly describe — the conventions being used! One reasonable starting point would be the curvature 2-forms, and the relations

Ωⁱ_j = ½ Rⁱ_jkl σ^k ∧ σ^l,   R_ij = R^m_imj,

where repeated indices are summed over.

In Sucuri: all of it: the components R_ij.

Problem 80 · Metric of AdS₃

T. Hartman, Lectures on Quantum Gravity and Black Holes (Cornell, 2015), §9.1 “Exercise: Metric of AdS₃”, eqs. (9.1)–(9.4), pp. 97–98 — hartmanhep.net.

Anti-de Sitter space is a constant-negative-curvature spacetime. It is the maximally symmetric solution of Einstein's equation with a negative cosmological constant. AdS_D can be realized as a hyperboloid embedded in a D + 1-dimensional geometry. In this section we will talk about AdS₃, which is the hyperboloid

X_AX^A = −ℓ²   (9.1)

where A = 0, 1, 2, 3 is an index in the space Minkowski₂ × Minkowski₂, with metric

H_AB dX^AdX^B = −dX₀² + dX₁² + dX₂² − dX₃².   (9.2)

To find intrinsic coordinates on AdS₃, we just need to solve (9.1). One way to solve this equation is by

X₀ = ℓ cosh ρ cos t,  X₁ = ℓ sinh ρ sin φ,  X₂ = ℓ sinh ρ cos φ,  X₃ = ℓ cosh ρ sin t.   (9.3)

(a) Check that this solves (9.1), and use (9.2) to find the induced metric on the hyperboloid.

Answer: ds² = ℓ²(−cosh²ρ dt² + dρ² + sinh²ρ dφ²).   (9.4)

These are global coordinates on AdS₃. Although on the hyperboloid (9.1) we can see from (9.3) that t is a periodic coordinate, when we say “AdS₃” we will always mean the space in which t is “unwrapped”, t ∈ (−∞, ∞) (the universal covering space of the hyperboloid).

(b) Find the cosmological constant in terms of the AdS radius ℓ.

In Sucuri: all of it: (a) the hyperboloid and the induced metric (9.4); (b) Λ = −1/ℓ², via the Ricci tensor.

Problem 81 · Geometry of hyperbolic space

C. Hirata, Ph 236 – Homework 16 (Caltech, 2012), problem 1 “Geometry of hyperbolic space”, p. 1 — tapir.caltech.edu.

Consider the hyperbolic space H³ with unit negative curvature,

ds² = dχ² + sinh²χ (dθ² + sin²θ dφ²).

(a) Show that this is the geometry that one obtains by taking 4-dimensional Minkowski space M⁴ and restricting to the spacelike 3-surface Σ ⊂ M⁴ given by −t² + x² + y² + z² = −1. What is the mapping Φ : H³ → Σ of the coordinates (χ, θ, φ) to (t, x, y, z)?

(b) Using your knowledge of infinitesimal Lorentz transformations, find the 6 Killing fields corresponding to Lorentz rotations and boosts, and express them as contravariant vectors in the χθφ coordinate system.

(c) Show that given two points P, Q ∈ H³ that the separation s between them (i.e. length of the shortest¹ geodesic that connects them) is given by cosh s = −Φ(P)·Φ(Q), where the dot product is that defined in M⁴.

(d) Use the result of (c) to prove that in the hyperbolic space, the law of cosines is given by cosh c = cosh a cosh b − sinh a sinh b cos γ, where we denote the side lengths of a “triangle” (i.e. set of 3 geodesics connecting the 3 vertices) as a, b, and c, and the angles opposite them as α, β, and γ.

(e) Taylor expand the result from (d) and show that if a, b, and c are ≪ 1 then the usual law of cosines is recovered.

¹ In hyperbolic space, the statement about the shortest geodesic is redundant: there is exactly one geodesic connecting any two points. This is actually true for any simply connected space of Euclidean signature (+ + +) and negative-semidefinite curvature, i.e. where R_αβγδξ^αη^βξ^γη^δ ≤ 0 for any vectors ξ and η.

In Sucuri, partly: (a) the induced metric and the map Φ, and (b) the six Killing fields — rotations and boosts of M⁴ restricted to the hyperboloid, and checked; (c) the distance, which needs invariance under the isometries, and with it (d) and (e), no.

Problem 82 · Ricci curvature of the N-sphere

G. 't Hooft, Introduction to General Relativity (Utrecht, 2013), ch. 14 “The Robertson-Walker metric”, exercise after eq. (14.8), with eq. (14.9), p. 60 — webspace.science.uu.nl.

[Context: for the three-dimensional isotropic space dω² = B(ϱ)dϱ² + ϱ²(dθ² + sin²θ dϕ²) (14.2), one requires R_ij = λg_ij (14.5), which leads to B = 1/(1 − ½λϱ²) (14.8).]

Exercise: show that with ϱ = √(2/λ) sin ψ, this gives the metric of the 3-sphere, in terms of its three angular coordinates ψ, θ, ϕ. Indeed, the metric for an N-sphere can be written as

dω_N² = dψ_N² + sin²ψ_N dω_N−1²,   R_ij^(N) = λ_Ng_ij;  λ_N = N − 1.   (14.9)

In Sucuri: all of it: the metric of the 3-sphere via ϱ = √(2/λ) sin ψ, and R_ij = (N − 1)g_ij.

Problem 85 · The shape of the Schwarzschild geometry (Flamm's paraboloid)

T. Dray, MTH 437/537 – HW #5 (Oregon State, Spring 2024), problem 1 “Shape of Schwarzschild geometry”, p. 1 — sites.science.oregonstate.edu.

(a) Plot the graph of 8mr = h² + 16m² with r as the horizontal coordinate. Equivalently, solve this equation for h, and plot h as a function of r.

(b) Express arclength along this graph in terms of r (and dr). Assume that (r, h) are rectangular, Euclidean coordinates.

(c) Consider the surface of revolution obtained by rotating your curve about the h-axis. What is the line element for this surface? Compare your answer with the Schwarzschild geometry.

(d) Find the Gaussian curvature of this surface. Is it positive or negative?

In Sucuri: all of it: (a) h(r), (b) the arclength, (c) the surface of revolution, which is the equatorial slice of Schwarzschild, and (d) K = −m/r³ < 0. The plot is left to the reader.

Problem 83 · Ricci scalar of a special metric

Y. Ali-Haïmoud, General Relativity (NYU, Fall 2019), Homework 6, Exercise 3, p. 1 — cosmo.nyu.edu.

At linear order in Φ, compute the 10 components of the Ricci tensor, as well as the Ricci scalar of the metric

ds² = −[1 + 2Φ(x⃗)] dt² + [1 − 2Φ(x⃗)] δ_ij dxⁱdx^j.   (3)

Note that Φ(x⃗) is assumed to depend only on the spatial coordinates (i.e. not on t).

In Sucuri: all of it: the ten components and the scalar, to first order.

Problem 84 · Spherical coordinates, III

T. Dray, MTH 434/534 – HW #6 (Oregon State, Winter 2024), problem 1, p. 1 — sites.science.oregonstate.edu.

Consider the sphere of radius r, in spherical coordinates (θ, φ), with line element ds² = r²(dθ² + sin²θ dφ²). [The original prints r²dθ² + sin²θ dφ², without the parentheses.]

(a) Find the connection 1-forms ωⁱ_j in this basis.

(b) Compute Ωⁱ_j = dωⁱ_j + ωⁱ_k ∧ ω^k_j for i, j = 1, 2 (and where there is an implicit sum over k).

(c) (Optional for MTH 434/Required for MTH 534.) Compare your answers (and your computations) with those from the previous homework assignment.

In Sucuri: all of it: (a) and (b); the comparison in (c) is left to the reader.

Problem 18 · The Lie bracket makes 𝔛(M) a Lie algebra

S. Richard, Differential geometry (Nagoya University, 2024), Exercise 2.2.3, §2.2, p. 19 — math.nagoya-u.ac.jp.

Context: The set 𝔛(M) has also the structure of a Lie algebra, meaning that there exists a Lie bracket [ , ] : 𝔛(M) × 𝔛(M) → 𝔛(M) satisfying:

(i) It is linear in each argument;
(ii) It is antisymmetric: [Y, X] = −[X, Y];
(iii) The Jacobi identity holds: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0.

[…] by defining [X, Y]_p(f) := X_p(Yf) − Y_p(Xf), this verifies the linearity property and the Leibniz rule. As a consequence, [X, Y]_p is an element of T_p(M) for each p ∈ M.

Exercise 2.2.3. Check that 𝔛(M) endowed with the product [X, Y] is a Lie algebra, see [2, p. 152–153].

In Sucuri: all of it: the properties of the bracket, index-free.

Problem 19 · Bracket of fields multiplied by functions

S. Richard, Differential geometry (Nagoya University, 2024), Exercise 2.2.4, §2.2, p. 19 — math.nagoya-u.ac.jp.

Exercise 2.2.4. For any X, Y ∈ 𝔛(M) and f, g ∈ C^∞(M), check the following equality:

[fX, gY] = fg[X, Y] + f(Xg)Y − g(Yf)X.

In Sucuri: all of it: the formula.

Problem 20 · Properties of the Lie derivative of vector fields

S. Richard, Differential geometry (Nagoya University, 2024), Exercise 2.3.12, §2.3, p. 23 — math.nagoya-u.ac.jp.

Context (Lemma 2.3.11): For any X, Y ∈ 𝔛(M), one has L_X(Y) = [X, Y].

Exercise 2.3.12. Prove (and understand) the following equality, for any X, Y, Z ∈ 𝔛(M) and f ∈ C^∞(M):

(i) L_X[Y, Z] = [L_XY, Z] + [Y, L_XZ],
(ii) L_X ∘ L_Y − L_Y ∘ L_X = L_[X,Y],
(iii) L_X(fY) = (L_Xf)Y + f(L_XY).

In Sucuri: all of it: (i) and (ii), with Jacobi as a hypothesis — ℒ_X is the bracket, and both are Jacobi rewritten —, and (iii), the Leibniz rule of the bracket, with no hypothesis.

Problem 21 · Torsion and curvature are C^∞(M)-linear

S. Richard, Differential geometry (Nagoya University, 2024), Exercise 5.4.3 (after Definition 5.4.2), §5.4, p. 49 — math.nagoya-u.ac.jp.

Context (Definition 5.4.2): For a Riemannian manifold M and an affine connection ∇ one sets for any X, Y ∈ 𝔛(M)

T(X, Y) := ∇_XY − ∇_YX − [X, Y] ∈ 𝔛(M)  and  R(X, Y) := ∇_X∇_Y − ∇_Y∇_X − ∇_[X,Y] ∈ End(𝔛(M)),

the former being called the torsion while the latter is called the curvature of the connection. It is then interesting to observe that the map (X, Y) ↦ T(X, Y) is C^∞(M)-linear in both arguments, and that the map (X, Y, Z) ↦ R(X, Y)Z is C^∞(M)-linear in the three arguments.

Exercise 5.4.3. Show the C^∞(M)-linearity mentioned above, see also [8, Prop. 6.3].

In Sucuri: all of it: the C^∞(M)-linearity of T and of R.

Problem 22 · The difference of two connections is a tensor

H. S. Reall, Part III General Relativity (2022), §3.1 “Introduction”, exercise right after eq. (3.10), p. 33 (p. 34 of the PDF) — damtp.cam.ac.uk.

Let ∇ and ∇̃ be two different connections on M. Show that ∇ − ∇̃ is a (1, 2) tensor field. You can do this either from the definition of a connection, or from the transformation law for the connection components.

In Sucuri: all of it: from the definition of a connection: (∇ − ∇̃)_XY is linear over functions in both slots — which makes it a (1, 2) tensor —, and, if both are torsion-free, symmetric.

Problem 23 · Koszul formula

S. Richard, Differential geometry (Nagoya University, 2024), Exercise 5.4.5, §5.4, p. 49 — math.nagoya-u.ac.jp.

Context: a connection is called torsion free if T(X, Y) = 0 for all X, Y, and is called Riemannian (or Levi-Civita) if, in addition,

X⟨Y₁, Y₂⟩ = ⟨∇_XY₁, Y₂⟩ + ⟨Y₁, ∇_XY₂⟩.   (5.4.1)

Exercise 5.4.5 (Koszul formula). For X, Y, Z ∈ 𝔛(M) set

⟨∇_XY, Z⟩ := ½ { X⟨Y, Z⟩ + Y⟨Z, X⟩ − Z⟨X, Y⟩ − ⟨X, [Y, Z]⟩ + ⟨Y, [Z, X]⟩ + ⟨Z, [X, Y]⟩ }.

Check that the connection defined by this relation is torsion free and satisfies (5.4.1). This equality is called Koszul formula and shows the existence of a Riemannian connection.

In Sucuri: all of it: both directions: the one asked for — with ∇ defined by the formula, it is torsion-free and metric-compatible — and the converse, uniqueness. From g(W, Z) = 0 for all Z follows W = 0: that is the metric being non-degenerate.

Problem 24 · First Bianchi identity

S. Richard, Differential geometry (Nagoya University, 2024), Exercise 6.2.2 (on Lemma 6.2.1), §6.2, p. 60 — math.nagoya-u.ac.jp.

Context (Lemma 6.2.1): Assume that the connection on smooth manifold is torsion free, then the following identity holds for the curvature: for any X, Y, Z ∈ 𝔛(M),

R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0,

where 0 means the 0-vector field.

Exercise 6.2.2. Prove Bianchi identity getting inspiration from [4, Prop. 4.1.3].

In Sucuri: all of it: the identity, from the definitions and zero torsion.

Problem 25 · Second Bianchi identity and Einstein manifolds

A. G. Kovalev, Part III: Riemannian Geometry — Example Sheets (Cambridge, Lent 2017), Example Sheet 1, Question 2, p. 1 — dec41.user.srcf.net.

2. (i) Let M be a Riemannian manifold. Show that the Levi–Civita covariant derivative of R(X, Y) ∈ Γ(End TM) is given by

∇_ZR(X, Y) = [∇_Z, R(X, Y)] − R(∇_ZX, Y) − R(X, ∇_ZY).

Deduce from this a version of the second Bianchi identity for the Levi–Civita connection

∇_XR(Y, Z) + ∇_YR(Z, X) + ∇_ZR(X, Y) = 0.   (∗)

(ii) When dim M ≥ 3, show, using (∗), that if Ric = f g for some smooth function f, then f is constant (M then is said to be an Einstein manifold).
(You might like to consider a map δ : Γ(Sym² T*M) → Γ(T*M) = Ω¹(M) defined by (δh)(X) = −∑_i=1ⁿ (∇_eih)(e_i, X), where {e_i} is any local orthonormal frame field on M, and put h = Ric.)

In Sucuri: all of it: (i) the second Bianchi identity, from the Ricci identity — the formula for ∇_ZR(X, Y) is the Leibniz rule for ∇ of a tensor, and the deduction comes out in indices; (ii) Ric = fg ⇒ ∇f = 0, with the condition d ≠ 2 stated by the proof itself.

Problem 26 · Lie group with a bi-invariant metric

A. G. Kovalev, Part III: Riemannian Geometry — Example Sheets (Cambridge, Lent 2017), Example Sheet 1, Question 5 (pp. 1–2 of the PDF) — dec41.user.srcf.net.

5. Let G be a Lie group endowed with a Riemannian metric g which is left and right invariant and let X, Y, Z be left invariant vector fields of G.

(i) Show that g([X, Y], Z) + g(Y, [X, Z]) = 0. (Consider the flow of X.)
(ii) Show that ∇_XX = 0. (Hint: consider g(Y, ∇_XX).)
(iii) Show that ∇_XY = ½[X, Y].
(iv) Prove that R(X, Y)Z = ¼[[X, Y], Z].
(v) Suppose that X and Y are orthonormal, and let K(σ) be the sectional curvature of the 2-plane σ spanned by X and Y. Prove that

K(σ) = ¼ |[X, Y]|_g².

In Sucuri, partly: (ii)–(v), with part (i) — g([X,Y], Z) + g(Y, [X,Z]) = 0 — as a hypothesis, and g(Y, Z) constant for left-invariant fields; (iv) and (v) in Kovalev's sign convention, stated as a definition. (i) itself, via the flow of X, no.

Problem 27 · dω(X, Y) for a 1-form

J. Ross, Part III: Differential Geometry, Example Sheet 2 (Cambridge, 2016, version 3 of 16/11/2016), question 9, p. 6 of the PDF — dec41.user.srcf.net.

9. (Differential Form Identity) Prove the identity

dω(X, Y) = Xω(Y) − Yω(X) − ω([X, Y]),

for a 1-form ω and vector fields X, Y. *Can you generalize this result to the case when ω is a p-form?

In Sucuri, partly: the formula for 1-forms, with dω_μν = ∂_μω_ν − ∂_νω_μ (the normalization is the formula's); the generalization to p-forms, the starred part, no.

Problem 53 · Commutator of vector fields via the covariant derivative

Y. Ali-Haïmoud, General Relativity (NYU, Fall 2019), Homework 6, Exercise 1, part (iii), p. 1 — cosmo.nyu.edu.

Given a vector field X, for any smooth function f : 𝓜 → ℝ, we can define the smooth function X(f) which, at each point p ∈ 𝓜, associates X|_p(f). Given two vector fields X and Y, we define their commutator [X, Y] such that

[X, Y](f) ≡ X(Y(f)) − Y(X(f)).   (1)

(iii) Recall the notation convention ∇_XY^α ≡ X^β∇_βY^α. Prove that [X, Y]^α = ∇_XY^α − ∇_YX^α.

In Sucuri: all of it: the identity.

Problem 58 · Lie derivative of vectors, covectors and (0,2) tensors; Killing condition

J. M. Evans, Part II General Relativity, Example Sheet 3 (Cambridge, Lent 2026), question 9, p. 2 — damtp.cam.ac.uk.

9. The Lie derivative (ℒ_ξV)^α of a vector field V^α with respect to a vector field ξ^α (assumed to be timelike) is defined by the following conditions: (i) If {x^α} is a coordinate system in which ξ^α = (1, 0, 0, 0), then

(ℒ_ξV)^α = ∂V^α/∂x⁰ = ξ^β ∂V^α/∂x^β,

and (ii) (ℒ_ξV)^α transforms as a vector. Show that, in a general coordinate system,

(ℒ_ξV)^α = ξ^β∇_βV^α − V^β∇_βξ^α.

Suppose, in addition, that the Lie derivative ℒ_ξφ of a scalar field φ with respect to a vector field ξ^α is defined in a general coordinate system {x^α} by

ℒ_ξφ = ξ^α ∂φ/∂x^α

and that the Lie derivative obeys the usual Leibniz rule when applied to a tensor product. Find the Lie derivative (ℒ_ξU)_α of a covector field U_α.

Write down an expression for the Lie derivative with respect to ξ^α of a (0, 2) tensor T_αβ and show that the condition for ξ^α to be a Killing vector field (as in question 6 above, i.e. ξ_α;β + ξ_β;α = 0) is (ℒ_ξg)_αβ = 0, where g_αβ is the metric tensor.

In Sucuri: all of it: ℒ_ξV from adapted coordinates, the covariant form (torsion-free), the covector and the (0, 2) — both as in problem 17 — and the Killing equation as ℒ_ξg = 0.

Problem 28 · Exterior derivative via a torsion-free connection

A. G. Kovalev, Part III: Riemannian Geometry, Example Sheet 3 (Cambridge, Lent 2017), question 3 — dec41.user.srcf.net.

Γ(T*M ⊗ Λ^p T*M) --alt--> Γ(Λ^{p+1} T*M), with d : Γ(Λ^p T*M) → Γ(Λ^{p+1} T*M), where alt(ξ ⊗ α) = ξ ∧ α denotes projection to the subspace of anti-symmetric tensors (p > 0). Deduce the formula for the exterior derivative of one-forms dα(X, Y) = (∇_X α)(Y) − (∇_Y α)(X) as stated in the Lectures. ∗ Show that these results hold for any torsion-free connection ∇ on M." -->

3. Show that for the Levi-Civita connection, the following diagram commutes:

Γ(Λ^pT*M) —∇→ Γ(T*M ⊗ Λ^pT*M) —alt→ Γ(Λ^p+1T*M),  with  d : Γ(Λ^pT*M) → Γ(Λ^p+1T*M),

where alt(ξ ⊗ α) = ξ ∧ α denotes projection to the subspace of anti-symmetric tensors (p > 0). Deduce the formula for the exterior derivative of one-forms

dα(X, Y) = (∇_Xα)(Y) − (∇_Yα)(X),

as stated in the Lectures. (*) Show that these results hold for any torsion-free connection ∇ on M.

In Sucuri, partly: the formula for dα for any torsion-free connection, with the formula for dα on vectors as a hypothesis; the diagram for p-forms, no.

Problem 29 · Contraction (interior product) as an antiderivation

J. Ross, Part III: Differential Geometry, Example Sheet 2 (Cambridge, 2016, version 2 of 14/10/2016), question 8 — dec41.user.srcf.net.

8. (Contractions) Let V be a vector space, X ∈ V and ω ∈ Λ^kV*. Define i_X(ω) by

i_X(ω)(Y₁, …, Y_k−1) = ω(X, Y₁, …, Y_k−1).

Prove that i_X(ω) ∈ Λ^k−1V*. Prove also that if ω ∈ Λ^kV* and η ∈ Λ^lV* then

i_X(ω ∧ η) = i_Xω ∧ η + (−1)^k ω ∧ (i_Xη).

Now on a smooth manifold M suppose X ∈ Vect(M) and ω ∈ Ω^p(M). Show how the formula

i_Xω|_p := i_Xp(ω_p)

gives an element of Ω^p−1(M).

In Sucuri: all of it: the antiderivation rule for ι_X; the rest of the problem statement is definition.

Problem 30 · Lie derivative of a 1-form and Cartan's formula

D. Tong, General Relativity: Example Sheet 1 (2019), question 5*, p. 2 — davidtong.org.

Use the Leibniz rule to derive the formula for the Lie derivative of a 1-form ω, valid in any coordinate basis:

(L_Xω)_μ = X^ν∂_νω_μ + ω_ν∂_μX^ν

[Hint: consider (L_Xω)(Y) for a vector field Y.] Show that the Lie derivative of a (0, 2) tensor g is

(L_Xg)_μν = X^ρ∂_ρg_μν + g_μρ∂_νX^ρ + g_ρν∂_μX^ρ

For a p-form η, define ι_Xη to be the (p − 1)-form that results from contracting a vector field X with the first index of η. Show that for a 1-form ω,

L_Xω = ι_X(dω) + d(ι_Xω).

In Sucuri: all of it: both formulas in coordinates, via the Leibniz rule, and Cartan's formula on a 1-form, from the Leibniz definition of ℒ_X and that of dω.

Problem 31 · Properties of the exterior derivative

H. S. Reall, Part 3 General Relativity (2022), §8.1 “Introduction” (ch. 8, differential forms), “Exercises (examples sheet 4)”, eqs. (8.7)–(8.9), p. 100 — damtp.cam.ac.uk.

Context: the exterior derivative of a p-form X is the (p + 1)-form dX defined in a coordinate basis by (dX)_μ1…μp+1 = (p + 1) ∂_[μ1X_μ2…μp+1] (8.5).

Exercises (examples sheet 4). Show that the exterior derivative enjoys the following properties:

d(dX) = 0 (8.7)

d(X ∧ Y) = (dX) ∧ Y + (−1)^p X ∧ dY (8.8)

(where Y is a q-form) and

d(φ^*X) = φ^* dX (8.9)

(where φ : N → M), i.e. the exterior derivative commutes with pull-back.

In Sucuri, partly: d(dX) = 0 and the graded Leibniz rule; the pull-back, no.

Problem 32 · Graded commutativity and associativity of the wedge product

H. S. Reall, Part III General Relativity (2022), §8.1 “Introduction”, exercise with eqs. (8.2)–(8.3), p. 99 — damtp.cam.ac.uk.

Definition. The wedge product of a p-form X and a q-form Y is the (p + q)-form X ∧ Y defined by

(X ∧ Y)_{a₁…apb₁…bq} = [(p + q)!/(p! q!)] X_[a₁…apY_b₁…bq].   (8.1)

Exercise. Show that

X ∧ Y = (−1)^pq Y ∧ X   (8.2)

(so X ∧ X = 0 if p is odd); and

(X ∧ Y) ∧ Z = X ∧ (Y ∧ Z),   (8.3)

i.e. the wedge product is associative so we don't need to include the brackets.

In Sucuri: all of it: (8.2) and (8.3).

Problem 33 · Hodge dual: ⋆⋆ and ⋆d⋆

H. S. Reall, Part 3 General Relativity (2022), §8.5 “Volume form”, Lemma with eqs. (8.56)–(8.57) (“Proof. Exercise (use (8.54))”), pp. 107–108 — damtp.cam.ac.uk.

Context: ε is the volume form of an oriented n-dimensional manifold with metric, and

εa1…apcp+1…cn εb1…bpcp+1…cn = ±p!(n − p)! δa1[b1 … δapbp] (8.54),

where the upper (lower) sign holds for Riemannian (Lorentzian) signature.

Definition. On an oriented manifold with metric, the Hodge dual of a p-form X is the (n − p)-form ⋆X defined by

(⋆X)_a1…an−p = (1/p!) ε_{a1…an−pb1…bp} X^b₁…b_p (8.55)

Lemma. For a p-form X,

⋆(⋆X) = ±(−1)^p(n−p) X (8.56)

(⋆d⋆X)_a1…ap−1 = ±(−1)^p(n−p) ∇^bX_a1…ap−1b (8.57)

where the upper (lower) sign holds for Riemannian (Lorentzian) signature.

Proof. Exercise (use (8.54)).

In Sucuri, partly: ⋆⋆ = ±(−1)^p(n−p), with the declared signature; ⋆d⋆ in terms of ∇, no.

Problem 47 · d commutes with the Lie derivative; Leibniz rule for L_X

C. Wendl, Differential Geometry I, Problem Set 7 (Humboldt-Universität zu Berlin, 2016–17), problem 1(a) — mathematik.hu-berlin.de.

1. The goal of this problem is to prove Cartan's formula for the Lie derivative of a differential form,

L_Xω = dι_Xω + ι_Xdω.   (1)

Recall that for a k-form ω ∈ Ω^k(M) on a smooth n-manifold M, L_Xω ∈ Ω^k(M) is defined as ∂_t φ_t*ω|_t=0 where φ_t : M → M denotes the time t flow of the vector field X ∈ Vec(M).

(a) Use the definition of the Lie derivative to prove the relation

d(L_Xω) = L_X(dω)

and the Leibniz rule

L_X(α ∧ β) = L_Xα ∧ β + α ∧ L_Xβ.   (2)

Observe that this formula determines the action of L_X on arbitrary differential forms if we know how it acts on 0-forms (i.e. smooth functions) and exact 1-forms (i.e. differentials of smooth functions).

In Sucuri, partly: d ℒ_X = ℒ_X d and the Leibniz rule for ℒ_X (the example under 48). But the exercise asks for both from the flow definition of ℒ_X, in order to then prove Cartan's formula; in Sucuri, ℒ_X on forms is defined by Cartan's formula, and that route cannot be walked.

Problem 48 · dι_X + ι_Xd satisfies the Leibniz rule (Cartan's formula)

C. Wendl, Differential Geometry I, Problem Set 7 (Humboldt-Universität zu Berlin, 2016–17), problem 1(d) — mathematik.hu-berlin.de.

Context: problem 1 aims to prove Cartan's formula L_Xω = dι_Xω + ι_Xdω (1). Part (a) proves the Leibniz rule L_X(α ∧ β) = L_Xα ∧ β + α ∧ L_Xβ (2). Part (c) proves that ι_v satisfies the graded Leibniz rule ι_v(α ∧ β) = ι_vα ∧ β + (−1)^|α| α ∧ ι_vβ.

(d) For any fixed X ∈ Vec(M), use the graded Leibniz rules satisfied by d and ι_X to show that the operator (d ∘ ι_X + ι_X ∘ d) : Ω^k(M) → Ω^k(M) also satisfies the Leibniz rule (2). Deduce that this operator matches L_X.

Author's note (footnote 2): in contrast to the exterior derivative and the interior product, the Leibniz rule satisfied by L_X does not include any annoying signs. This is consistent with thinking of L_X : Ω^k(M) → Ω^k(M) as an object of degree zero (hence even), of d as an object of degree one (hence odd), and of ι_X : Ω^k(M) → Ω^k−1(M) as an object of degree −1 (hence odd).

In Sucuri: all of it: that dι_X + ι_Xd satisfies the Leibniz rule.

Problem 51 · d² = 0 on ℝ³: curl grad = 0 and div curl = 0

C. Wendl, Differential Geometry I, Problem Set 6 (Humboldt-Universität zu Berlin, 2016–17), problem 6(c) — mathematik.hu-berlin.de.

Context (preamble of problem 6 and part (b)): given a volume form μ ∈ Ωⁿ(M) on an n-manifold M, the divergence of X ∈ Vec(M) is the unique function div(X) : M → ℝ such that L_Xμ = div(X) μ. On M = ℝ³, with μ = dx ∧ dy ∧ dz and X = X^x∂_x + X^y∂_y + X^z∂_z, one has div(X) = ∂_xX^x + ∂_yX^y + ∂_zX^z, an expression sometimes also denoted by ∇ · X.

(c) Recall that on ℝ³, the gradient of a function f : ℝ³ → ℝ is the vector field

grad(f) = ∇f := (∂_xf)∂_x + (∂_yf)∂_y + (∂_zf)∂_z,

and the curl of a vector field X = X^x∂_x + X^y∂_y + X^z∂_z is the vector field

curl(X) = ∇ × X := (∂_yX^z − ∂_zX^y)∂_x + (∂_zX^x − ∂_xX^z)∂_y + (∂_xX^y − ∂_yX^x)∂_z.

Using the relations of these operations to differential forms and the exterior derivative, deduce from d² = 0 the formulas

∇ × (∇f) = 0   and   ∇ · (∇ × X) = 0

for all f ∈ C^∞(ℝ³) and X ∈ Vec(ℝ³).

In Sucuri: all of it: curl grad = 0 and div curl = 0, with forms and with indices.

Problem 46 · ℒ_[X,Y] = [ℒ_X, ℒ_Y] on forms

C. Wendl, Differential Geometry I, Problem Set 7 (Humboldt-Universität zu Berlin, 2016–17), problem 4(a), p. 3 — mathematik.hu-berlin.de.

4. (a) Use Leibniz rules as in Problem 1 to show that for all X, Y ∈ Vec(M) and ω ∈ Ω^k(M),

ℒ_[X,Y]ω = ℒ_Xℒ_Yω − ℒ_Yℒ_Xω.

In Sucuri, partly: on functions — it is the definition of the bracket — and on 1-forms, by Leibniz and Jacobi; on higher-degree forms, the argument that two derivations agreeing on functions and on df agree everywhere, no.

Problem 49 · dω(X, Y, Z) for a 2-form

C. Wendl, Differential Geometry I, Problem Set 6 (Humboldt-Universität zu Berlin, 2016–17), problem 4 (preamble) and part (c), pp. 2–3 — mathematik.hu-berlin.de.

4. Recall that if (x¹, …, xⁿ) : U → ℝⁿ is a chart defined on an open subset in some n-manifold M, any k-form ω ∈ Ω^k(M) can be written on U as

ω = ω_i1…ik dx^i₁ ⊗ … ⊗ dx^i_k = (1/k!) ω_i1…ik dx^i₁ ∧ … ∧ dx^i_k = ∑_i1<…<ik ω_i1…ik dx^i₁ ∧ … ∧ dx^i_k,

where the first two expressions use the Einstein summation convention and the third one does not. Here the component functions ω_i1…ik : U → ℝ can be written in terms of the coordinate vector fields ∂₁, …, ∂_n as ω_i1…ik = ω(∂_i1, …, ∂_ik). In order to write down a coordinate formula for the exterior derivative, we introduce the following notation: given any collection of functions T_i1…ik on U labeled by the indices i₁, …, i_k, define

T_[i1…ik] := (1/k!) ∑_σ∈Sk (−1)^|σ| T_{iσ(1)…iσ(k)},

so for instance if T_i1…ik are the components of a tensor field T, then Alt(T)_i1…ik = T_[i1…ik], and the wedge product of α ∈ Ω^k(M) and β ∈ Ω^ℓ(M) can now be written in coordinates as

(α ∧ β)_{i1…ikj1…jℓ} = α_[i1…ikβ_j1…jℓ].

[…]

(c) It now follows from Problem Set 4 #1(a) that the exterior derivative of a 1-form λ can also be written as

dλ(X, Y) = ℒ_X(λ(Y)) − ℒ_Y(λ(X)) − λ([X, Y]).

Indeed, the right hand side is C^∞-linear with respect to vector fields X, Y ∈ Vec(M) and thus defines a tensor field, whose component functions we've seen match the formula from part (b). Prove the corresponding formula for the exterior derivative of a 2-form,

dω(X, Y, Z) = ℒ_X(ω(Y, Z)) + ℒ_Y(ω(Z, X)) + ℒ_Z(ω(X, Y)) − ω([X, Y], Z) − ω([Y, Z], X) − ω([Z, X], Y).

Remark: Similar formulas exist for the exterior derivatives of k-forms for all k > 2, though I cannot recall ever having needed to use them.

In Sucuri: all of it: the formula, in components, with (dω)_λμν = ∂_λω_μν + ∂_μω_νλ + ∂_νω_λμ, its normalization.

Problem 34 · Maurer–Cartan equations

J. Ross, Part III: Differential Geometry, Example Sheet 2 (Cambridge, 2016, version 3 of 16/11/2016), question 11, p. 7 of the PDF — dec41.user.srcf.net.

11. Let G be Lie group that is a subgroup of GL_n(ℝ) and X_i, i = 1, …, d = dim G, be linearly independent left-invariant vector fields on G induced by a basis of T_IG. Show that the condition that ωⁱ(X_j) = δⁱ_j identically on G defines a system of pointwise linearly independent smooth 1-forms ωⁱ on G. Show further that the 1-forms ωⁱ are left-invariant in the sense that

L_g^*(ωⁱ) = ωⁱ,   for every g ∈ G.

Let C^k_ij be a set of real constants determined by [X_i, X_j] = ∑_k C^k_ij X_k. Deduce from the identity of the previous question the formula

dω^k = −½ ∑_i,j C^k_ij ωⁱ ∧ ω^j.

In Sucuri, partly: the formula dω^k = −½ C^k_ij ωⁱ∧ω^j, deduced from the identity of the previous question — first without indices, dω(X, Y) = −ω([X, Y]) for constant ω(X), ω(Y); then with indices, summing over the basis. That the ωⁱ exist and are smooth, independent and left-invariant is argument, and does not count.

Problem 50 · The divergence via ℒ_Xμ = div(X) μ

C. Wendl, Differential Geometry I, Problem Set 6 (Humboldt-Universität zu Berlin, 2016–17), problem 6 (preamble) and part (b), p. 3 — mathematik.hu-berlin.de.

6. Given a volume form μ ∈ Ωⁿ(M) on an n-manifold M, one can define volumes of compact regions U ⊂ M by

Vol(U) := ∫_U μ.

The divergence of a vector field X ∈ Vec(M) can then be defined in terms of the Lie derivative of μ with respect to X: let div(X) : M → ℝ be the unique real-valued function such that

ℒ_Xμ = div(X) μ.

Note that this is well defined since ℒ_Xμ is an n-form and the space ΛⁿT_p^*M of n-forms at each point p ∈ M is 1-dimensional. Observe also that dμ = 0 since Ωⁿ⁺¹(M) = {0}, so Cartan's formula implies div(X) μ = dι_Xμ, which matches the formula we saw in lecture for the case M = ℝⁿ.

[…]

(b) Show that in the case M = ℝ³ with μ = dx ∧ dy ∧ dz and X = X^x∂_x + X^y∂_y + X^z∂_z ∈ Vec(ℝ³) in standard Cartesian coordinates (x, y, z),

div(X) = ∂_xX^x + ∂_yX^y + ∂_zX^z.

The latter expression is sometimes also denoted by ∇ · X.
Note: One can show more generally that if μ = dx¹ ∧ … ∧ dxⁿ on ℝⁿ then div(X) = ∂_iXⁱ.

In Sucuri: all of it: div(X) = ∂_xX^x + ∂_yX^y + ∂_zX^z, from ℒ_Xμ.

Problem 52 · Contact forms on 3-manifolds

C. Wendl, Differential Geometry I, Problem Set 12 (Humboldt-Universität zu Berlin, 2016–17), problem 4, parts (a)–(d), pp. 3–4 — mathematik.hu-berlin.de.

4. Suppose M is an oriented 3-manifold and λ ∈ Ω¹(M) is nowhere zero, i.e. for all p ∈ M there exist vectors X ∈ T_pM with λ(X) ≠ 0. Then at every p ∈ M, the kernel ker λ_p = {X ∈ T_pM | λ(X) = 0} is a 2-dimensional subspace of T_pM, and the union of these for all p defines a smooth 2-dimensional distribution

ξ := ker λ ⊂ TM.

(a) Show that the following conditions are equivalent:
i. λ ∧ dλ ≡ 0;
ii. For all p ∈ M and X, Y ∈ T_pM, dλ(X, Y) = 0 [sic; from context, read X, Y ∈ ξ_p];
iii. ξ is integrable.
Hint: In Problem Set 6 #4(c), you will find a useful formula for dλ as a C^∞-bilinear form on vector fields. Combine this with the Frobenius theorem.

The 1-form λ is called a contact form if λ ∧ dλ is a volume form; the distribution ξ (called the contact structure) is then “as non-integrable as possible.” An example on ℝ³ is shown in the figure in the original. Such examples can be constructed by the following trick. Let (ρ, φ, z) denote the standard cylindrical coordinates on ℝ³, so x = ρ cos φ and y = ρ sin φ. Choose smooth real-valued functions f(ρ), g(ρ) and define λ at (ρ, φ, z) by

λ = f(ρ) dz + g(ρ) dφ.   (2)

(b) Since the coordinates (ρ, φ, z) are not well defined at ρ = 0, there is of course some danger that the 1-form defined in Equation (2) might be singular at the z-axis. Show that λ is in fact smooth on all of ℝ³ and satisfies λ ∧ dλ ≠ 0 near the z-axis if we assume f(ρ) = 1 and g(ρ) = ρ² for ρ sufficiently close to 0. Hint: Convert to Cartesian coordinates.

(c) Assuming f and g take the form described above for ρ near 0, show that λ is a contact form if and only if

f(ρ)g′(ρ) − f′(ρ)g(ρ) ≠ 0

for all ρ > 0. What does this mean geometrically about the curve ρ ↦ (f(ρ), g(ρ)) ∈ ℝ²? Interpret this in terms of “twisting” of the planes ξ_p as p ∈ ℝ³ moves along radial paths away from the z-axis.

(d) By a fundamental result in contact geometry known as Gray's theorem, contact structures have the following remarkable “stability” property: if M is a closed 3-manifold and {ξ_t}_t∈[0,1] is any smooth family of contact structures on M, then they are all “equivalent” in the sense that there exists a smooth family of diffeomorphisms {ϕ_t : M → M}_t∈[0,1] such that ϕ₀ = Id and Tϕ_t(ξ₀) = ξ_t for all t ∈ [0, 1]. Show that this is not true in general for arbitrary smooth families of distributions, e.g. it becomes false if we assume that ξ₀ is integrable but ξ_t is a contact structure for each t > 0.

In Sucuri, partly: (c): λ∧dλ = (f g′ − f′ g) dρ∧dφ∧dz, and λ is contact if and only if that coefficient does not vanish; the equivalences in (a), the regularity on the axis in (b) and the stability in (d), no.

Problem 54 · Hodge dual in spherical coordinates

T. Dray, MTH 434/534, HW #3 (Oregon State University, winter 2024), problem 1 — sites.science.oregonstate.edu.

1. Hodge dual in spherical coordinates. Consider spherical coordinates in 3-dimensional Euclidean space with the usual orientation, namely ω = r² sin θ dr ∧ dθ ∧ dφ.
Warning: these are “physics” conventions: θ is the angle from the north pole (colatitude), and φ is the angle in the xy-plane (longitude).

(a) Determine the Hodge dual operator ∗ on all forms (expressed in spherical coordinates) by computing its action on basis forms at each rank.

(b) Compute the dot and cross products of two generic 1-forms in spherical coordinates using the expressions

α · β = ∗(α ∧ ∗β),
α × β = ∗(α ∧ β).

You may express your results either with respect to an orthonormal basis or with respect to a “coordinate” (non-orthonormal) spherical basis; make sure you know which you're doing. (“Generic” means for any two 1-forms, e.g. in terms of their components.)

In Sucuri: all of it: (a) ⋆ on all basis forms, and (b) α·β and α×β of two generic 1-forms — in the coordinate basis, which the problem allows, and says so.

Problem 55 · Hodge dual in Minkowski space

T. Dray, MTH 434/534, HW #3 (Oregon State University, winter 2024), problem 2 — sites.science.oregonstate.edu.

2. Hodge dual in Minkowski space. 4-dimensional Minkowski space has an orthonormal, oriented basis of 1-forms given by

{dx, dy, dz, dt},

with g(dt, dt) = −1, g(dx, dx) = g(dy, dy) = g(dz, dz) = 1, and all others zero. The “volume element” (choice of orientation) is given by ω = dx ∧ dy ∧ dz ∧ dt.

(a) Determine the Hodge dual operator ∗ on all forms by computing its action on basis forms at each rank.

(b) How does your answer change if the opposite orientation is chosen, namely

ω = dt ∧ dx ∧ dy ∧ dz?

In Sucuri: all of it: (a) ⋆ on the basis forms, with ω = dx∧dy∧dz∧dt, and (b) the opposite orientation, which flips the sign of all of them.

Problem 56 · Gradient, curl, divergence and Laplacian with d and ∗

T. Dray, MTH 434/534, HW #4 (Oregon State University, winter 2024), problem 1 — sites.science.oregonstate.edu.

1. Orthogonal coordinates. Choose any orthogonal coordinate system in 3-dimensional Euclidean space ℝ³ other than Cartesian, cylindrical, or spherical coordinates. (You may see me for suggestions.) Working in an orthonormal basis, compute the gradient and Laplacian of an arbitrary function, and the curl and divergence of an arbitrary “vector field” (again, really a 1-form), using the expressions

∇f = df,
∇ × α = ∗dα,
∇ · α = ∗d∗α,
△f = ∇ · ∇f = ∗d∗df.

You may check your answer in standard reference books, but you should use exterior differentiation and Hodge duality in your computation.

In Sucuri: all of it: gradient, curl, divergence and Laplacian from the four formulas, in parabolic coordinates — and in the orthonormal cobasis, as the problem asks.

Problem 57 · Decomposable forms

T. Dray, MTH 434/534 – HW #2 (Oregon State, winter 2024), problem 1 “Decomposable forms”, p. 1 — sites.science.oregonstate.edu.

Denote the p-forms in ℝⁿ by ⋀^p(ℝⁿ). A typical 1-form in ℝ² would therefore take the form F = F_x dx + F_y dy ∈ ⋀¹(ℝ²). A p-form β ∈ ⋀^p(ℝⁿ) is called decomposable if there exist 1-forms α_i ∈ ⋀¹(ℝⁿ) with β = α₁ ∧ … ∧ α_p.

(a) Show that all elements of ⋀²(ℝ³), that is, all 2-forms in ℝ³, are decomposable. In other words, show that H = H_x dy∧dz + H_y dz∧dx + H_z dx∧dy is decomposable. Hint: consider the previous assignment! You may cite your solution to the previous assignment without proof, so long as an explicit reference is given (“see HW #1”). If you do this, it wouldn't hurt to include a copy of your previous assignment.

(b) Find an example of an indecomposable 2-form γ ∈ ⋀^p(ℝⁿ). Hint: don't work in ℝ³…

(c) Is γ ∧ γ = 0? Should it be? Can it be?

(d) MTH 534 students only (or extra credit): show that all 3-forms are decomposable in ℝ⁴.

Extra credit: can you argue that all elements of ⋀ⁿ⁻¹(ℝⁿ) are decomposable?

In Sucuri: all of it: (a) every 2-form on ℝ³ is α∧β — the decomposition, for H_x ≠ 0, checked; the other cases permute the coordinates —, (b) and (c), and (d): every 3-form on ℝ⁴ is α∧β∧γ, the same way. The extra credit, in ℝⁿ, is left to the reader.