《黎曼几何》课程教学资源（英文讲义）PSet1 VARIOUS STRUCTURES.pdf_大学文库

PROBLEM SET 1: VARIOUS STRUCTURES DUE: APRIL 09, 2024 Instruction: Please work on at least SIX problems of your interest, and in each problem you are supposed to work on at least TWO sub-problems. 1. [Distance and Length] (a) Let (X, d) be a metric space. For any continuous curve γ : [0, 1] → X we define its length to be Ld(γ) = sup (Xn i=1 d(γ(ti−1), γ(ti)) | 0=t0 < t1 <· · ·< tn = 1 is a subdivision of [0, 1]) which could be +∞. Then define a new “induced intrinsic metric” dI on X via dI (x, y) := inf {Ld(γ) | γ is a continuous curve joining x and y } which, again, could be +∞. (i) Prove: d(x, y) ≤ dI (x, y), and if dI (x, y) < +∞ for all x, y, then dI is a metric on X. (ii) A metric space (M, d) with d = dI is called a length space. Endow S 1 with the metric d inherited from the Euclidean distance. Is (S 1 , d) a length space? (b) Let (M, g) be a connected Riemannian manifold, and d the Riemannian distance defined in Lecture 3. (i) Prove: For any smooth curve γ : [0, 1] → M, one has Lengthg (γ) = Ld(γ), where Lengthg (γ) is the length defined in Lecture 3. (ii) Prove: d = dI . (c) Again let (M, g) be a connected Riemannian manifold. Let C be the set of all piecewise smooth curves γ : [0, 1] → M, endowed with the uniform convergence topology. Prove: the “length functional” L : C → R, γ 7→ Lengthg (γ) is lower semi-continuous, i.e. if γk ∈ C and γk → γ uniformly, then lim inf k→∞ Lengthg (γk) ≥ Lengthg (γ). 2. [Warped products] Let (M, g), (N, h) be Riemannian manifolds, and ψ a positive smooth function on M. Define a warped product metric g ×ψ h on M × N via (g ×ψ h)((Xp, Yq),(X′ p , Y ′ q )) = gp(Xp, X′ p ) + ψ 2 (p)hq(Yq, Y ′ q ). (a) Prove: g ×ψ h is a Riemannian metric. (b) Identify R + × S 1 with R 2 \ {0} via the polar coordinates, i.e. R + × S 1 → R 2 \ {0}, (r, θ) 7→ (r cos θ, r sin θ). Prove: The warped product metric on R + × S 1 with ψ(r) = r coincides with the standard Euclidean metric on R 2 \ {0}. 1

2PROBLEM SET 1:VARIOUS STRUCTURESDUE:APRIL09,2024(c) Identify (0, π) × sm-1 with sm - [N, S) (where N, S are the north/south poles of smrespectively)viathemap(0, 元) × sm-1 → sm _(N, S) cR × Rm, (r,2) (cosr, (sinr)z)Prove: The warped product metric on (O,) × sm-1 with (r)= sinr coincides with thestandard round metric on sm - [N, S].3.[Riemannian Covering]Suppose M, N are connected smooth manifolds, and π : M → N is a smooth covering map.(a) Given any Riemannian metric g on N, we may endow with M the induced metric *g(called the covering metric) and call π : (M,n*g) - (N,g) a Riemannian covering map.Prove: The covering metric *g is invariant under Deck transformations. (Recall that a decktransformation is a diffeomorphism : M -→+ M such that o = r.)(b) Let π : (M,元*g) → (N,g) be a Riemannian covering map, : (P,h) → (N,g) is a localisometry, and : P -→ M is a lift of . Prove: is a local isometry.(c) Conversely, suppose : M -→ N is a smooth normal covering map (i.e. the group of Decktransformations acts transitively on each fiber), g is a Riemannian metric that is invariantunder all Deck transformations, then there exists a Riemannian metric +g on N such thatπ : (M, π*g) → (N,g) is a Riemannian covering map.(d) Let ei,**,em be a basis of Rm, andT=[kiei +...+kmem I kiEZ]be the lattice generated by these vectors. Starting with the Euclidean metric go on Rm, wemay get an induced metric gr on the torus Rm/ so that π : (Rm,go)→ (Rm/,gr) is aRiemannian covering map. Prove: Two metrics gr and gr are isometric if and only if thereexists an isometry of (IRm, go) that sends to I'.4.[The holonomy group]Let M be a connected smooth manifold with a connection V. Consider the holonomy groupHol,(T,M) = [Po,i I : [0, 1] → M is a piecewise smooth closed curve with (0) = (1) = p)which is obviously a subgroup of GL(T,M). Prove:(a) For any p+ q, Holp(TpM) = (Po.1)-1Holg(TqM)Po.1, where T is any piecewise smooth curvefrom p = r(0) to g = t(1).(b) If M is simply connected, then Hol,(T,M) is connected.(c) If (M,g) is a Riemannian manifold and metric compatible, then Hol,(T,M) C O(T,M)(d) Find the holonomy groups of the standard Rn, sm and Hm(e) Find the relation between the holonomy groups of (Mi × M2, gi × g2) and (Mi, gi).5.[MoreRiemannianmetrics](a) An immersion f : N -→ Rm+1 of an m dimensional smooth manifold N into Rm+1 is calleda hypersurface. Suppose [U,ul,..-,um] is a local chart on U so that the map f can beexpressed locally asah=fh(ul,..,um),1≤k≤m+1

2 PROBLEM SET 1: VARIOUS STRUCTURES DUE: APRIL 09, 2024 (c) Identify (0, π) × S m−1 with S m − {N, S} (where N, S are the north/south poles of S m respectively) via the map (0, π) × S m−1 → S m − {N, S} ⊂ R × R m, (r, z) 7→ (cos r,(sin r)z). Prove: The warped product metric on (0, π) × S m−1 with ψ(r) = sin r coincides with the standard round metric on S m − {N, S}. 3. [Riemannian Covering] Suppose M, N are connected smooth manifolds, and π : M → N is a smooth covering map. (a) Given any Riemannian metric g on N, we may endow with M the induced metric π ∗ g (called the covering metric) and call π : (M, π∗ g) → (N, g) a Riemannian covering map. Prove: The covering metric π ∗ g is invariant under Deck transformations. (Recall that a deck transformation is a diffeomorphism φ : M → M such that π ◦ φ = π. ) (b) Let π : (M, π∗ g) → (N, g) be a Riemannian covering map, φ : (P, h) → (N, g) is a local isometry, and φe : P → M is a lift of φ. Prove: φe is a local isometry. (c) Conversely, suppose π : M → N is a smooth normal covering map (i.e. the group of Deck transformations acts transitively on each fiber), g is a Riemannian metric that is invariant under all Deck transformations, then there exists a Riemannian metric π∗g on N such that π : (M, π∗ g) → (N, g) is a Riemannian covering map. (d) Let ⃗e1, · · · , ⃗em be a basis of R m, and Γ = {k1⃗e1 + · · · + km⃗em | ki ∈ Z} be the lattice generated by these vectors. Starting with the Euclidean metric g0 on R m, we may get an induced metric gΓ on the torus R m/Γ so that π : (R m, g0) → (R m/Γ, gΓ) is a Riemannian covering map. Prove: Two metrics gΓ and gΓ′ are isometric if and only if there exists an isometry of (R m, g0) that sends Γ to Γ′ . 4. [The holonomy group] Let M be a connected smooth manifold with a connection ∇. Consider the holonomy group Holp(TpM) = {P γ 0,1 | γ : [0, 1] → M is a piecewise smooth closed curve with γ(0) = γ(1) = p} which is obviously a subgroup of GL(TpM). Prove: (a) For any p ̸= q, Holp(TpM) = (P τ 0,1 ) −1Holq(TqM)P τ 0,1 , where τ is any piecewise smooth curve from p = τ (0) to q = τ (1). (b) If M is simply connected, then Holp(TpM) is connected. (c) If (M, g) is a Riemannian manifold and ∇ metric compatible, then Holp(TpM) ⊂ O(TpM). (d) Find the holonomy groups of the standard R n , S m and Hm. (e) Find the relation between the holonomy groups of (M1 × M2, g1 × g2) and (Mi , gi). 5. [More Riemannian metrics] (a) An immersion f : N → R m+1 of an m dimensional smooth manifold N into R m+1 is called a hypersurface. Suppose {U, u1 , · · · , um} is a local chart on U so that the map f can be expressed locally as x k = f k (u 1 , · · · , um), 1 ≤ k ≤ m + 1

4 PROBLEM SET 1: VARIOUS STRUCTURES DUE: APRIL 09, 2024 Define a map g : B m → U m, (x 1 , · · · , xm) 7→ (1 − (x 1 ) 2 − · · · − (x m) 2 , 2x 2 , · · · , 2x m) (x 1 − 1)2 + (x 2) 2 + · · · + (xm) 2 . Prove: g is an isometry. 8. [The Laplace-Beltrami operator] Let (M, g) be a Riemannian manifold. (a) Let f, h be smooth functions on (M, g). Check: ∆(fh) = f∆h − 2g(∇f, ∇h) + h∆f. (b) Let g = e 2φg be a Riemannian metric on M that is conformal to g. Prove: ∆f = e −2φ [∆f − (m − 2)g(∇φ, ∇f)] , where ∆ denote the Laplace-Beltrami operator associated to g. (c) Find the relation between ∆Rm and ∆Sm−1 . (d) Find the relation between ∆M×N and ∆M, ∆N . 9. [Basic Spectral Geometry] We say that λ is an eigenvalue of ∆ if there exist smooth function φ ̸= 0 such that ∆φ = λφ. The function φ is called an eigenfunction of ∆ corresponding to λ. Now suppose (M, g) is a connected compact Riemannian manifold. Prove: (a) All eigenvalues of ∆ are non-negative real numbers. (b) λ = 0 is always an eigenvalue, whose eigenfunctions are constants. (c) If φ is an eigenfunction of λ > 0, then R M φ dVol = 0. (d) If φ1 and φ2 are eigenfunctions of different eigenvalues, then R M φ1φ2 dVol = 0. (e) Equip S 1 × S 1 with the standard metric. Find all its eigenvalues and corresponding eigenfunctions. 10. [Riemannian manifold with boundary] In Lecture 4 (from bottom of page 9 to top of page 10) we briefly discussed Riemannian manifold with boundary. (a) Prove Divergence theorem II. (b) Prove Green’s formula II. (c) We call λ ∈ R a Dirichlet eigenvalue of ∆ if there exist smooth function φ ̸= 0 such that ∆φ = λφ in M and φ = 0 on ∂M. (i) Prove: All Dirichlet eigenvalues of ∆ are positive real numbers. (ii) Prove: If φ1 and φ2 are eigenfunctions of different eigenvalues, then R M φ1φ2 dVol = 0. 11. [Killing vector fields] Any vector field X defines a local family of diffeomorphisms ϕ X t : U ⊂ M → ϕ(U) ⊂ M, p 7→ ϕ X t (p) = γp,Xp (t) for −ε < t < ε, which satisfies ϕ X t ◦ ϕ X s = ϕ X t+s for t, s, t + s ∈ (−ε, ε). Now suppose (M, g) is a Riemannian manifold. We say X is a Killing vector field if these ϕ X t ’s are isometries

PROBLEM SET 1: VARIOUS STRUCTURES DUE: APRIL 09, 2024 5 (a) Let X, Y, Z ∈ Γ∞(TM). Prove: The following statements are equivalent: (i) X is a Killing vector field. (ii) LXg = 0. (iii) X⟨Y, Z⟩ = ⟨[X, Y ], Z⟩ + ⟨Y, [X, Z]⟩. (iv) ⟨∇Y X, Z⟩ + ⟨∇ZX, Y ⟩ = 0. (v) The (1, 1)-tensor field ∇X defined by ∇X : Y 7→ ∇Y X is skew-symmetric. (b) Let X be a Killing vector field, and p ∈ M is a given point. Prove: X is uniquely determined by X(p) and ∇X(p). (c) Prove: The set of Killing vector fields, denoted by iso(M, g), is a Lie subalgebra of (Γ∞(M), [·, ·]). (d) Prove: dim iso(M, g) ≤ (m+1)m 2 , where m = dim M. (e) Now suppose M is compact, in which case the isometry group Iso(M, g) is known to be a Lie group. Assume this. Prove: iso(M, g) is the the Lie algebra of Iso(M, g). 12. [Bi-invariant metric on Lie groups] A Riemannian metric on a Lie group G is left-invariant if ⟨Xg, Yg⟩g = ⟨(dLh)gXg,(dLh)gYg⟩hg holds for all g, h ∈ G and all Xg, Yg ∈ TgG. (a) Take an arbitrary inner product ⟨·, ·⟩e on TeG. Check: ⟨Xg, Yg⟩g := ⟨(dLg−1 )gXg,(dLg−1 )gYg⟩e defines a left-invariant Riemannian metric on G. (b) Write down the definition for a Riemannian metric to be right-invariant, and construct a right-invariant Riemannian metric on G. (c) A Riemannian metric on a Lie group G is bi-invariant if it is both left-invariant and right-invariant. Prove: If ⟨·, ·⟩ is a bi-invariant Riemannian metric on G, then ⟨·, ·⟩e is a conjugation-invariant inner product on TeG. (d) Conversely, show that if ⟨·, ·⟩e is a conjugation-invariant inner product on TeG, then the left-invariant Riemannian metric defined in (a) is bi-invariant. (e) Now suppose G is compact, so that there exists a Haar measure on G, which can be think of as a volume form ω that is both left-invariant and right-invariant: L ∗ gω = ω = R ∗ gω, ∀g ∈ G. Prove: For any left-invariant metric on G, the new inner product ⟨Xg, Yg⟩ new g = Z G ⟨(dRh)gXg,(dRh)gYg⟩ old gh ω, ∀g ∈ G, and Xg, Yg ∈ TgG defines a bi-invariant Riemannian metric on G