Exercises—Global Analysis 1. Consider the cylinder in R3 given by the equation M := {(x, y, z) ∈ R3 : x2 + y2 = R2 }, where R > 0. Show that M is a 2-dimensional submanifold in R3 . Moreover, give formula for local parametrizations and local trivializations, and a description of M as a local graph. 2. Consider a double cone given by rotating a line through 0 of slope α around the z-axis in R3 . It is given by the equation z2 = (tan α)2 (x2 + y2 ). At which points is the double cone a smooth submanifold of R3 ? Around the points where it is give a formula for local parametrizations and trivializations, and a description of it as a local graph. 3. Denote by Hom(Rn , Rm ) the nm-dimensional vector space of linear maps from Rn to Rm . Consider the subset Homr(Rn , Rm ) of linear maps in Hom(Rn , Rm ) of rank r. Show that Homr(Rn , Rm ) is a submanifold of dimension of r(n + m − r) in Hom(Rn , Rm ). Hint: Let T0 ∈ Homr(Rn , Rm ) be a linear map of rank r and decompose Rn and Rm as follows Rn = E ⊕ E⊥ and Rm = F ⊕ F⊥ , (0.1) where F equals the image of T0 and E⊥ the kernel of T0, and (·)⊥ denotes the orthogonal complement. Note that dim E = dim F = r. With respect to (0.1) any T ∈ Hom(Rn , Rm ) can be viewed as a matrix T = A B C D , where A ∈ Hom(E, F), B ∈ Hom(E⊥ , F), C ∈ Hom(E, F⊥ ) and D ∈ Hom(E⊥ , F⊥ ). Show that the set of matrices T with A invertible defines an open neighbourhood of T0 and characterize the elements in this neighbourhood that have rank r (equivalently, the ones that have an (n − r)-dimensional kernel). 1 2 4. For i = 1, ...n let (Mi, Ai) be a smooth manifolds. Suppose M := M1 × ... × Mn is endowed with the product topology. Then show that A := {(U1 × ... × Un, u1 × ... × un) : (Ui, ui) ∈ Ai} defines a smooth atlas on M and that the projections pri : M → Mi are smooth. Moreover show that, for any smooth manifold N, a map f : N → M is smooth if and only if fi := pri ◦ f : N → Mi is smooth for all i, and show that this property characterizes the smooth manifold structure on M uniquely. 5. Suppose (Mi, Ai) are smooth manifolds for i ∈ I, where I is countable. Consider the disjoint union M := i∈IMi = ∪i∈I{(x, i) : x ∈ Mi} endowed with the disjoint union topology and denote by inji : Mi → M the canonical injections (inji(x) = (x, i)). Show that A := ∪i∈IAi defines a smooth atlas on M and that the injections inji are smooth. Moreover, show that for any smooth manifold N, a map f : M → N is smooth if and only if fi := f ◦ inji : Mi → N is smooth for all i, and show that this property characterizes the smooth manifold structure on M uniquely. 6. Suppose U ⊂ Rm is open and f : U → Rn a smooth map such that Dxf : Rm → Rn is of rank r for all x ∈ U. Show that for any x0 ∈ U there exists a diffeomorphism φ between an open neighbourhood of x0 and an open neighbourhood of 0 ∈ Rm and a diffeomorphism ψ between an open neighbourhood of y0 = f(x0) and an open neighbourhood of 0 in Rn such that the locally defined map ψ ◦ f ◦ φ−1 : Rr × Rm−r → Rr × Rn−r has the form (x1, ..., xr, ..., xm) → (x1, ..., xr, 0, ..., 0). Hint: The idea is that f locally around x0 looks like Dx0 f, which is a linear map Rm → Rn of rank r, which up to a basis change has the form (x1, .., xm) → (x1, ..., xr, 0, ..., 0). (a) Set E2 := ker(Dx0 f) ⊂ Rm and E1 := E⊥ 2 , and F1 := Im(Dx0 f) ⊂ Rn and F2 := F⊥ 1 . Decompose Rm = E1 ⊕ E2 and Rn = F1 ⊕ F2, and consider f as a map f = (f1, f2) : E1 ⊕ E2 → F1 ⊕ F2 defined on U ⊂ E1 ⊕ E2 = Rm . (b) Show that φ : E1 ⊕ E2 → F1 ⊕ E2 given by φ(x1 , x2 ) = (f1(x1 , x2 ) − f1(x1 0, x2 0), x2 − x2 0) is a local diffeomorphism around x0 = (x1 0, x2 0) whose local inverse will be the required map. 3 (c) Show that g := f ◦ φ−1 : F1 ⊕ E2 → F1 ⊕ F2 has the form g(y1 , y2 ) = (g1((y1 , y2 ), g2((y1 , y2 )) = (y1 + y1 0, g2(y1 , 0)). Now ψ is easily seen to be...? 7. Suppose M and N are are manifolds of dimension m respectively n and let f : M → N be a smooth map of constant rank r. Deduce from (1) that for any fixed y ∈ f(M) the preimage f−1 (y) ⊂ M is a submanifold of dimension m − r in M. 8. Consider the Grassmannian of r-planes in Rn : Gr(r, n) := {E ⊂ Rn : E is a r-dimensional subspace of Rn }. Denote by Str(Rn ) the set of r-tuples of linearly independent vectors in Rn . Identifying an element X ∈ Str(Rn ) with a n × r matrix X = (x1 , ...., xr ) xi ∈ Rn , shows that Str(Rn ) equals the subset of rank r matrices in the vector space Mn×r(R), which we know from Tutorial 1 is an open subset. Write π : Str(Rn ) → Gr(r, n) for the natural projection given by π(X) = span(x1 , ..., xr ) and equip Gr(r, n) with the quotient topology with respect to π. (a) Fix E ∈ Gr(r, n) and let F ⊂ Rn be a subspace of dimension n − r such that Rn = E ⊕ F. Show that U(E,F) = {W ∈ Gr(r, n) : W ∩ F = {0}} ⊂ Gr(r, n) is an open neighbourhood of E. (b) Show that any element W ∈ U(E,F) determines a unique linear map W : E → F such that its graph equals W, i.e. W = {(x, Wx) : x ∈ E}. (c) Show that the map uE,F : U(E,F) → Hom(E, F) given by uE,F (W) = W is a homeomorphism. (d) Show that A := {(U(E,F), u(E,F)) : E, F ⊂ Rn complimentary subspaces of dimension r resp. n−r} is a smooth atlas for Gr(r, n). 9. For a topological space M denote by C0 (M) the vector space of continuous realvalued functions f : M → R. Any continuous map F : M → N between topological spaces M and N induces a map F∗ : C0 (N) → C0 (M) given by F∗ (f) := f ◦ F : M → R. 4 (a) Show that F∗ is linear. (b) If M and N are (smooth) manifolds, show that F : M → N is smooth ⇐⇒ F∗ (C∞ (N)) ⊂ C∞ (M). (c) If F is a homeomorphism between (smooth) manifolds, show that F is a diffeomorphism ⇐⇒ F∗ (C∞ (N)) ⊂ C∞ (M) and F∗ : C∞ (N) → C∞ (M) is an isomorphism. 10. We have seen in the first tutorial that Homr(Rn , Rm ) is a submanifold of Hom(Rn , Rm ) of dimension r(n + m − r) in. For X ∈ Homr(Rn , Rm ) compute the tangent space TXHomr(Rn , Rm ) ⊂ TXHom(Rn , Rm ) ∼= Hom(Rn , Rm ). 11. We have seen in the first tutorial that the Grassmannian manifold Gr(r, n) can be realized as a submanifold of Hom(Rn , Rn ) of dimension r(n−r). For E ∈ Gr(r, n) compute the tangent space TEGr(r, n) ⊂ TEHom(Rn , Rn ) ∼= Hom(Rn , Rn ). 12. Consider the general linear group GL(n, R) and the special linear group SL(n, R). We have seen that they are submanifolds of Mn(R) = Rn2 (even so called Lie groups) and that TIdGL(n, R) ∼= Mn(R) = Rn2 . (a) Compute the tangent space TIdSL(n, R) of SL(n, R) at the identity Id. (b) Fix A ∈ SL(n, R) and consider the conjugation conjA : SL(n, R) → SL(n, R) by A given by conjA(B) = ABA−1 . Show that conjA is smooth and compute the derivative TIdconjA : TIdSL(n, R) → TIdSL(n, R). (c) Consider the map Ad : SL(n, R) → Hom(TIdSL(n, R), TIdSL(n, R)) given by Ad(A) := TIdconjA. Show that Ad is smooth and compute TIdAd. 13. Consider Rn equipped with the standard inner product of signature (p, q) (where p + q = n) given by x, y := p i=1 xiyi − n i=p+1 xiyi and the group of linear orthogonal transformation of (Rn , ·, · ) given by O(p, q) := {A ∈ GL(n, R) : Ax, Ay = x, y ∀x, y ∈ Rn }. (a) Show that O(p, q) = {A ∈ GL(n, R) : A−1 = Ip,qAt Ip,q}, where Ip,q = Idp 0 0 −Idq , and that O(p, q) is a submanifold of Mn(R). What is its dimension? 5 (b) Show that O(p, q) is a subgroup of GL(n, R) with respect to matrix multiplication µ and that µ : O(p, q) × O(p, q) → O(p, q) is smooth (i.e. that O(p, q) is a Lie group.) (c) Compute the tangent space TIdO(p, q) of O(p, q) at the identity Id. 14. Suppose M = R3 with standard coordinates (x, y, z). Consider the vector field ξ(x, y, z) = 2 ∂ ∂x − ∂ ∂y + 3 ∂ ∂z . How does this vector field look like in terms of the coordinate vector fields associated to the cylindrical coordinates (r, φ, z), where x = r cos φ, y = r sin φ and z = z? Or with respect to the spherical coordinates (r, φ, θ), where x = r sin θ cos φ, y = r sin θ cos φ and z = r cos θ? 15. Consider R3 with coordinates (x, y, z) and the vector fields ξ(x, y, z) = (x2 − 1) ∂ ∂x + xy ∂ ∂y + xz ∂ ∂z η(x, y, z) = x ∂ ∂x + y ∂ ∂y + 2xz2 ∂ ∂z . Are they tangent to the cylinder M = {(x, y, z) ∈ R3 : x2 + y2 = 1} ⊂ R3 with radius 1 (i.e. do they restrict to vector fields on M)? 16. Suppose M = R2 with coordinates (x, y). Consider the vector fields ξ(x, y) = y ∂ ∂x and η(x, y) = x2 2 ∂ ∂y on M. We computed in class their flows and saw that they are complete. Compute [ξ, η] and its flow? Is [ξ, η] complete? 17. Let M be a (smooth) manifold and ξ, η ∈ X(M) two vector fields on M. Show that (a) [ξ, η] = 0 ⇐⇒ (Flξ t )∗ η = η, whenever defined ⇐⇒ Flξ t ◦ Flη s = Flη s ◦ Flξ t , whenever defined. (b) If N is another manifold, f : M → N a smooth map, and ξ and η are f-related to vector fields ˜ξ resp. ˜η on N, then [ξ, η] is f-related to [˜ξ, ˜η]. 18. Consider the general linear group GL(n, R). For A ∈ GL(n, R) denote by λA : GL(n, R) → GL(n, R) λA(B) = AB ρA : GL(n, R) → GL(n, R) ρA(B) = BA left respectively right multiplication by A, and by µ : GL(n, R) × GL(n, R) → GL(n, R) the multiplication map. (a) Show that λA and ρA are diffeomorphisms for any A ∈ GL(n, R) and that TBλA(B, X) = (AB, AX) TBρA(B, X) = (BA, XA), where (B, X) ∈ TBGL(n, R) = {(B, X) : X ∈ Mn(R)}. 6 (b) Show that T(A,B)µ((A, B), (X, Y )) = TBλAY + TAρB X = (AB, AY + XB) where (A, B) ∈ GL(n, R) × GL(n, R) and (X, Y ) ∈ Mn(R) × Mn(R). (c) For any X ∈ Mn(R) ∼= TIdGL(n, R) consider the maps LX : GL(n, R) → TGL(n, R) LX(B) = TIdλB(Id, X) = (B, BX). RX : GL(n, R) → TGL(n, R) RX(B) = TIdρB(Id, X) = (B, XB). Show that LX and RX are smooth vector field and that λ∗ ALX = LX and ρ∗ ARX = RX for any A ∈ GL(n, R). What are their flows? Are these vector fields complete? (d) Show that [LX, RY ] = 0 for any X, Y ∈ Mn(R). 19. Suppose G is a Lie group, i.e. a manifold, which is also a group, where the group multiplication µ : G × G → G is smooth. Denote by λg : G → G and ρg : G → G the left resp. right multiplication by g ∈ G, i.e. λg(h) = µ(g, h) and ρg(h) = µ(h, g). (a) Show that the tangent map of µ at (g, h) ∈ G × G is given by T(g,h)µ (ξ, η) = Thλgη + Tgρhξ, where ξ ∈ TgG and η ∈ ThG. (b) Show that the inversion ι(g) = g−1 is smooth and that its tangent map at g is given by Tgι = −Teρg−1 ◦ Tgλg−1 = −Teλg−1 ◦ Tgρg−1 , where e ∈ G denotes the neutral element in G. In particular, Teι = −Id. 20. Suppose (G, µ, e) is a Lie group as in the previous example. A vector field ξ ∈ X(G) is called left- resp. right-invariant, if λ∗ hξ = ξ resp. ρ∗ hξ = ξ for all h ∈ G. (a) Show that for any X ∈ TeG, LX(g) = TeλgX and RX(g) = TeρgX define a smooth left- resp. right-invariant vector field on G. Moreover, show that RX = ι∗ (L−X). (b) Show that any left- resp. right-invariant vector field ξ ∈ X(G) is of the form LX resp. RX for some X ∈ TeG. (c) Show that for any X ∈ TeG the vector fields LX and RX are complete. (d) Show that [LX, RY ] = 0 for any X, Y ∈ TeG. 21. Suppose αi j for i = 1, ..., k and j = 1, ..., n are smooth real-valued functions defined on some open set U ⊂ Rn+k satisfying ∂αi j ∂xr + k =1 αr ∂αi j ∂z = ∂αi r ∂xj + k =1 αj ∂αi r ∂z , 7 where we write (x, z) = (x1 , ..., xn , z1 , ..., zk ) for a point in Rn+k . Show that for any point (x0, z0) ∈ U there exists an open neighbourhood V of x0 in Rn and a unique C∞ -map f : V → Rk such that ∂fi ∂xj (x1 , ..., xn ) = αi j(x1 , ..., xn , f1 (x), ..., fk (x)) and f(x0) = z0. In the class/tutorial we proved this for k = 1 and j = 2. 22. Which of the following systems of PDEs have solutions f(x, y) (resp. f(x, y) and g(x, y)) in an open neighbourhood of the origin for positive values of f(0, 0) (resp. f(0, 0) and g(0, 0))? (a) ∂f ∂x = f cos y and ∂f ∂y = −f log f tan y. (b) ∂f ∂x = exf and ∂f ∂y = xeyf . (c) ∂f ∂x = f and ∂f ∂y = g; ∂g ∂x = g and ∂g ∂y = f. 23. Suppose E → M is a (smooth) vector bundle of rank k over a manifold M. Then E is called trivializable, if it isomorphic to the trivial vector bundle M × Rk → M. (a) Show that E → M is trivializable ⇐⇒ E → M admits a global frame, i.e. there exist (smooth) sections s1, ..., sk of E such that s1(x), ..., sk(x) span Ex for any x ∈ M. (b) Show that the tangent bundle of any Lie group G is trivializable. (c) Recall that Rn has the structure of a (not necessarily associative) normed division algebra over R for n = 1, 2, 4, 8. Use this to show that the tangent bundle of the spheres S1 ⊂ R2 , S3 ⊂ R4 and S7 ⊂ R8 is trivializable. 24. Let V be a finite dimensional real vector space and consider the subspace of rlinear alternating maps Λr V ∗ = Lr alt(V, R) of the vector space of r-linear maps Lr (V, R) = (V ∗ )⊗r . Show that for ω ∈ Lr (V, R) the following are equivalent: (a) ω ∈ Λr V ∗ (b) For any vectors v1, ..., vr ∈ V one has ω(v1, ..., vi, ..., vj, ..., vk) = −ω(v1, ..., vj, ..., vi, ..., vk) (c) ω is zero whenever one inserts a vector v ∈ V twice. (d) ω(v1, ..., vk) = 0, whenever v1, ..., vk ∈ V are linearly dependent vectors. 25. Let V be a finite dimensional real vector space. Show that the vector space Λ∗ V ∗ := r≥0 Λr V ∗ is an associative, unitial, graded-anticommutative algebra with respect to the wedge product ∧, i.e. show that the following holds: (a) (ω ∧ η) ∧ ζ = ω ∧ (η ∧ ζ) for all ω, η, ζ ∈ Λ∗ V ∗ . 8 (b) 1 ∈ R = Λ0 V ∗ satisfies 1 ∧ ω = ω ∧ 1 = 1 for all ω ∈ Λ∗ V ∗ . (c) Λr V ∗ ∧ Λs V ∗ ⊂ Λr+s V ∗ . (d) ω ∧ η = (−1)rs η ∧ ω for ω ∈ Λr V ∗ and η ∈ Λs V ∗ . Moreover, show that for any linear map f : V → W the linear map f∗ : Λ∗ W∗ → Λ∗ V ∗ is a morphism of graded unitlal algebras, i.e. f∗ 1 = 1, f∗ (Λr W∗ ) ⊂ Λr V ∗ and f∗ (ω ∧ η) = f∗ ω ∧ f∗ η. 26. Let V be a finite dimensional real vector space. Show that: (a) If ω1, ..., ωr ∈ V ∗ and v1, ..., vr ∈ V , then ω1 ∧ ... ∧ ωr(v1, ..., vr) = det((ωi(vj))1≤i,j≤r). In particular, ω1, ..., ωr are linearly independent ⇐⇒ ω1 ∧ ... ∧ ωr = 0. (b) If {λ1, ..., λn} is a basis of V ∗ , then {λi1 ∧ ... ∧ λir : 1 ≤ i1 < ... < ir ≤ n} is a basis of Λr V ∗ . 27. Let V be a finite dimensional real vector space. An element µ ∈ Lr (V, R) is called symmetric, if µ(v1, ..., vr) = µ(vσ(1), ..., vσ(r)) for any vectors v1, ..., vr ∈ V and any permutation σ ∈ Sr . Denote by Sr V ∗ ⊂ µ ∈ Lr (V, R) the subspace of symmetric elements in the vector space Lr (V, R). (a) For µ ∈ Lr (V, R) show that µ ∈ Sr V ∗ ⇐⇒ µ(v1, ..., vi, ..., vj, ..., vr) = µ(v1, ..., vj, ..., vi, ..., vr), for any vectors v1, ..., vr ∈ V . (b) Consider the map Sym : Lr (V, R) → Lr (V, R) given by Sym(µ)(v1, ..., vr) = 1 r! σ∈Sr µ(vσ(1), ..., vσ(r)). Show that Image(Sym) = Sr V ∗ and that µ ∈ Sr V ∗ ⇐⇒ Sym(µ) = µ. 28. Let V be a finite dimensional real vector space and set S(V ∗ ) := ⊕∞ r=0Sr V ∗ with the convention S0 V ∗ = R and S1 V ∗ = V ∗ . For µ ∈ Sr V ∗ and ν ∈ St V ∗ define their symmetric product by µ ν := Sym(µ ⊗ ν) ∈ Sr+t V ∗ . By blinearity, we extend this to a R-bilinear map : S(V ∗ ) × S(V ∗ ) → S(V ∗ ). Show that S(V ∗ ) is an unitial, associative, commutative, graded algebra with respect to the symmetric product . 9 29. Suppose p : E → M and q : F → M are vector bundles over M. Show that their direct sum E ⊕ F := x∈M Ex ⊕ Fx → M and their tensor product E ⊗ F := x∈M Ex ⊗ Fx → M are again vector bundles over M. 30. Suppose E ⊂ TM is a smooth distribution of rank k on a manifold M of dimension n and denote by Ω(M) the vector space of differential forms on M. (a) Show that locally around any point x ∈ M there exists (local) 1-forms ω1 , ..., ωn−k such that for any (local) vector field ξ one has: ξ is a (local) section of E ⇐⇒ ωi(ξ) = 0 for all i = 1, ..., n − k. (b) Show that E is involutive ⇐⇒ whenever ω1 , ..., ωn−k are local 1-forms as in (a) then there exists local 1-forms µi,j for i, j = 1, ..., n − k such that dωi = n−k j=1 µi,j ∧ ωj . (c) Show ΩE(M) := {ω ∈ Ω(M) : ω|E = 0} ⊂ Ω(M) is an ideal of the algebra (Ω(M), ∧). Here, ω|E = 0 for a -form ω means that ω(ξ1, ..., ξ ) = 0 for any sections ξ1, ...ξ of E. (d) An ideal J of (Ω(M), ∧) is called differential ideal, if d(J ) ⊂ J . Show that ΩE(M) is a differential ideal ⇐⇒ E is involutive. 31. Suppose M is a manifold. Then a graded derivation of the algebra (Ω(M), ∧) of degree r is a linear map D : Ω(M) → Ω(M) such that • D maps Ωk (M) to Ωk+r (M), and • for any ω ∈ Ωk (M) and any η ∈ Ω (M), D(ω ∧ η) = D(ω) ∧ η + (−1)rk ω ∧ D(η). In class we have seen that d and Lξ for ξ ∈ X(M) are graded derivations of degree 1 respectively 0. (a) Show that for two graded derivations D1 and D2 of (Ω(M), ∧) of degree r1 respectively r2, [D1, D2] := D1 ◦ D2 − (−1)r1r2 D2 ◦ D1 is a graded derivation of degree r1 + r2. (b) Suppose D is a graded derivation of (Ω(M), ∧). Let ω ∈ Ωk (M) be a differential form and U ⊂ M an open subset. Show that ω|U = 0 implies D(ω)|U = 0. Hint: Think about writing 0 as fω for some smooth function f and use the defining properties of a graded derivation. 10 (c) Suppose D and ˜D are two graded derivations such that D(f) = ˜D(f) and D(df) = ˜D(df) for all f ∈ C∞ (M, R). Show that D = ˜D. 32. Suppose M is a manifold and ξ, η ∈ Γ(TM) vector fields. (a) Show that the insertion operator iξ : Ωk (M) → Ωk−1 (M) is a graded derivation of degree −1 of (Ω(M), ∧). (b) Recall from class that [d, d] = 0. Verify (the remaining) graded-commutator relations between d, Lξ, iη: (i) [d, Lξ] = 0. (ii) [d, iξ] = d ◦ iξ + iξ ◦ d = Lξ. (iii) [Lξ, Lη] = L[ξ,η]. (iv) [Lξ, iη] = i[ξ,η]. (v) [iξ, iη] = 0. Hint: Use (c) from previous exercise 33. Prove the Poincar´e Lemma: Suppose ω ∈ Ωk (Rm ) is a closed k-form, where k ≥ 1. Show that there exists τ ∈ Ωk−1 (Rm ) such that dτ = ω. Hint: Show that for any k-form ω = i1<...) is a surface in Euclidean space. Let u : U → u(U) be a local chart for M with corresponding local parametrization v = u−1 : u(U) → U. With respect to the frame { ∂ ∂x1 , ∂ ∂x2 } of TR2 , we can write v∗ g and v∗ II as matrices E F F G and ˜E ˜F ˜F ˜G , 12 where E = g( ∂ ∂u1 , ∂ ∂u1 ) ◦ v F = g( ∂ ∂u1 , ∂ ∂u2 ) ◦ v G = g( ∂ ∂u2 , ∂ ∂u2 ) ◦ v, and ˜E = II( ∂ ∂u1 , ∂ ∂u1 ) ◦ v ˜F = II( ∂ ∂u1 , ∂ ∂u2 ) ◦ v ˜G = II( ∂ ∂u2 , ∂ ∂u2 ) ◦ v. Compute in terms of E, F, G, ˜E, ˜F and ˜G, the Weingarten map L ◦ v, the Gauß curvature K ◦ v, the mean curvature H ◦ v, and the principal curvatures κ1 ◦ v and κ2 ◦ v. 42. Let us write (x1 , x2 , x3 ) for the coordinates in R3 . Take a circle of radius r > 0 in the (x1 , x3 )-plane and rotate it around a circle of radius R > r in the (x1 , x2 )-plane. The result is a 2-dimensional torus M in R3 . If I ⊂ R is an open interval of length < 2π the map v : I × I → R3 given by v(φ, θ) = ((R + r cos θ) cos φ, (R + r cos θ) sin φ, r sin θ) defines a local parametrization of M. With respect to v, compute, using the previous exercise, the metric g on M induced by the Euclidean metric on R3 , the 2nd fundamental form, the Gauß and the mean curvature, the principal curvatures and the principal curvature directions of the surface (M, g) in R3 . Hint: Note that ν(φ, θ) = (cos φ cos θ, sin φ cos θ, sin θ) defines a local unit normal vector field for M. 43. Suppose (M, g) ⊂ (Rm+1 , g) = (Rm+1 , geuc ) is a connected oriented hypersurface in Euclidean space. Show that all points in M are umbilic if and only if M is part of an affine hyperplane or a sphere. Hint: For , =⇒ show the following: • Fix a global unit normal vector field ν : M → Rm+1 . Then, by assumption, for any x ∈ M there exists λ(x) ∈ R such that Lx = λ(x)IdTxM . Since λ = g(L(ξ),ξ) g(ξ,ξ) for any local vector field ξ on M, λ : M → R is smooth. Show that λ is constant, by, for instance, picking a chart and computing the left-hand-side of [ ∂ ∂ui , ∂ ∂uj ] · ν = 0. • If λ = 0, show that any curve in M is contained in an affine hyperplane with (constant) normal vector ν. • If λ = 0, show that f : M → Rm+1 , given by f(x) = x − 1 λ ν(x), is constant. 44. Suppose is an affine connection on a manifold M. 13 (a) Show that its curvature, given by, R(ξ, η)(ζ) = ξ ηζ − η ξζ − [ξ,η]ζ, for vector fields ξ, η, ζ ∈ X(M) defines a 1 3 -tensor on M. (b) Show that, if is torsion-free, the Bianchi identity holds: R(ξ, η)(ζ) + R(η, ζ)(ξ) + R(ζ, ξ)(η) = 0, for any ξ, η, ζ ∈ X(M). 45. Suppose E → M is a vector bundle over a manifold M equipped with a linear connection , that is, a R-bilinear map : Γ(TM) × Γ(E) → Γ(E) (ξ, s) → ξs such that for ξ ∈ Γ(TM), s ∈ Γ(E) and f ∈ C∞ (M, R) one has • fξs = f ξs • ξfs = f ξs + (ξ · f)s. (a) Show that : Γ(TM)×Γ(E∗ ) → Γ(E∗ ) (typically also denoted by ) given by ( ξµ)(s) = ξ · µ(s) − µ( ξs), for µ ∈ Γ(E∗ ), ξ, ∈ Γ(TM), s ∈ Γ(E) defines a linear connection on the dual vector bundle E∗ → M. (b) Suppose ˜E → M is another vector bundle equipped with a linear connection ˜ . Show the vector bundle E ⊗ ˜E → M admits a linear connection characterized by ξ(s ⊗ ˜s) = ξs ⊗ ˜s + s ⊗ ˜ ξ ˜s for ξ ∈ Γ(TM), s ∈ Γ(E) and ˜s ∈ Γ( ˜E). 46. Suppose is an affine connection on a manifold M. Then the previous exercise shows that induces a linear connection : Γ(TM) × T p q (M) → T p q (M) on all tensor bundles. Show that it also induces a linear connection on the bundles Λk T∗ M for k = 1, ... dim(M) characterized by ξ(ω ∧ µ) = ξω ∧ µ + ω ∧ ξµ for ω ∈ Γ(Λk T∗ M) and µ ∈ Γ(Λ T∗ M) and give a formula. 47. Suppose (M, g) is a Riemannian manifold. 14 (a) For vector fields ξ, η ∈ X(M), let ξη ∈ X(M) be the unique vector field such that g( ξη, ζ) = 1 2 ξ·g(η, ζ)+η·g(ζ, ξ)−ζ·g(ξ, η)+g([ξ, η], ζ)−g([ξ, ζ], η)−g([η, ζ], ξ) for all ζ ∈ X(M). Show that defines a torsion-free affine connection satis- fying ξ · g(η, ζ) = g( ξη, ζ) + g(η, ξζ) for ξ, η, ζ ∈ X(M). (b) The connection in (a) is called the Levi-Civita connection of (M, g). Show that its curvature satisfies: • g(R(ξ, η)(ζ), µ) = −g(R(ξ, η)(µ), ζ), • g(R(ξ, η)(ζ), µ) = g(R(ζ, µ)(ξ), η), for ξ, η, ζ, µ ∈ X(M). (c) Suppose (U, u) is a chart for M and let R be the Riemann curvature, i.e. the curvature of the Levi-Civita connection of (M, g). Compute R( ∂ ∂ui , ∂ ∂uj )( ∂ ∂uk ) in terms of the Christoffel symbols. 48. Suppose H = {(x, y) ∈ R2 : y > 0} is the upper-half plane and equip it with the Riemannian metric g = 1 y2 dx ⊗ dx + 1 y2 dy ⊗ dy. (a) Compute the Christoffel symbols of g. (b) Compute the geodesics of g. (c) Compute the Riemann curvature. 49. Identifying H = {(x, y) ∈ R2 : y > 0} = {z = x + iy ∈ C : y > 0} in the previous example, we may write g as g = 1 Im(z)2 Re(dz ⊗ d¯z) = 4 |z − ¯z|2 Re(dz ⊗ d¯z), where dz = dx + idy, d¯z = dx − idy, Im and Re denote imaginary and real part, and |−| is the absolute value of complex numbers. Consider SL(2, R) = {A ∈ GL(2, R) : det A = 1}. For A = a b c d ∈ SL(2, R) and z ∈ C let fA(z) = az + b cz + d . (a) Show that fA is a diffeomorphism from H to itself for any A ∈ SL(2, R) and that fAB = fA ◦ fB. 15 (b) Show that fA is an isometry of H. (c) Show that for any two points z, z ∈ H there exists A ∈ SL(2, R) such that fA(z) = z . (d) Characterize the elements A ∈ SL(2, R) such that fA(i) = i.