PROBLEM SET 6,PART 1:TENSORS AND DIFFERENTIAL FORMSDUE:DEC.14(1)[Tedious multi-linear algebra]Prove Lemma 5.1.19 and Proposition 5.1.21(2)[Decomposable tensors]A tensor T e ViV@..-@Ve is called decomposable ifthere exists ui E Vi, **, Uk E Vkso that T = ui ...vk.[Indecomposable tensors areknown as entangled states inquantum information theory.](a) Let Vi,v2 be linearly independent vectors in V.Show that the 2-tensor ivi+V2v2 EVV is not decomposable.(b) Let V,W be vector spaces, with basis [ui) and [wi) respectively. Prove:AnonzeroT=Eiaijviwi V@Wisdecomposable if and onlyif the coefficientmatrix (aij) has rank 1.(c) [not required] For any vector space V, we let P(V) = V - [0)/(u ~ Au) be theprojective space of V.Prove:The mapP(Vi) ×... ×P(Ve) -→P(Vi @..@ Ve),([ui], .., [vkl) [ui @..@v]is an embedding.This is known as the (generalized)Segre embedding.(d)Supposek≥2,anddimV:≥2foreach1≤i≤k.Prove:Anydecomposablek-tensor T eVi..V lines in the image of the generalized Segre embeddingabove. Conclude that most k-tensors are indecomposable.(3)[Tensor product of vector spaces]Let V, W be finite dimensional vector spaces, with basis [u;] and [w;] respectively.(a) Prove: VW*L(W,V) as vector spaces. Then answer the following problems:(i) What is the trace of T e V V* ~ L(V, V)?(i)Give a conceptional proof of thefact that the contraction defined in Defi-nition 5.1.5 is well-defined.(b) [not required] [Universal Property] Prove:There exists a blinear map :V×W→VW, so that the following universal property holds:for any vectorspace Z and any bilinear map h : V × W -→ Z, there exists a unique linear maph :V W-→Z so that h=h o.Moreover, such a pair (VW, ) is uniqueup to isomorphism.[Lie derivative on differential forms] [Not required](4)[(a) Read section 5.2.3.(b)Prove Proposition 5.2.11.1
PROBLEM SET 6, PART 1: TENSORS AND DIFFERENTIAL FORMS DUE: DEC. 14 (1) [Tedious multi-linear algebra] Prove Lemma 5.1.19 and Proposition 5.1.21. (2) [Decomposable tensors] A tensor T ∈ V1⊗V2⊗· · ·⊗Vk is called decomposable if there exists v1 ∈ V1, · · · , vk ∈ Vk so that T = v1 ⊗ · · · ⊗ vk. [Indecomposable tensors are known as entangled states in quantum information theory.] (a) Let v1, v2 be linearly independent vectors in V . Show that the 2-tensor v1 ⊗v1 + v2 ⊗ v2 ∈ V ⊗ V is not decomposable. (b) Let V, W be vector spaces, with basis {vi} and {wj} respectively. Prove: A nonzero T = P ij aijvi⊗wj ∈ V ⊗W is decomposable if and only if the coefficient matrix (aij ) has rank 1. (c) [not required] For any vector space V , we let P(V ) = V − {0}/{v ∼ λv} be the projective space of V . Prove: The map P(V1) × · · · × P(Vk) → P(V1 ⊗ · · · ⊗ Vk), ([v1], · · · , [vk]) 7→ [v1 ⊗ · · · ⊗ vk] is an embedding. This is known as the (generalized) Segre embedding. (d) Suppose k ≥ 2, and dim Vi ≥ 2 for each 1 ≤ i ≤ k. Prove: Any decomposable k-tensor T ∈ V1 ⊗ · · · ⊗ Vk lines in the image of the generalized Segre embedding above. Conclude that most k-tensors are indecomposable. (3) [Tensor product of vector spaces] Let V, W be finite dimensional vector spaces, with basis {vi} and {wj} respectively. (a) Prove: V ⊗W∗ ≃ L(W, V ) as vector spaces. Then answer the following problems: (i) What is the trace of T ∈ V ⊗ V ∗ ≃ L(V, V )? (ii) Give a conceptional proof of the fact that the contraction defined in Definition 5.1.5 is well-defined. (b) [not required] [Universal Property] Prove: There exists a blinear map φ : V × W → V ⊗ W, so that the following universal property holds: for any vector space Z and any bilinear map h : V × W → Z, there exists a unique linear map h˜ : V ⊗ W → Z so that h = h˜ ◦ φ. Moreover, such a pair (V ⊗ W, φ) is unique up to isomorphism. (4) [Lie derivative on differential forms] [Not required] (a) Read section 5.2.3. (b) Prove Proposition 5.2.11. 1
2PROBLEMSET6,PART1:TENSORSANDDIFFERENTIALFORMSDUE:DEC.14(5)[Maxwell's equations]The famousMarwell's eguations in thetheory of electromagnetism areof theformaBV.B= 0,+V×E=0,tDEV·E=4元p,-V×B= -4元J,otwhere E = (Er,Ey,X2) is the electricity field, B = (Br,By, B) is the magnetismfield,pisthe chargedensity,and -(Jr, Jy,J.) is the currentdensity.Allfunctions,e.g. Er, Ey, Br, p, Jr, are functions of a, y, z and t.Define the Faraday 2-form F on R4 (with coordinates t, ,y,z) asF=Erda^dt+Eydy^dt+E,dz^dt+Brdy^dz+Bydzda+B,da^dy,and define the current l-form to beJ= pdt + Jrdr + Jydy + Jzdz.We also define the Hodge star operators on R4 (with respect to the Minkowski metric)tobethelinearoperators*: 2*(R) → 24-k(R4),0≤k≤4,so that on 2-forms one has **=-1 and*(da^dt)=dy^dz，*(dy^dt)=-ddz,*(dz^dt)=dr ^dywhile on 1-forms one has dt ^ (*dt) = dr ^ (*dar) =. = dt ^ dr ^ dy dz, e.g.*dt=dr^dy^dz,*dr=-dtdy^dz,*dy=dt^dr^dzetc.Prove:ThesystemofMaxwellequationsisequivalenttodF=0,d*F=4元*J.(6)[Thecanonical symplecticform onR2n]Denoteby (r',.",r", y',...,y")the coordinatefunctions on R2n.Let wdrindyi2(a) What is dw?(b)What iswn =w Λ...Λw (wedge n times)?(c) Find t*w for the following three cases:(i) t : IR2n-2 IR2n be the embedded submanifold of R2n defined by r" =yn = 0.(i) t : Rn -→ R2n be the submanifold of R2n defined by y' =... = y" = 0.(i) t: Tn → R2n be the submanifold of R2n defined by (r)? + (y')? = 1,1 ≤i<n(d) For any f e Co(IR2n), find a vector field Xy so that ixjw = df.(e) Let X, be as above. Prove: Cx, f = 0 and Cx,w = 0
2 PROBLEM SET 6, PART 1: TENSORS AND DIFFERENTIAL FORMS DUE: DEC. 14 (5) [Maxwell’s equations] The famous Maxwell’s equations in the theory of electromagnetism are of the form ∇ · B⃗ = 0, ∂B⃗ ∂t + ∇ × E⃗ = 0, ∇ · E⃗ = 4πρ, ∂E⃗ ∂t − ∇ × B⃗ = −4πJ, ⃗ where E⃗ = ⟨Ex, Ey, Xz⟩ is the electricity field, B⃗ = ⟨Bx, By, Bz⟩ is the magnetism field, ρ is the charge density, and J⃗ = ⟨Jx, Jy, Jz⟩ is the current density. All functions, e.g. Ex, Ey, Bx, ρ, Jx, are functions of x, y, z and t. Define the Faraday 2-form F on R 4 (with coordinates t, x, y, z) as F = Exdx ∧ dt + Eydy ∧ dt + Ezdz ∧ dt + Bxdy ∧ dz + Bydz ∧ dx + Bzdx ∧ dy, and define the current 1-form to be J = ρdt + Jxdx + Jydy + Jzdz. We also define the Hodge star operators on R 4 (with respect to the Minkowski metric) to be the linear operators ∗ : Ωk (R 4 ) → Ω 4−k (R 4 ), 0 ≤ k ≤ 4, so that on 2-forms one has ∗∗ = −1 and ∗(dx ∧ dt) = dy ∧ dz, ∗(dy ∧ dt) = −dx ∧ dz, ∗(dz ∧ dt) = dx ∧ dy, while on 1-forms one has dt ∧ (∗dt) = dx ∧ (∗dx) = · · · = dt ∧ dx ∧ dy ∧ dz, e.g. ∗dt = dx ∧ dy ∧ dz, ∗dx = −dt ∧ dy ∧ dz, ∗dy = dt ∧ dx ∧ dz etc. Prove: The system of Maxwell equations is equivalent to dF = 0, d ∗ F = 4π ∗ J. (6) [The canonical symplectic form on R 2n ] Denote by (x 1 ,· · ·, xn , y1 ,· · ·, yn ) the coordinate functions on R 2n . Let ω = Pn i=1 dxi∧dyi . (a) What is dω? (b) What is ω n = ω ∧ · · · ∧ ω (wedge n times)? (c) Find ι ∗ω for the following three cases: (i) ι : R 2n−2 ,→ R 2n be the embedded submanifold of R 2n defined by x n = y n = 0. (ii) ι : R n ,→ R 2n be the submanifold of R 2n defined by y 1 = · · · = y n = 0. (iii) ι : T n ,→ R 2n be the submanifold of R 2n defined by (x i ) 2 + (y i ) 2 = 1, 1 ≤ i ≤ n. (d) For any f ∈ C∞(R 2n ), find a vector field Xf so that ιXfω = df. (e) Let Xf be as above. Prove: LXf f = 0 and LXfω = 0
3PROBLEMSET6,PART1:TENSORSANDDIFFERENTIALFORMSDUE: DEC. 14[Involutive distributions revisited](7)(a) Let w=yda-dzbe a smooth 1-form on M=R3.For each p EM, letVp := ker(wr,M). Prove: V is a distribution of rank 2 on M. Is it integrable?(b) Let V be a smooth distribution of rank k on M. Prove: For each p E M, thereexists a neighborhood U of p and smooth 1-forms wi,*-*,wn-k on U such thatfor each q e U, Va = ker(wilaM) n...n ker(wn-klT,M).Definition 0.1. We say a differential form n E i(M) annihilates a smoothdistribution V if n(Xi,...,X,) = 0 for all X, E V[Not required] Let Vbe a smooth distribution of rank k on M, and wi,..,wn-k(c)are as described above. Prove: n E 2i(M) annihilates V if and only if there exists(j - 1)-forms a1,* ,an-k so that n = Er-- w; Aai.[Not required] Prove: a smooth distribution V is involutive if and only if the(d)following criteria issatisfied:If n is a 1-form that annihilates V on an open set U, then dn alsoannihilates V on U.(e)[Not required] Let wi, : ,wn-k be as in (b). Prove: V is involutive if and onlyif there exists 1-forms α, (1 ≤ i, j ≤ n - k) so thatn-kdwi=EwiAag,1<i<n-kj=l(Ref:John Lee,Introduction toSmooth Manifolds page493-495.)
PROBLEM SET 6, PART 1: TENSORS AND DIFFERENTIAL FORMS DUE: DEC. 14 3 (7) [Involutive distributions revisited] (a) Let ω = ydx − dz be a smooth 1-form on M = R 3 . For each p ∈ M, let Vp := ker(ω|TpM). Prove: V is a distribution of rank 2 on M. Is it integrable? (b) Let V be a smooth distribution of rank k on M. Prove: For each p ∈ M, there exists a neighborhood U of p and smooth 1-forms ω1, · · · , ωn−k on U such that for each q ∈ U, Vq = ker(ω1|TqM) ∩ · · · ∩ ker(ωn−k|TqM). Definition 0.1. We say a differential form η ∈ Ω j (M) annihilates a smooth distribution V if η(X1, · · · , Xj ) = 0 for all Xi ∈ V. (c) [Not required] Let V be a smooth distribution of rank k on M, and ω1, · · · , ωn−k are as described above. Prove: η ∈ Ω j (M) annihilates V if and only if there exists (j − 1)-forms α1, · · · , αn−k so that η = Pn−k i=1 ωi ∧ αi . (d) [Not required] Prove: a smooth distribution V is involutive if and only if the following criteria is satisfied: If η is a 1-form that annihilates V on an open set U, then dη also annihilates V on U. (e) [Not required] Let ω1, · · · , ωn−k be as in (b). Prove: V is involutive if and only if there exists 1-forms α i j (1 ≤ i, j ≤ n − k) so that dωi = nX−k j=1 ω j ∧ α i j , 1 ≤ i ≤ n − k. (Ref: John Lee, Introduction to Smooth Manifolds page 493-495.)