r = x 2 + y 2 and h = z y = r sin sin ϕ

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1 Homewok 4. Solutions Calculate the Chistoffel symbols of the canonical flat connection in E 3 in a cylindical coodinates x cos ϕ, y sin ϕ, z h, b spheical coodinates. Fo the case of sphee ty to make calculations at least fo components Γ, Γ θ, Γ ϕ, Γ θθ,..., Γ ϕϕ Remak One can calculate Chistoffel symbols using Levi-Civita Theoem. Thee is a thid way to calculate Chistoffel symbols: It using appoach of Lagangian. This is the easiest way. see the Homewok 6 In cylindical coodinates, ϕ, h we have { x cos ϕ y sin ϕ z h x 2 + y 2 and ϕ actna y x h z We know that in Catesian coodinates all Chistoffel symbols vanish. Hence in cylindical coodinates see in detail lectue notes: Γ 2 x 2 x + 2 y 2 y + 2 z 2 z 0, since Γ ϕ Γ ϕ 2 x ϕ x + 2 y ϕ y + 2 z sin ϕ cos ϕ + sin ϕ cos ϕ 0. ϕ z Γ ϕϕ 2 x 2 ϕ x + 2 y 2 ϕ y + 2 z 2 ϕ z xx y y. Γ ϕ 2 x ϕ 2 x + 2 y ϕ 2 y + 2 z ϕ 2 z 0. Γ ϕ ϕ Γ ϕ ϕ 2 x ϕ ϕ x + 2 y ϕ ϕ y + 2 z ϕ sin ϕ y ϕ z 2 + cos ϕ x 2 All symbols Γ h, Γ h vanish 2 x h... 2 y h... Γ ϕ ϕϕ 2 x ϕ 2 ϕ x + 2 y ϕ 2 ϕ y + 2 z ϕ 2 ϕ z x x 2 y y z h... 0 Fo all symbols Γ h Γ h 2 z all symbols Γ h vanish. Γ h Γ h Γ hh Γ ϕh Γ hϕ Γ ϕ h dots 00 h since x h h y 0 and y. On the othe hand all 2 z vanish. Hence b spheical coodinates { x sin cos ϕ y sin sin ϕ z cos θ x 2 + y 2 + z 2 z θ accos ϕ actan y x x 2 +y 2 +z 2 We aleady know the fast way to calculate Chistoffel symbol using Lagangian of fee paticle and this method wok fo a flat connection since flat connection is a Levi-Civita connection fo Euclidean metic So pefom now bute foce calculations only fo some components. Then late in homewok 6 we will calculate using vey quickly Lagangian of fee paticle.

2 Γ 0 since 2 x i 2 0. Γ θ Γ θ 2 x θ x + 2 y θ y + 2 z θ z cos θ cos ϕx + cos θ sin ϕy sin θ z 0, Γ θθ 2 x 2 θ x + 2 y 2 θ y + 2 z 2 θ z sin θ cos ϕx sin θ sin ϕy cos θ z Γ ϕ Γ ϕ 2 x ϕ x + 2 y ϕ y + 2 z ϕ z sin θ sin ϕx + sin θ cos ϕy 0 and so on... 2 a Conside a connection such that its Chistoffel symbols ae symmetic in a given coodinate system: Γ i km Γi mk. Show that they ae symmetic in an abitay coodinate system. b Show that the Chistoffel symbols of connection ae symmetic in any coodinate system if and only if X Y Y X [X, Y] 0, fo abitay vecto fields X, Y. c Conside fo an abitay connection the following opeation on the vecto fields: and find its popeties. Hence Solution a Let Γ i km Γi mk But Γ i km Γi mk and SX, Y X Y Y X [X, Y]. to pove that Γi k m Γi m k Γ i k m Γ i m k x x x m xk x k x xi x i xi x i x k x k x m x m m. Hence x m x m Γ i km + x k x k Γ i mk + x x i x k x m x. x x i x m x k x Γ i m k xi x i x m x m x k x k Γ i mk + x x i x m x k x xi x i x m x m x k x k Γ i km + x x i x k x m x Γi k m. b The elation X Y Y X [X, Y] 0 holds fo all fields if and only if it holds fo all basic fields. One can easy check it using axioms of connection see the next pat. Conside X x, Y i x then since [ j i, j ] 0 we have that X Y Y X [X, Y] i j j i Γ k ij k Γ k ji k Γ k ij Γ k ji k 0 We see that commutato fo basic fields X Y Y X [X, Y] 0 if and only if Γ k ij Γk ji 0. c One can easy check it by staightfowad calculations o using axioms fo connection that SX, Y is a vecto-valued bilinea fom on vectos. In paticulaly SfX, Y fsx, Y fo an abitay smooth function. Show this just using axioms defining connection: SfX, Y fx Y Y fx [fx, Y] f X Y f Y X Y fx + [Y, fx] 2

3 f X Y f Y X Y fx + Y fx + f [Y, X] f X Y Y X [X, Y] fsx, Y 3 Let, 2 be two diffeent connections. Let Γ i km and 2 Γ i km be the Chistoffel symbols of connections and 2 espectively. a Find the tansfomation law fo the object : Tkm i Γ i km 2 Γ i km unde a change of coodinates. Show that it is tenso. 2 b? Conside an opeation 2 on vecto fields and find its popeties. Chistoffel symbols of both connections tansfom accoding the law. The second tem is the same. Hence it vanishes fo thei diffeence: T i k m Γ i k m 2 Γ i k m xi x i x k x k We see that T i km tansfoms as a tenso of the type 2 x m Γ i x m km 2 Γ i km xi x i. x k x k x m x m T i km b One can do it in invaiant way. Using axioms of connection study T 2 is a vecto field. Conside T X, Y X Y 2X Y Show that T fx, Y ft X, Y fo an abitay smooth function, i.e. it does not possesses deivatives: T fx, Y fx Y 2fX Y X fy + f X Y X fy f 2X Y ft X, Y. 4 a Conside t m Γ i im. Show that the tansfomation law fo t m is t m b Show that this law can be witten as t m xm x m t m + xm x m t m + x m 2 x x k x m x k x. x log det x. Solution. Using tansfomation law we have that xi x i x k x i t m Γ i i m δk i. Hence xi x k x i x i x m x m Γ i km + x x i x i x m x t m Γ i i m xi x k x i x i x m Γ i x km+ x x i m x i x m x δk i x m Γ i x km+ x x i m x i x m x xm t m + x x i x m x i x m x. b When calculating x log det x m x use vey impotant fomula: δ det A det AT A δa δ log det A T A δa. Hence x m x log det x xi 2 x x x i x m 3

4 and we come to tansfomation law fo. To deduce the fomula fo δ det A notice that deta + δa det A det + A δa and use the elation: det + δa + T δa + Oδ 2 A 5 Calculate Chistoffel symbols of the connection induced on the suface M in E n equipped with canonical flat connection. a M S in E 2 b M paabola y x 2 in E 2 c M cylinde,cone,sphee in E 3. d saddle z xy Solution. a Conside pola coodinate on S, x R cos ϕ, y R sin ϕ. to define the connection on S induced by the canonical flat connection on E 2. It suffices to define ϕ ϕ Γϕ ϕϕ ϕ. Recall the geneal ule. Let u α : x i x i u α is embedded suface in Euclidean space E n. The basic u α vectos. To take the induced covaiant deivative X Y fo two vectos X, Y we take a usual deivative of vecto Y along vecto X the deivative with espect to canonical flat connection: in Catesian coodiantes is just usual deivatives of components then we take the component of the answe, since in geneal deivative of vecto Y along vecto X is not to suface: u u α u α u β Γγ αβ u γ canonical α u β 2 u u α u β canonical α u β is just usual deivative in Euclidean space since fo canonical connection all Chistoffel symbols vanish. In the case of -dimensional manifold, cuve it is just ial acceleation!: u u Γu uu u canonical u u Fo the cicle S, x R cos ϕ, y R sin ϕ, in E 2. ϕ ϕ x ϕ x + y ϕ y ϕ d 2 u du 2 R sin ϕ x + R cos ϕ y, ϕ Γϕ ϕϕ ϕ canonic. ϕ ϕ ϕ ϕ R sin ϕ ϕ x + ϕ R cos ϕ R cos ϕ y x R sin ϕ y a since the vecto R cos ϕ x R sin ϕ y is othogonal to the vecto ϕ. In othe wods it means that acceleation is centipetal: ial acceleation equals to zeo. We see that in coodinate ϕ, Γ ϕ ϕϕ 0. Additional wok: Pefom calculation of Chistoffel symbol in steeogaphic coodinate t: x 2tR2 R 2 + t 2, y Rt2 R 2 t 2 + R ,

5 In this case t t t x t t Γt tt t x + y t y 2R 2 R 2 + t 2 2 canonic. t t R 2 t 2 t t 4t t 2 + R 2 2R 2 t + R 2 + t 2 2 2t x + 2R y x + 2tR, x tt In this case tt is not othogonal to velocity: to calculate tt we need to extact its othogonal component: tt tt tt, n t n n t R 2 + t 2 2tRx + t 2 R 2 y, whee t, n 0. Hence tt, n t 4t t 2 + R 2 t + We come to the answe: 4R3 t 2 +R 2 and 2 2R 2 R 2 + t 2 2 2t x + 2R + y t t tt tt tt, n t n 4R 3 t 2 + R 2 2 2t t 2 + R 2 t, i.e.γ t tt 2t t 2 + R 2 R 2 + t 2 2tRx + t 2 R 2 y 2t t 2 + R 2 t Of couse we could calculate the Chistoffel symbol in steeogaphic coodinates just using the fact that we aleady know the Chistoffel symbol in pola coodinates: Γ ϕ ϕϕ 0, hence Γ t tt dt dϕ dϕ dϕ dx dx Γϕ ϕϕ + d2 ϕ dt 2 dt dϕ d2 ϕ dt dt 2 dϕ It is easy to see that t R tan π 4 + ϕ 2, i.e. ϕ 2 actan t R π 2 and Γ t tt d2 ϕ dt 2 d dt dϕ 2 ϕ dt 2 dϕ dt 2t t 2 + R 2. b Fo paabola x t, y t 2 t t t x t x + y t t Γt tt t canonic. t t t t y x + 2t y, tt 2 y To calculate tt we need to extact its othogonal component: tt tt tt, n t n, whee n is an othogonal unit vecto: n, t 0, n, n : n t tt tt tt, n t n 2 y + 4t 2 2t x + y. 2 y, 2t x + y + 4t 2 + 4t 2 2t x + y 5

6 We come to the answe: 4t + 4t 2 x + 8t2 + 4t 2 y 4t + 4t 2 x + 2t y 4t + 4t 2 t t t 4t + 4t 2 t, i.e.γ t tt 4t + 4t 2 Remak Do not be supised by esemblance of the answe to the answe fo cicle in steeogaphic coodinates. c cylinde, cone and sphee a Cylinde{ x a cos ϕ h, ϕ: y a sin ϕ. z h h h Calculate Hence Γ h hh Γϕ hh 0 0 0, ϕ ϕ Hence Γ h hϕ Γh ϕh Γϕ hϕ Γϕ ϕh 0. a sin ϕ a cos ϕ 0 h h Γ h hh h + Γ ϕ hh ϕ h ϕ ϕ h Γ h hϕ h + Γ ϕ hϕ ϕ ϕ ϕ Γ h ϕϕ h + Γ ϕ ϕϕ ϕ 2 h 2 0 since hh since hϕ 0 hϕ 2 a cos ϕ a sin ϕ ϕϕ 0 a cos ϕ since the vecto ϕϕ a sin ϕ is othogonal to the suface of cylinde. Hence Γ h hϕ Γh ϕh Γϕ hϕ 0 Γ ϕ ϕh 0 We see that fo cylinde all Chistoffel symbols in cylindical coodinates vanish. This is not big supise: in cylindical coodinates metic equals dh 2 a 2 dϕ 2. This due to Levi-Ciovita theoem one can see that Levi-Civita which equals to induced connection vanishes since allcoefficients ae constants. Fo cone: see Cousewok poblem 3. Fo the sphee θ, ϕ: θ θ { x R sin θ cos ϕ y R sin θ sin ϕ z R cos θ R cos θ cos ϕ R cos θ sin ϕ, R sin θ, we have ϕ ϕ 0 R sin θ sin ϕ R sin θ cos ϕ R sin θ cos ϕ, n R sin θ sin ϕ 0 R cos θ Calculate θ θ Γ θ θθ θ + Γ ϕ θθ ϕ 2 θ 2 0 6

7 since 2 θ Rn is othogonal to the sphee. Hence Γ θ 2 θθ Γϕ θθ 0. Now calculate θ ϕ Γ θ θϕ θ + Γ ϕ 2 θϕ ϕ. θϕ hence Γ θ θϕ 0, Γϕ θϕ cotan θ hence Now calculate θ ϕ Γ θ θϕ θ + Γ ϕ θϕ ϕ 2 θϕ cotan θ ϕ, ϕ θ Γ θ ϕθ θ + Γ ϕ ϕθ ϕ θ ϕ Γ θ θϕ θ + Γ ϕ θϕ ϕ 2 cotan θ ϕ, i.e. θϕ 2 ϕθ cotan θ ϕ, 2. ϕθ 2 cotan θ ϕ, i.e. θϕ Γ θ ϕθ 0, Γϕ ϕθ cotan θ. Of couse we did not need to pefom these calculations: since is symmetic connection and ϕ θ θ ϕ, i.e. and finally Γ θ ϕθ Γ θ θϕ 0 Γ ϕ ϕθ Γϕ θϕ cotan θ. ϕ ϕ Γ θ ϕϕ θ + Γ ϕ ϕϕ ϕ 2 ϕ 2. 2 R sin θ cos ϕ R cos θ cos ϕ sin θ cos ϕ ϕ 2 sin θ cos θ R sin 2 θ sin θ cos θ θ R sin 2 θn, hence R sin θ sin ϕ 0 Γ θ ϕϕ sin θ cos θ, Γ ϕ ϕϕ 0. ϕ ϕ Γ θ ϕϕ θ + Γ ϕ ϕϕ ϕ R cos θ sin ϕ R sin θ sin θ sin ϕ cos θ 2 sin θ cos θ θ, i.e. ϕϕ { x u Fo saddle z xy: u, v: y v, u u 0, v v 0 It will be useful also z uv v u to use the nomal unit vecto n v +u u. 2 +v 2 Calculate: u u Γ u uu u + Γ v 2 uu v u 2 uu 0 since uu 0. 7

8 Hence Γ u uu Γ v uu 0. Analogously Γ u vv Γ v vv 0 since vv 0. Now calculate Γ u uv, Γ v uv, Γ u vu, Γ v vu: u v v u Γ u uv u + Γ v uv v uv 0 0 Using nomal unit vecto n we have: uv uv uv, n n Γ u uv u + Γ v uv v 0 0 v + u 2 + v 2 u u 2 + v , v u v u + u2 + v 2 + u2 + v 2 v + u 2 + v 2 0 v + u + u 2 + v 2 0 u u v u + u v + u 2 + v 2. Hence Γ u uv Γ u v vu +u 2 +v and Γ v 2 uv Γ v u vu +u 2 +v. 2 Sue one may calculate this connection as Levi-Civita connction of the induced Riemannian metic using explicit Levi-Civita fomula o using method of Lagangian of fee paticle. 8