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Appendix a May a Torus with Null Riemann Curvature Exist on E3? Appendix A May a Torus with Null Riemann Curvature Exist on E3? 1 1 1 We can also give a flat Riemann curvature tensor for T3 S × S T1 T2. First parametrize 1 2 1 2 3 1 2 1 2 T3 by (x ,x ), x ,x ∈ R, such that its coordinates in R are x(x ,x )=x(x +2π, x +2π)and x(x1,x2)=(h(x1)cosx2,h(x1)sinx2,l(x1), (A.1) with h(x1)=R + r cos x1, l(x1)=r sin x1, (A.2) where here R and r are real positive constants and R>r(See Figure A.1). 3 Figure A.1: Some Possible GSS for the Torus T3 living in E where the grid defines the parallelism 1 Recall that T1 and T2 have been defined in Section 1.1. 110 Appendix A. May a Torus with Null Riemann Curvature Exist on E3? 1 2 Now, T T3 has a global coordinate basis {∂/∂x ,∂/∂x }, and the induced metric on T3 is easily found as g = r2dx1 ⊗ dx1 +(R + r cos x1)2dx2 ⊗ dx2 (A.3) Now, let us introduce a connection ∇ on T3 such that j ∇∂/∂xi ∂/∂x =0. (A.4) With respect to this connection, it is immediately verified that its torsion and curvature tensors are null, but the nonmetricity of the connection is non null. Indeed, 1 1 ∇∂/∂xi g = −2(R + r cos x )sinx . (A.5) So, in this case, we have a particular Riemann-Cartan-Weyl GSS (T3, g, ∇)withA =0 , Θ=0, R =0. We can alternatively define the parallelism rule on T3 by introducing a metric compatible con- nection ∇ as follows. First, we introduce the global orthonormal basis {ei} for T T3 with 1 ∂ 1 ∂ e1 = , e2 = , (A.6) r ∂x1 (R + r cos x1) ∂x2 and, then, postulate that ∇ ej ei =0. (A.7) Since r sin x1 [e1, e2]= e2 (A.8) r(R + r cos x1) we immediately get that the torsion operator of ∇ is 1 R sin x τ (e1, e2)=[e1, e2]= e2, (A.9) r(R + r cos x1) 2 2 2 and the non null components of the torsion tensor are T12 and T21 = −T12, 2 2 2 2 T12 = Θ(θ , e1, e2)=−θ (τ(e1, e2)) = −θ (−[e1, e2]) R sin x1 = − . (A.10) r(R + r cos x1) Also, it is trivial to see that the curvature operator is null and that ∇g = 0, and, in this case, we have a particular Riemann-Cartan MCGSS (T3, g, ∇ ), with Θ =0andwithanullcurvature tensor. Appendix B Levi-Civita and Nunes Connections on S˚2 First, consider S2, a sphere of radius R = 1 embedded in R3.Let(x1,x2)=(ϑ, ϕ)0<ϑ<π, 0 <ϕ<2π, be the standard spherical coordinates of S2, which cover all the open set U which is S2 with the exclusion of a semi-circle uniting the north and south poles. Introduce the coordinate bases μ μ {∂μ}, {θ = dx } (B.1) ∗ ∗ a ∗ ∗ for T U and T U. Next, introduce the orthonormal bases {ea}, {θ } for T U and T U with 1 e1 = ∂1, e2 = ∂2, (B.2a) sin x1 θ1 = dx1, θ2 =sinx1dx2. (B.2b) Then, k [ei, ej]=cijek, (B.3) 2 2 1 c12 = −c21 = − cot x . 0 2 Moreover, the metric g ∈ sec T2 S inherited from the ambient Euclidean metric is: g = dx1 ⊗ dx1 +sin2 x1dx2 ⊗ dx2 = θ1 ⊗ θ1 + θ2 ⊗ θ2. (B.4) ρ The Levi-Civita connection D of g has the following non null connections coefficients Γμν in the coordinate basis (just introduced): ∂ ρ ∂ D∂μ ν =Γμν ρ, 2 ϕ 2 ϕ 1 ϑ − Γ21 =Γθϕ =Γ12 =Γϕθ =cotϑ,Γ22 =Γϕϕ = cos ϑ sin ϑ. (B.5) 112 Appendix B. Levi-Civita and Nunes Connections on S˚2 { } k Also, in the basis ea , Dei ej = ωijek and the non null coefficients are: 2 1 ω21 =cotϑ, ω22 = − cot ϑ. (B.6) The torsion and the (Riemann) curvature tensors of D are k k k − − Θ(θ , ei, ej)=θ (τ(ei, ej)) = θ Dej ei Dei ej [ei, ej] , (B.7) a a − − R(ek,θ , ei, ej)=θ Dei Dej Dej Dei D[ei, ej] ek , (B.8) 1 1 2 2 which results in T = 0 and the non null components of R are R121= −R112 = R112= −R112 = −1. Since the Riemann curvature tensor is non null, the parallel transport of a given vector depends on the path to be followed. We say that a vector (say v0) is parallel transported along a generic 3 path R ⊃I → γ(s) ∈ R (say, from A = γ(0) to B = γ(1)) with tangent vector γ∗(s)(atγ(s)) if it determines a vector field V along γ satisfying Dγ∗ V =0, (B.9) 1 and such that V(γ(0)) = v0. When the path is a geodesic of the connection D, i.e., a curve 3 R ⊃I → c(s) ∈ R with tangent vector c∗(s)(atc(s)) satisfying Dc∗ c∗ =0, (B.10) the parallel transported vector along c forms a constant angle with c∗.Indeed,fromEq.(B.9)itis · γ∗ Dγ∗ V = 0. Then, taking into account Eq.(B.10) it follows that g · Dγ∗ (γ∗ V)=0. g i.e., γ∗ · V = constant. This is clearly illustrated in Figure B.1 (adapted from [1]). g Consider next the manifold S˚2 = {S2\north pole + south pole}⊂R3, which is a sphere of 0 2 unitary radius R = 1, but this time excluding the north and south poles. Let again g ∈ sec T2 S˚ be the metric field on S˚2 inherited from the ambient space R3 and introduce on S˚2 the Nunes (or navigator) connection ∇ defined by the following parallel transport rule: a vector at an arbitrary point of S˚2 is parallel transported along a curve γ, if it determines a vector field on γ, such that, at any point of γ the angle between the transported vector and the vector tangent to the latitude line passing through that point is constant during the transport. This is clearly illustrated in Figure B.2. And to distinguish the Nunes transport from the Levi-Civita transport, we also ask the reader to study the caption of Figure B.1. We recall that, from the calculation of the Riemann tensor R, it follows that the structures ˚2 2 (S , g,D,τg)andalso(S , g,D,τg) are Riemann spaces of constant curvature. We now show that 1We recall that a geodesic of D also determines the minimal distance (as given by the metric g) between any two points on S2. Appendix B. Levi-Civita and Nunes Connections on S˚2 113 V a 2 r V a 1 s a a V0 p q Figure B.1: Levi-Civita and Nunes transport of a vector v0 starting at p through the paths psr and pqr. Levi-Civita transport through psr leads to v1 whereas Nunes transport leads to v2.Alongpqr both Levi-Civita and Nunes transport agree and leads to v2. ˚2 2 the structure (S , g, ∇,τg)isateleparallel space , with zero Riemann curvature tensor, but non zero torsion tensor. Indeed, from Figure B.2, it is clear that (a) if a vector is transported along the infinitesimal quadrilateral pqrs composed of latitudes and longitudes, first starting from p along pqr,andthen starting from p along psr, the parallel transported vectors that result in both cases will coincide (study also the caption of Figure B.2). 2 Now, the vector fields e1 and e2 in Eq.(B.2a) define a basis for each point p of TpS˚ and ∇ is clearly characterized by: ∇ ej ei =0. (B.11) The components of the curvature operator are: a a ∇ ∇ −∇ ∇ −∇ R(ek,θ , ei, ej)=θ ei ej ej ei [ei,ej] ek =0, (B.12) andthetorsionoperatoris ∇ −∇ − τ(ei, ej)= ej ei ei ej [ei, ej] =[ei, ej], (B.13) 2 2 2 which gives for the components of the torsion tensor, T12 = −T12 =cotϑ. It follows that S˚ , ˚2 considered as part of the structure (S , g, ∇,τg), is flat (but has torsion)! If you need more details to grasp this last result, consider Figure B.2 (b), which shows the standard parametrization of the points p, q, r, s in terms of the spherical coordinates introduced above. According to the geometrical meaning of torsion, its value at a given point is determined 2As recalled in Section 1, a teleparallel manifold M is characterized by the existence of global vector fields, which 2 is a basis for TxM for any x ∈ M. The reason for considering S˚ for introducing the Nunes connection is that, as is well known (see, e.g., [2]), S2 does not admit a continuous vector field that is nonnull at all its points. 114 Appendix B. Levi-Civita and Nunes Connections on S˚2 a b (J – dJ, j) (J – dJ, j + dj) s r r = 2 r 1 a s r p q p q (J,j) (J,j + dj) Figure B.2: Characterization of the Nunes connection. 3 by calculating the difference between the (infinitesimal) vectors pr1and pr2. If the vector pq is 1 transported along ps, one gets (recalling that R = 1) the vector v = sr1 such that |g(v, v)| 2 = sin ϑϕ. On the other hand, if the vector ps is transported along pq, one gets the vector qr2 = qr. Let w = sr. Then, |g(w, w)| =sin(ϑ −ϑ)ϕ sin ϑϕ − cos ϑϑϕ, (B.14) Also, 1 ∂ u = r1r2 = −u( ), u = |g(u, u)| =cosϑϑϕ.
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