UNIVERSITY OF DUBLIN

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1 UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429 & MA4448: Differential Geometry & General Relativity day???, May??? RDS??? Dr. Sergey Frolov and Dr. Andrei Parnachev FOR THOSE TAKING MA3429 AND MA4448, ATTEMPT SIX QUESTIONS: THREE FROM PART I AND THREE FROM PART II. PLEASE SUBMIT ANSWERS TO PART I AND PART II ON SEPARATE ANSWER BOOKS. FOR THOSE TAKING MA3429 ONLY, ATTEMPT THREE QUESTIONS FROM PART I. Log tables are available from the invigilators, if required. Non-programmable calculators are permitted for this examination, please indicate the make and model of your calculator on each answer book used.

2 Page 2 of 10 Part I: Module MA3429 Credit will be given for the best three questions answered. Each question is worth 20 marks. Sample I 1. (a) Let (M, g) be a manifold endowed with a metric. Let be the covariant derivative: j ξ i = j ξ i + ξ k Γ i kj, j ξ i = j ξ i ξ k Γ k ij. Show that a torsion-free connection that is compatible with the metric ( k g ij = 0) defines a unique connection, known as the Levi-Civita (or metric) connection. Obtain an expression for the Levi-Civita connection coefficients Γ k ij. [7 marks] (b) The line-element on the unit 2-sphere in spherical coordinates is given by ds 2 = dθ 2 + sin 2 θdφ 2. i. Calculate the components of the Levi-Civita connection. [3 marks] ii. A covector field on this 2-sphere with components A m in the coordinate system (θ, φ) is taken with initial components (X, Y ) and is carried by parallel transport with respect to the metric connection along an arc of the circle θ = α = const of length φ sin α. Show that the components A m attain final values A θ = X cos(φ cos α) + Y csc α sin(φ cos α), A φ = X sin α sin(φ cos α) + Y cos(φ cos α). [10 marks] 2. The Riemann curvature tensor R i jkl is defined by means of the formula [ k, l ]ξ i = R i jkl ξ j + T j kl jξ i. (1)

3 Page 3 of 10 (a) Use the formula to find explicit formulae for Rjkl i and T j kl in terms of the connection coefficients Γ i jk. What is T j kl? (b) Let the manifold is endowed with a metric, and let the connection be torsion-free and compatible with the metric. List the algebraic symmetries the components of the Riemann tensor satisfy. How many linearly independent components does R ijkl have in arbitrary dimensions? Prove the formula. (c) By computing the curvature tensor, determine whether the space-time described by the line-element ds 2 = du 2 u 2 dv 2 (2) is flat. (d) Prove Bianchi s identities for the curvature tensor of a symmetric connection m R n ikl + l R n imk + k R n ilm = A hyperbolic n-space, H n, is one sheet of the hyperboloid of two sheets realised as a surface n x x 2 i = 1, x 0 > 0, i=1 in the Minkowski space R 1,n with the standard metric ds 2 = dx n dx 2 i. i=1 (a) Plot H 1 in R 2 (where it is a curve) by taking the x 0 -axis as a vertical one. Plot a straight line through the point (x 0, x 1 ) = ( 1, 0) and a point of H 1. [1 mark] (b) Prove that H n is topologically an n-dimensional disc by considering the stereographic projection from the point (x 0, x 1,..., x n ) = ( 1, 0,..., 0) onto the plane x 0 = 0, i.e. by drawing a straight line through the point ( 1, 0,..., 0) and a point of H n. Let u i be the coordinates of the point of intersection of the line with the plane x 0 = 0. Express x 0, x 1,..., x n of H n in terms of u i. The stereographic

4 Page 4 of 10 coordinates u i of the n-dimensional disc provide the Poincaré disc model of H n. What is the radius of the disc? (c) The metric on the hyperbolic space H n is the metric induced from the ambient Minkowski metric. Find the metric, ds 2 H n, on Hn and the volume form, Ω H n, of H n in terms of the coordinates u i. (d) Use the coordinates u i to calculate the components of the Levi-Civita connection of H n. (e) Calculate the components of the Riemann tensor, Ricci tensor, and scalar curvature of H n. 4. A smooth non-singular surface M of dimension m in n-dimensional Euclidean space R n is given by a set of n m equations f i (x 1,..., x n ) = 0, i = 1,..., n m, (3) where f i are smooth functions, and for all x the matrix ( f i x j ) has rank n m. (a) Define a differentiable n-dimensional manifold. (b) Prove that M is a smooth m-dimensional manifold. (c) What is an orientable manifold? Prove that M is orientable. [2 marks] [8 marks] (d) Let ξ and η be two vector fields on R n. Define the commutator (Lie Bracket) of the vector fields ξ and η. Show that if vector fields ξ and η are both tangent to M, then their commutator is also tangent to the surface. What does it say about the linear space of vector fields tangent to a surface? [6 marks]

5 Page 5 of 10 Sample II 1. (a) Let (M, g) be a manifold endowed with a metric. Let be the covariant derivative: j ξ i = j ξ i + ξ k Γ i kj, j ξ i = j ξ i ξ k Γ k ij. Show that a torsion-free connection that is compatible with the metric ( k g ij = 0) defines a unique connection, known as the Levi-Civita (or metric) connection. Obtain an expression for the Levi-Civita connection coefficients Γ k ij. [7 marks] (b) Given that Γ i jk is a symmetric connection compatible with the metric, show that g kl Γ i kl = 1 g x k ( g g ik ). Γ i ki = 1 g 2g x. k [3 marks] (c) Let M be a surface in Euclidean n-space R n, let π be the linear operator which projects R n orthogonally onto the tangent space to M at an arbitrary fixed point of M, and let X, Y be vector fields in R n tangent to the surface M. Show that the connection on M compatible with the induced metric on M satisfies X Y = π(x k Y x k ). [10 marks] 2. The Riemann curvature tensor R i jkl is defined by means of the formula [ k, l ]ξ i = R i jkl ξ j + T j kl jξ i. (4) (a) Use the formula to find explicit formulae for Rjkl i and T j kl in terms of the connection coefficients Γ i jk. What is T j kl? (b) Let the manifold is endowed with a metric, and let the connection be torsion-free and compatible with the metric. List the algebraic symmetries the components of the Riemann tensor satisfy. How many linearly independent components does R ijkl have in arbitrary dimensions?

6 Page 6 of 10 (c) Prove that in 3D, the curvature tensor is given by R abcd = R ac g bd R ad g bc + g ac R bd g ad R bc 1 2 R(g acg bd g ad g bc ). (d) Use Bianchi s identities for the curvature tensor of a symmetric connection to show that if dim (M) 3 and R abcd = 1 2 K(g acg bd g ad g bc ), where K is a scalar function, then K must be a constant. 3. Let an n-dimensional sphere S n of radius 1 be realised as a surface n+1 x 2 i = 1, i=1 in the Euclidean space R n+1 with the standard metric n+1 ds 2 = dx 2 i. i=1 (a) Prove that S n is a manifold by introducing two subsets U N = S n (0,..., 0, 1) and U S = S n (0,..., 0, 1), and considering the stereographic projections from the north and south poles onto the plane x n+1 = 0. (b) The metric on the n-sphere is the metric induced from the ambient Euclidean metric. Find the metric, dω 2 n, on the n-sphere and the volume form, Ω S n, of S n in terms of the stereographic coordinates on U N. (c) Use stereographic coordinates on U N Civita connection of S n. to calculate the components of the Levi- (d) Calculate the components of the Riemann tensor, Ricci tensor, and scalar curvature of S n.

7 Page 7 of Consider the set SL(2, R) of all transformations of the real line of the form x x + 2πa + 1 i ln 1 ze ix 1 ze ix, where x R, a R, z C, z < 1 and ln is the main branch of the natural logarithmic function, i.e. the continuous branch determined by ln 1 = 0. (a) Show that SL(2, R) is a connected 3-dimensional Lie group. (b) Calculate the Lie algebra sl(2, R) of the Lie group SL(2, R). [10 marks] (c) Calculate the Lie algebra of SL(2, R) and show that it is isomorphic to sl(2, R). [6 marks]

8 Page 8 of 10 Sample III 1. (a) Let (M, g) be a manifold endowed with a metric. Let be the covariant derivative: j ξ i = j ξ i + ξ k Γ i kj, j ξ i = j ξ i ξ k Γ k ij. Show that a torsion-free connection that is compatible with the metric ( k g ij = 0) defines a unique connection, known as the Levi-Civita (or metric) connection. Obtain an expression for the Levi-Civita connection coefficients Γ k ij. [7 marks] (b) Consider the 2-dimensional de Sitter space with the metric ds 2 = dt 2 + cosh 2 tdθ 2. i. Calculate the components of the metric connection. [3 marks] ii. Show that the geodesics x m (τ) satisfy θ cosh 2 t = K, ṫ 2 cosh 2 t = K 2 L cosh 2 t, where a dot denotes differentiation with respect to τ, and K and L are constants that should be identified. iii. The first equation above allows one to choose θ as a parameter of the geodesics. Then the second equation takes the form t 2 K 2 cosh 2 t = K2 L cosh 2 t, where denotes differentiation with respect to θ. variable t: t = f(v) which brings the equation to the form v 2 = M 2 v 2, Find the change of the where the constant M should be identified. iv. Determine and sketch the geodesics (timelike, null and spacelike) passing through (t, θ) = (0, 0). [2 marks]

9 Page 9 of The Riemann curvature tensor R i jkl is defined by means of the formula [ k, l ]ξ i = R i jkl ξ j + T j kl jξ i. (5) (a) Use the formula to find explicit formulae for Rjkl i and T j kl in terms of the connection coefficients Γ i jk. What is T j kl? (b) Let the manifold is endowed with a metric, and let the connection be torsion-free and compatible with the metric. List the algebraic symmetries the components of the Riemann tensor satisfy. How many linearly independent components does R ijkl have in arbitrary dimensions? (c) Let ξ(ɛ) be the result of parallel-transporting a vector ξ = (ξ k ) around the boundary of a square of side ɛ with its sides parallel to the coordinates x i - and x j -axes. Prove that lim ξk (ɛ) ξ k = R ɛ 0 ɛ lijξ k l. 2 Assume for simplicity that the connection is torsion-free. [10 marks] 3. Let M be a manifold, and let G be a Lie group. (a) Define the left action of G on M, define the transitive action of G on M, and define a homogeneous space of G. [2 mark] (b) Define the isotropy group H x of the point x M, and prove that all isotropy groups H x of points x of a homogeneous space are isomorphic. (c) Prove that there is a one-to-one correspondence between the points of a homogeneous space M of a group G, and the left cosets gh of H in G, where H is the isotropy group and G is assumed to act on the left. (d) Stiefel manifolds. For each n, k the Stiefel manifold V n,k has as its points all orthonormal frames x = (e 1,..., e k ) of k vectors in R n. i. Prove that the Stiefel manifold V n,k is a nonsingular surface of dimension nk k(k + 1)/2 in R nk. ii. Define the left action of O(n) on V n,k and prove that it is transitive. [2 marks]

10 Page 10 of 10 iii. Determine the isotropy group of V n,k. iv. Which manifolds are V n,n, V n,n 1, and V n,1 diffeomorphic to? [2 marks] [1 marks] 4. Let ξ be a vector field on M. (a) Define the flow F t generated by the vector field. (b) Define the Lie derivative of a tensor (T α 1 α p β 1 β q ) along a vector field ξ. [2 marks] (c) Derive an explicit formula for the Lie derivative of a tensor (T α 1 α p β 1 β q ) along a vector field ξ. [9 marks] (d) Specialise the formula for the Lie derivative to scalars, vectors, covectors, and tensors of type (0, 2). [2 marks] (e) Calculate the Lie derivative of the volume element Ω = g ɛ α1 α n, and express it in terms of the strain tensor. [3 marks] c UNIVERSITY OF DUBLIN 2015