Two dimensional manifolds

Transcription

1 Two dimensional manifolds We are given a real two-dimensional manifold, M. A point of M is denoted X and local coordinates are X (x, y) R 2. If we use different local coordinates, (x, y ), we have x f(x, y) and y g(x, y) where f and g are smooth and are such that the Jacobian determinant det(j) f x g y g x f y is non-zero. Here the Jacobian matrix is: J f x f y g x g y. For example on the plane M R 2 we might use affine coordinates (x, y) which are valid globally, or polar co-ordinates (x, y ) (r, θ) which have r > 0 and π < θ < π and which are co-ordinates for R 2 with the non-positive x-axis deleted. Then the transition formulas are: r x 2 + y 2, θ π 2, if x 0 and y > 0, θ π, if x 0 and y < 0, 2 Then the Jacobian matrix is: θ arctan(yx 1 ), if x > 0, θ arctan(yx 1 ) + π, if x < 0 and y > 0, θ arctan(yx 1 ) π, if x < 0 and y < 0. J x x 2 +y 2 y x 2 +y 2 y x 2 +y 2 x x 2 +y 2 cos(θ) sin(θ) sin(θ) r cos(θ) r Then det(j) (x 2 + y 2 ) 1 2 r 1, so is everywhere non-zero. The inverse transformation is: x r cos(θ), y r sin(θ). The Jacobian of this transformation is: J cos(θ) r sin(θ) sin(θ) r cos(θ) Then det(j ) r 0. Also the matrix J is the inverse of the matrix J: JJ I J J, where I is the 2 2 identity matrix, as is easily checked. 1..

2 The tangent bundle The tangent bundle TM is the collection of all tangent directions at each point. Intuitively these are the possible velocity vectors of particles moving along curves in the manifold. At each point X, the collection of tangent vectors forms a two dimensional real vector space called T X. A point of the tangent bundle is denoted (X, V ) with V T X a tangent vector at X. The deleted tangent bundle, TM is the collection of all non-zero tangent vectors at every point, so is all pairs (X, V ) TM with V 0. The projective tangent bundle PM is the space of tangent directions at each point. If V 0, its direction is denoted [V ]. If e 1 and e 2 are basis vectors for T X at a point X, each vector at X is expressible as V pe 1 + qe 2, for unique real numbers p and q. Then, when V 0, the direction [V ] is represented by the ratio p : q, or if we write p r cos(θ) and q r sin(θ), with r > 0, then the angle θ represents the direction [V ]. Here we think of θ as defined modulo 2π, so, in particular, opposite directions have values of θ differing by π. So PM is a circle bundle over the space M. There is a natural map π which takes TM to PM, which assigns to each non-zero vector, its direction. If we use local coordinates (x, y) for M, the corresponding local co-ordinates for TM are (x, y, dx, dy), such that if we move to a new coordinate patch (x, y, dx, dy ), we have: x f(x, y), y g(x, y), dx f x dx + f y dy, dy g x dx + g y dy. Note that the Jacobian matrix here is: f x f y 0 0 g x g y 0 0 f xx dx + f xy dy f xy dx + f yy dy f x f y g xx dx + g xy dy g xy dx + g yy dy g x g y This has non-zero determinant (f x g y g x f y ) 2 J(f, g) 2, so the transformation is invertible, with the inverse transformation homogeneous linear in the variables dx and dy. In view of these formulas, we also write the tangent bundle as all pairs (X, dx) with dx a tangent vector at X M. 2

3 Riemann structures We want to associate to our space a notion of length for the tangent vectors. We require that it scale appropriately so, for example, if we double a vector, we double the length. Also the length of a vector and its negative will be the same. We make the appropriate definitions: A length function on TM is a positive function L(X, dx) such that L(X, tdx) t L(X, dx), for any t 0. A Riemann structure for M is a lift of PM into TM, so is a map σ : PM TM, such that π σ id PM. Every length function defines a lift σ : PM TM, which maps the point (X, [dx]) of PM to the unique point (X, dx) where L(X, dx) 1, so σ is well-defined wherever L > 0. Conversely, every lift σ gives rise to a length function, whose value at a point (X, dx) is given by the formula: L(X, dx) t where dx tσ(x, [dx]). Henceforth we assume given a smooth Riemann structure for M. We now use it to define the length of a curve in the space. So let X(t) be a smooth curve in the manifold M from a point P M with t a to a point Q M with t b. Let V (t) 0 be its corresponding tangent vector. Then we can define the length S of the curve X(t) as the integral S b L(X(t), dx(t)). Under a a change of parameter t h(t) with h (t) > 0, we have: dx(t) dx(t ), so the length S is invariant. By adding the pieces, we can extend the notion of length of a curve to curves that are piecewise smooth. In local co-ordinates, we have L(X, dx) L(x, y, dx, dy) and the integral is: S tb ta L(x(t), y(t), dx(t), dy(t)). If the curve is a graph over the x-axis, then using the fact that L is homogeneous, we can rewrite this with x as the curve parameter: xx1 S L(x, y(x), 1, y (x)) dx L(x, y, p)dx. xx 0 yy(x), py (x) Here the prime denotes the derivative with respect to x. Also we have put L(x, y, p) L(x, y, 1, p) and we have taken x to increase from x x 0 to x x 1, so dx dx. 3

4 The geodesic equations A geodesic from P to Q is a curve such that if we perturb the curve slightly, the length does not change to first order, so it is a critical point of the length functional. We consider the geodesic equations for the length functional: S Q P L(x, y, p)dx. We replace the curve y y(x) by a family of nearby curves y y(x)+tδy(x), where δy(x) vanishes at x x 0 and x x 1. The rate of change of S, with respect to t is: x1 x 0 ds Q dt P x1 x 0 ( L y d ) dx (L p) x1 x 0 d dt (L(x, y + tδy, y + tδy )dx ( ) d L y δy + L p dx (δy) dx δy dx (here we used integration by parts) (L y L px L py p L pp q) δy dx. Here we have put q y, the second derivative of y with respect to x. Then the curve y(x) is a critical point of the length functional if ds dt vanishes for any choice of the perturbation δy. This entails that the coefficient of δy in the integral be zero. This gives the geodesic equations for the Riemann structure L: L y L px L py p L pp q 0. Our final requirement for a proper Riemann structure is that the second derivative L pp be everywhere non-vanishing. Then the differential equation for a geodesic for the Riemann structure defined by L(x, y, p) is the second order (generally non-linear) ordinary differential equation: L pp y L y L px L py p. By the standard properties of second order differential equations, there is, at least locally, a unique solution curve given an initial point X and an initial tangent direction [V ], so here by initial values for the triple (x, y, y ). 4

5 The geodesic equations in the energy formalism We study an energy integral: E b a L(x i, v i )dt, x i x i (t), v i ẋ i dxi dt. We make a variation x i (t) x i (t) + sy i (t) and require that de ds s0 0. We require that the variation keep the end-points fixed, so we have y i (a) y i (b) 0. Then we have: de ds s0 b a ( b a b a ) d ds L(xi (t) + sy i (t), v i (t) + sẏ i (t))dt s0 (L x j(x i, v i )y j + L v j(x i, v i )ẏ j )dt ( y j L x j(x i, v i ) + d dt (yj L v j(x i, v i )) y j d ) dt L v j(xi, v i ) dt [ y j (t)l v j(x i, v i ) ] b a + b b a a ( (y j L x j(x i, v i ) d )) dt L v j(xi, v i ) dt ( (y j L x j(x i, v i ) d )) dt L v j(xi, v i ) dt. Here the index j is summed over. Since y i (t) is essentially arbitrary, the critical point condition is the condition: L x j(x i, v i ) d dt L v j(xi, v i ). If we write this out more explicitly, using the chain rule, the condition is: L x j v i L x i v j + v i L vi v j. Here the index i is summed over. With the equation v i x i this is a system of second order equations for x i (t). We always assume here that the hessian matrix L vi v j is invertible, so that we can solve for the unknown v j. The resulting equations have unique solutions, locally, for x i (t), given the initial position x i (0) and initial velocity v i (0) x i (0). These curves are called the geodesics of the energy functional. 5

6 Finally we can calculate the rate of change of L along a geodesic curve. We have: dl dt vi L x i + v i L v i. Suppose that L is homogeneous of degree k in v i. Then we have: v j L v j kl. Differentiating both sides of this equation with respect to v i, we get, since v iv j δ j i : v j L v iv j + L v i kl v i, vj L v iv j (k 1)L v i. If instead we differentiate with respect to x i, we get: v j L x i v j kl x i. Contracting this equation with v i and using the geodesic equations, we get: So we get: (k 1) v i L v i v j vi L v i v j v j L x j v j v i L x i v j (1 k)vj L x j. (k 1) dl dt 0. So in the case that L is homogeneous of degree k in v i, we have that L is a constant λ, along the trajectory of the geodesic, unless k 1. Then the energy of the trajectory is just λ(b a). 6

7 The metric case Consider the case that L(x k, v k ) 2 1 g ij (x k )v i v j, where the indices i and j are summed over (henceforth we use the Einstein summation convention) and the coefficient metric tensor g ij is symmetric and invertible. Denote by g ij the inverse matrix of g ij, so we have g ij g jk δ k i. Then the critical equations for the energy are: L x k v i L x i v j + v i L vi v j, 2 1 v p v q k g pq v i g ik + v p v q p g qk, v i g ik v p v q (2 1 k g pq (p g q)k ). Summarizing, the geodesic equations are the differential system: ẋ i v i, vi + v p v q Γpq i 0, Γ i pq 1 2 ( pg qk + q g pk k g pq ). The quantities Γpq i are called the Christoffel symbols of the second kind. Note that, computationally, the main difficulty in studying these equations is the need to invert the metric. Now for the metric case, the function L is homogeneous of degree two in the velocities, so we have that L is a constant along the trajectory. In the case of general relativity, where g ij has Lorentzian signature (1, n 1) (so n 3 for standard relativity), the geodesic is called timelike if L > 0, null if L 0 and space-like if L < 0. The timelike trajectories are the trajectories of free particles, so, for example to a good approximation the path of the Earth around the Sun is a timelike geodesic. The null geodesic trajectories in space-time represent the trajectories of light rays. 7

8 The two-dimensional case In two dimensions, we have the matrix: g E F F G, det(g) EG F 2 0. Here E(x, y), F (x, y) and G(x, y) are smooth functions. Note that the metric is Riemannian if and only if EG F 2 > 0 and Lorentzian if and only if EG F 2 < 0. Put v i (p, q). Then we have: L 2 1 (g ij v i v j ) 2 1 (Ep 2 + 2F pq + Gq 2 ), L x pl xp + ql yp + ṗl pp + ql pq pl xp + ql yp + ṗe + qf, L y pl xq + ql yq + ṗl pq + ql qq pl xp + ql yp + ṗf + qg This gives us the matrix equations: E F F G ṗ q L x pl xp ql yp L y pl xq ql yq 1 2 E x p 2 + 2F x pq + G x q 2 2E x p 2 2F x pq 2E y pq 2F y q 2 E y p 2 + 2F y pq + G y q 2 2F x p 2 2G x pq 2F y pq 2G y q 2 E F F G ṗ q 1 2 E x p 2 2E y pq + (G x 2F y )q 2 (E y 2F x )p 2 2G x pq G y q 2 If we want to we can solve explicitly for ṗ and q. Put EF G 2 0. ṗ q (2 ) 1 G F F E E x p 2 2E y pq + (G x 2F y )q 2 (E y 2F x )p 2 2G x pq G y q 2 (2 1 )p 2 GE x F E y + 2F F x F E x + EE y 2EF x + 1 pq GE y + F G x F E y EG x + (2 ) 1 q 2 GG x 2GF y + F G y F G x + 2F F y EG y. 8

9 We can now check by direct calculation that L is preserved along the geodesics: dl dt (Ep+F q)ṗ+(f p+gq) q+2 1 p(e x p 2 +2F x pq+g x q 2 )+2 1 q(e y p 2 +2F y pq+g y q 2 ) 2 1 p q p q E F F G ṗ q +2 1 p(e x p 2 +2F x pq+g x q 2 )+2 1 q(e y p 2 +2F y pq+g y q 2 ) E x p 2 2E y pq + (G x 2F y )q 2 (E y 2F x )p 2 2G x pq G y q p(e x p 2 +2F x pq+g x q 2 )+2 1 q(e y p 2 +2F y pq+g y q 2 ) 2 1 p( E x p 2 2E y pq + (G x 2F y )q 2 + E x p 2 + 2F x pq + G x q 2 ) +2 1 q((e y 2F x )p 2 2G x pq G y q 2 + E y p 2 + 2F y pq + G y q 2 ) p((f x E y )pq + (G x F y )q 2 ) + q((e y F x )p 2 + (F y G x )pq) 0. 9

10 Parallel transport We define parallel transport of a vector w i along a vector field v j : D ei e j Γ k ije k. Here D is the connection and e i is the i-th co-ordinate vector. Then if we write v v i e i and w w j e j, we have: D v w v i D ei w j e j (v(w k ) + v i w j Γ k ij)e k. So w j is parallelly transported along v if we have: For a metric, we have: v(w k ) + v i w j Γ k ij 0. Γ k ij 1 2 gkm ( j g im + i g jm m g ij. v j j w i v j Γ i jkw k. If v j is tangent to a curve, x i (t), so v j ẋ j, this equation is: d dt wj ẋ j Γ i jkw k. This is a first order system, the solution is specified by a choice of w j at an initial value of t. More generally we can define the covariant derivative D v of w j in the direction of v i : D v w j v j j w i v j Γ i jkw k. The choice of the coefficients Γ i jk fixes the particular covariant derivative. The totality of all derivatives in all possible directions is called the connection. Then a vector w i is parallel propagated along v if D v w 0: its covariant deivativ along v is zero. If a curve γ goes from a point P at t a to a point Q at t b, then parallel propagation along γ gives a map from the tangent vector given by w i (a) at the initial point P to the tangent vector given by w i (b) at the final 10

11 point Q. Since the propagation equation for w i along the curve is homogenous linear in w i, the map from vectors at P to vectors at Q is a linear isomorphism. This map is called the holonomy map γ Q P ; it depends only on the connection and on the choice of the curve connecting P to Q. For any three points P, Q and R we have the composition formula: δ R Q γ Q P (δγ)r P. Here γ is the curve from P to Q, δ is the curve from Q to R, γ Q P and δr Q are the corresponding holonomy maps. Also δγ is the path from P to R given by concatenating the paths γ and δ, and (δγ) R P is the corresponding holonomy map. In particular, we can consider the holonomy around a closed curve γ from a point P to itself. This is then an isomorphism of the tangent space at P to itself. The collection of all such isomorphism as γ varies, forms a group called the holonomy group of the connection. If we move to a different point Q connected to P, the holonomy changes by a group conjugation, so the holonomy groups are isomorphic. 11

12 Surfaces of revolution ds 2 dz 2 + dr 2 + r 2 dθ 2. Parametrize by r f(u), z g(u), θ v. We have X u (f (u), 0, g (u)) and X v [0, 1, 0]. A vector perpendicular to this is ( g (u), 0, f (u)). This is unit if f (u) 2 + g (u) 2 1. Then we have: ds 2 du 2 + f 2 (u)dv 2. Then we may take u x, v θ y and: Then the geodesic equation is: L 1 + f 2 (x)p 2. ql pp L px. But we have: L p f 2 (1 + f 2 p 2 ) 1 2, L pp pf 4 (1 + f 2 p 2 ) 3 2 L px 2ff (1 + f 2 p 2 ) 1 2 f 3 f p 2 (1 + f 2 p 2 ) 3 2 ff (1 + f 2 p 2 ) 3 2 (2 + f 2 p 2 ) Written out this is: pq f 3 f (2 + f 2 p 2 ). p dv du, 2p dp df 2(f 1 dp ) du 2f 3 (2 + f 2 p 2 ), d(p 2 ) df 4f 3 + 2f 1 p 2, p 2 Af 2 f 2, The curve is z g(u), r f(u) and θ θ(u). The tangent is: f (u) r + g (u) z + p θ. 12

13 Geodesics in the plane and in the sphere For the plane, we have L(x, y, dx, dy) dx 2 + dy 2. This gives L(x, y, p) 1 + p 2. Then L y, L px and L py all vanish, but we have L pp (1 + p 2 ) 3 2, so the differential equation for the geodesics is just y 0, giving the geodesics as the collection of all straight lines in the plane. Next consider the metric of the sphere, using stereographic co-ordinates, where the point X of the sphere is: We have: This has: L pp X [X, Y, Z] 1 x 2 + y [2x, 2x, x2 + y 2 1]. ds dx.dx 2 dx 2 + dy x 2 + y 2. L p x 2 + y 2, 2yL L y 1 + x 2 + y, L 2xL 2 x 1 + x 2 + y, L 2 p pl 1 + p, 2 L (1 + p 2 ), L 2xpL 2 xp (1 + p 2 )(1 + x 2 + y 2 ), L 2ypL yp (1 + p 2 )(1 + x 2 + y 2 ) (1 + p 2 )(1 + x 2 + y 2 )L 1 (L y L xp pl yp ) 2y(1 + p 2 ) + 2xp + 2yp 2 2y + 2xp q(1 + x2 + y 2 ) 1 + p 2. So the differential equation for the geodesics is: y 2( y + xy )(1 + (y ) 2 ) 1 + x 2 + y 2. To solve this we go to polar coordinates: ds 2 dx 2 + dy 2 2 dr2 + r 2 dθ 2 du2 + dθ 2, r e u. 1 + x 2 + y r 2 cosh(u) 13

14 This is a special case of the family of Riemann functions: L(x, y, p) 1 + p 2. f(x) Here we have x u, f(x) cosh(u) r + r 1 2 We have L y 0. Also we have: L 1 L pp So the required differential equation is: d(ln(f(x))) and p dy dx dθ du. 1 (1 + p 2 ), 2 L 1 L px f (x)p f(x)(1 + p 2 ), dp p(1 + p 2 ) dp p p 1 + p 2 A 1 f(x), f(x) dp dx f (x)p(1 + p 2 ), y + B pdp 1 + p 2 d ( ln(p) 1 ) 2 ln(1 + p2 ), p 2 (A 2 f 2 (x)) f 2 (x), f(x) A2 f 2 (x) dx. Putting in f(x) cosh(x), we get: cosh(x)dx y + B dx A 2 cosh 2 (x) d(sinh(x)) A 2 1 sinh 2 (x) ( ) sinh(x) dx arcsin arcsin A2 1 Here we have put A cosh(c) 1, with C 0. Also we have y θ and e x r, so we get: 2 sinh(c) sin(θ + B) 2 sinh(x) e x e x r r 1. ( ) sinh(x). sinh(c) 14

15 Multiplying by r and converting to x, y coordinates with r cos(θ) x and r sin(θ) y and r 2 x 2 + y 2, we get: r 2 1 2r sinh(c) sin(θ + B) 2 sinh(c) (r sin(θ) cos(b) + r cos(θ) sin(b)) 1 x 2 + y 2 2 sinh(c) (y cos(b) + x sin(b)), 1 (x sinh(c) sin(b)) 2 + (y sinh(c) cos(b)) 2 sinh 2 (C), (x sinh(c) sin(b)) 2 + (y sinh(c) cos(b)) 2 cosh 2 (C). This is the equation of a circle center sinh(c)(sin(b), cos(b)) and radius cosh(c). Note that the origin is always inside the circle. Finally we parametrize the circle: x sinh(c) sin(b) + cosh(c) cos(t), y sinh(c) cos(b) + cosh(c) sin(t). In particular we have, as shown above: x 2 + y sinh(c) sin(b)x + 2 sinh(c) cos(b)y. Then we map back to the sphere: X [X, Y, Z] We see that the image points obey: 1 (x 2 + y 2 + 1) [2x, 2y, x2 + y 2 1], Z sinh(c) sin(b)x + sinh(c) cos(b)y. This is the equation of a great circle on the sphere. The unit normal of the plane of the circle is: [tanh(c) sin(b), tanh(c) cos(b), sech(c)]. As B and C vary we get all possible great circles, except those passing through the North and South poles, the lines of fixed longitude. The missing geodesics, those passing through the poles, project into the (x, y)-plane as the straight lines with θ constant. These are obtained here in the limit as C, with B fixed. They are the intersection of the sphere with the planes with unit normals: [sin(b), cos(b), 0]. 15