The prototypes of smooth manifolds

Transcription

1 The prototypes of smooth manifolds The prototype smooth manifolds are the open subsets of R n. If U is an open subset of R n, a smooth map from U to R m is an m-tuple of real valued functions (f 1, f 2,..., f m ), such that each f i is infinitely differentiable, so all its partial derivatives of all orders exist. Slightly more generally, a smooth map from U to an open subset V of R m is a smooth map from U to R m, whose range is in V. Consider for example the polar coordinate map: (x, y) = f(r, θ) = (r cos(θ), r sin(θ)), defined for r > 0 and θ real. Then f is smooth as a map from R + R R 2 {(0, 0)} (and is actually surjective and locally invertible). Here R + is the subset of R consisting of all positive real numbers. If U R n is an open set, its tangent bundle TU is defined to be the product U R n. If also V R m is an open set and f : U V is smooth, then we define: f : TU TV by the formula: f (x, v) = (f(x), df(v)). Here df is the Jacobian matrix of f, whose (i, j) entry is j f i, where j is an abbreviation for the partial derivative with respect to the j-th entry of x. So for the example of the polar co-ordinate map given above, we have: f (r, θ, p, q) = (r cos(θ), r sin(θ), cos(θ)p r sin(θ)q, sin(θ)p + r cos(θ)q) Here we have, for v = (p, q): df(v) = x r y r x θ y θ df = p q x r y r x θ y θ = cos(θ) r sin(θ) = sin(θ) r cos(θ) cos(θ) r sin(θ) sin(θ) r cos(θ) p q =, cos(θ)p r sin(θ)q sin(θ)p + r cos(θ)q). The matrix df is everywhere invertible, since its determinant is r > 0. It is a theorem that f is locally smoothly invertible if and only if df is invertible. Finally the chain rule for the composition of smooth maps f : U R n V R m and g : V W R k, where U, V and W are open, is the formula: d(g f) = (dg) (df), or equivalently (g f) = g f, so the composition map has its Jacobian matrix the product of the Jacobian matrices for each term of the composition. 1

2 The category of smooth manifolds The category of smooth manifolds is a collection of spaces called smooth manifolds and for each pair of (X, Y) of smooth manifolds, a collection of maps from X to Y called smooth maps. These obey the properties that the composition of smooth maps is smooth, and the identity map for any manifold is smooth. Also the Cartesian product X Y of smooth manifolds is required to be smooth as are the natural projections from X Y to X and Y. The reals are required to be a smooth manifold and the operations of addition, subtraction and multiplication of reals are required to be smooth. Also any open interval is required to be a smooth manifold and the natural inclusion of one interval into a larger interval is required to be smooth. Then taking the reciprocal of a non-zero real number is required to be a smooth mapping from the manifold of non-zero real numbers to itself. More generally, we require that any open subset X of R n be a smooth manifold and if f : X R m is a smooth map with range in Y an open subset of R m, then f : X Y is required to be smooth. The differential character of the manifold X is reflected in the requirement that for each smooth manifold, there be associated a smooth manifold called TX, the tangent bundle of X, together with a smooth projection denoted π X from TX to X and such that if f : X Y is smooth, then there is a smooth map denoted f from TX to TY such that π Y f = f π X. Then we require that if also g : Y Z is a smooth map of manifolds, then (g f) = g f. Here TX has the structure of a vector bundle: given x X, put TX x the set of elements of TX that map to x under the projection π X. Then we require that TX x, called the tangent space at x, has the natural structure of a vector space, such that the vector space operations vary smoothly with x. Further we require that the tangent bundle of any open subset U of R n be the product TU = U R n with the natural projection π U to U, such that if f : U V R m, then f is the Jacobian mapping as discussed above. If f : X V R m, with V open, is smooth, then f maps TX U R m. So if v TX, we can define df(v) R m by the formula f (x, v) = (f(x), df(v)), where π X (v) = x, so v TX x. This gives the differential of the function f, a linear operator denoted df on each TX x, varying smoothly with x, whose value at any v is df(v). If m = 1, so f is just one function, then the map (v, df) df(v) is called the dual pairing. 2

3 We say that a smooth map x : X V R n with V open is a local coordinate map at x if dx : TX x R n is a basis. The number n, which depends only on x, not on the map f, is called the dimension of X at x. For a smooth manifold, we require that each point x of X have a local co-ordinate map and we usually require that the dimension be constant: that dimension is then called the dimension of X. If x is a local coordinate map at x, then it is still a local coordinate map in a neighborhood of x. An open set in X for which x is a local co-ordinate map at each point is called a local co-ordinate patch. So, for example, for the sphere S 2, whose equation is x 2 + y 2 + z 2 = 1 for (x, y, z) R 3, the following maps N and S are local co-ordinate patches: ( ) x N : (x, y, z) (u, v) = 1 z, y, z 1, 1 z ( ) x S : (x, y, z) (s, t) = 1 + z, y, z z Here N is stereographic projection from the North Pole and S is steregraphic projection from the South Pole. The tangent bundle of the sphere may be regarded as all tangent vectors to the sphere in the usual sense, so the tangent vectors at a point (x, y, z) are represented by the triples (dx, dy, dz) such that (x, y, z).(dx, dy, dz) = 0. We have: (dn)(dx, dy, dz) = 1 dz (dx, dy)+ 1 z (1 z) (x, y) = 1 ((1 z)dx+xdz, (1 z)dy+ydz). 2 (1 z) 2 Note that this vanishes when dx = x y dz and dy = 1 z 1 z dy. But then we have: ( (x 2 + y 2 ) 0 = xdx+ydy+zdz = 1 z ) ( + z dz = ( ) 1 z 2 1 z ) + z dz = ( (1+z)+z)dz = dz. Since dz = 0, it follows that dx = dy = 0 also, so dn is an isomorphism, since its kernel is zero. Similarly ds is an isomorphism. Explicitly, we have: (ds)(dx, dy, dz) = 1 dz (dx, dy) 1 + z (1 + z) (x, y) = 1 ((1+z)dx xdz, (1+z)dy ydz). 2 (1 + z) 2 So both N and S are local coordinate patches and S 2 has dimension 2. (s, t) (u, v) Then away from the poles we have: (u, v) = and (s, t) = s 2 + t2 u 2 + v. 2 These relations valid for (u, v) (0, 0) and (s, t) (0, 0), are called the patching relations for these co-ordinates. 3

4 A vector field on a manifold X is a smooth assignment of a vector v(x) TX x at each point x X. Then if f is a smooth function, the dual pairing df(v) gives a smooth function on X. If f is a local coordinate map, then the vector field v i such that df(v i ) = e i is called the i-th coordinate vector field. Dually, a smooth curve in a manifold X is a smooth map f from an open subset U of R into the manifold. Then f maps TU = U R to TX. For a real number t in U, we can write f (t, u) = (x(t), v(t)u), for some unique vector v(t) at x(t). Then v(t) is called the velocity vector of the curve x(t) at the point with parameter t. 4

5 The differential and vector fields If f is a function on a manifold M its differential is the co-vector field or one-form given in local co-ordinates by: df = f i dx i, f i = f x i = if, i = f x i. If v is a vector field, then the directional derivative of f in the direction of v is: v(f) = (df)(v) = v i f i. We write v = v i i, where e i = i x j = δ j i are the basis of vector fields dual to the basis e j = dx j of differentials. Note that we can compute the coefficients v i from the action of the vector field v on the co-ordinate functions: v i = v(x i ) = dx i (v). The differential obeys all the usual rules of calculus, for any smooth functions f and g: d(c) = 0 and d(cf) = cdf, if c is a constant function. If f is defined on an open subset R and if S is an open subset of R, then we have: (df) S = d(f S ). Here f S is the restriction of f from the domain R to the domain S. d(f ± g) = df ± dg d(fg) = gdf + fdg, ( ) 1 d = dg g g, 2 ( ) f gdf fdg d = g g 2 d(f (u)) = F (u)du. df (u 1, u 2,... u k ) = F u j duj. 5

6 Differential forms The algebra of differential forms is just the exterior algebra of the covariant tensor algebra of the manifold, so in local co-rodiantes is the algebra of all polynomials in the differentials dx i, subject to the commutation rule dx i dx j = dx j dx i. A differential form which is the sum of monomials formed from the product of p dx s is said to be of degree p. In the case that p = 0, the form is just a function. In the case that p = 1, a form α is just a differential or co-vector field: α = α i dx i. The algebra of differential forms has some additional structure, coming from the fact that the differentials are co-vectors. First we have the exterior derivative: d. This maps k-forms to (k + 1)-forms for any integer k and obeys the rules, valid for any forms S and T and for any function f: df is the usual differential of f. d(s ± T ) = ds ± dt, d(fs) = (df)s + fds, d(st ) = (ds)t + ( 1) p SdT, where S has degree p, d 2 = 0. If g : X Y is a smooth map of manifolds, then in local co-ordinates, the map is given by a formula y i = g i (x j ), where y i, i = 1, 2,... m are the coordinates for Y and x j, j = 1, 2,..., n ate the co-ordinates for X. Also the functions g i (x j ) are smooth. Then we can take the differential of the map: dy i = G i j(x k )dx j, where G i j(x k ) = yi is the Jacobian matrix of the x j transformation. More generally, if α = α i (y j )dy i is a one-form or co-vector field, where the coefficients α i (y j ) are smooth functions of the variables y j, then we can define its pull-back along g, g (α) = α(y m (x n ))G i j(x k )dx j, so the pullback is one-form on X. 6

7 Closed and exact forms; cohomology A form S is said to be closed if ds = 0 and to be exact if S = dt, for some form T. Every exact form is closed. The Poincare Lemma says that every closed form is locally exact. The standard example of a closed form which is not exact on its domain is the one-form: α = xdy ydx x 2 + y 2. It is easy to check that dα = 0. If α were exact, α = df, then by the Fundamental Theorem of Calculus, we would have: α = df = 0. Γ Here Γ is the circle x 2 + y 2 = 1, traced once counter-clockwise. But in fact we have, putting x = cos(t) and y = sin(t), with 0 t 2π, since we have also dx = sin(t)dt and dy = cos(t)dt: α = cos(t)(cos(t)dt) sin(t)( sin(t)dt) cos 2 (t) + sin 2 (t) So α is closed, but not exact on its domain. Γ = dt, Γ α = 2π 0 dt = 2π 0. 7

8 The interior product and the Lie derivative Given a vector field, v, we introduce an operator δ v on forms. This maps k-forms to k 1-forms, so in particular maps all functions to zero. Then it acts on a one-form α by δ v (α) = α(v). More generally, it acts on a product of one-forms by a Liebniz rule. If α i, i = 1, 2,..., k are one-forms, we have: δ v (α 1 α 2 α 3... α k ) = v(α 1 )α 2 α 3... α k v(α 2 )α 1 α 3... α k +v(α 3 )α 1 α 2... α k + +( 1) k 1 v(α k )α 1 α 2... α k 1. The properties characterizing δ v, valid for any smooth functions f and forms S and T : δ v maps k-forms to (k 1)-forms for any k. δ v (f) = 0 δ v (S ± T ) = δ v (S) ± δ v (T ). δ v (ST ) = δ v (S)T + ( 1) k Sδ v (T ), where S is a k-form δ v (df) = v(f). In local co-ordinates, if v = v i i and S = s i1 i 2 i 3...i k dx i 1 dx i 2 dx i 3... dx i k, with s i1 i 2 i 3...i k completely skew, then we have: δ v (S) = kv i 1 s i1 i 2 i 3...i k dx i 2 dx i 3... dx i k. If w = w i i is a second vector field, then we have, since s i1 i 2 i 3...i k the arguments i 1 and i 2 : is skew in (δ v δ w + δ w δ v )Sk)(k 1)(v i 1 w i 2 + w i 1 v i 2 )s i1 i 2 i 3...i k dx i 3... dx i k = 0. So the interior products anti-commute; for any vector fields v and w, we have: δ v δ w + δ w δ v = 0, δ 2 v = 0. 8