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Seminar “Differential Forms and Their Use” Differentiable Manifolds

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Seminar “Differential Forms and Their Use” Differentiable Manifolds Seminar “Differential forms and their use" Differentiable manifolds Sandra Schluttenhofer and Danilo Ciaffi December 18, 2014 1 Definitions and properties Motivation 1. 3 3 In R we defined a regular surface S as a subset S ⊂ R such that for every point p 2 S there 2 exists a neighbourhood V of p and a map fα : Uα ⊂ R −! V \ S with the property that (i) fα is a differentiable homeomorphism. 3 (ii) the differential (dfα)q : Tq(Uα) −! R is injective for all q in Uα. The map fα is called a parametrization of S around p. A fundamental property of regular surfaces is the fact that a change of parametrizations is differentiable, i.e. for any two parametrizations fα : Uα −! S, fβ : Uβ −! S with W := fα(Uα) \ fβ(Uβ) 6= ; −1 −1 2 fβ ◦ fα : fα (W ) −! R is differentiable. Proof of this fundamental property. As a composition of homeomorphisms f −1 ◦ g is again a homeomorphism. The idea is to use −1 the inverse function theorem to proof that fα ◦ fβ is differentiable. First, we choose a point −1 −1 x 2 fβ (W ), then let (fα ◦ fβ)(x) = q. Since the differential dfα is injective for all q 2 Uα, it has rank 2 and by - if necessary - rearranging the axes, we can assume that 1 1 " @fα @fα #! @x jq @y jq det 2 2 6= 0 @fα @fα @x jq @y jq at point q. Now we define an extension of fα to which we can apply the inverse function theorem. 3 F : Uα × R −! R 1 2 3 F (x; y; z) = (fα(x; y); fα(x; y); fα(x; y) + z) We immediately notice that F (x; y; 0) = fα(x; y) and 2 1 1 1 3 @fα @fα @fα @x @y @z 2 03 2 2 2 6 @fα @fα @fα 7 dFq = 6 @x @y @z 7 = 4dfα 05 4 3 3 3 5 @fα+z @fα+z @fα+z 1 @x @y @z 1 and therefore (Laplace expansion along the last column) 1 1 " @fα @fα #! @x jq @y jq det(dFq) = det 2 2 6= 0 @fα @fα @x jq @y jq The inverse function theorem gives us the existence of a neighbourhood M of fα(q) such that −1 −1 F exists and is differentiable on M. We can now restrict F to a neighbourhood fβ(N) ⊂ M, where N is a neighbourhood of x in Uβ. In this neighbourhood −1 −1 F ◦ fβjN = fα ◦ fβjN −1 is differentiable as the composition of differentiable maps. Since x was arbitrary fα ◦ fβ is −1 differnetiable on fβ (W ). We are going to use this property to formulate a definition that is independent of the ambient space. Definition 1 (n-dimensional differentiable manifold). An n-dimensional differential manifold is a set M with a family of injective maps n n fα : Uα ⊂ R −! M of open sets Uα ⊂ R into M, such that S 1. α fα(Uα) = M −1 −1 2. For each pair α; β with fα(Uα) \ fβ(Uβ) = W = ;, the sets fα (W ) and fβ (W ) are open n −1 −1 −1 −1 sets in R and the maps fβ ◦ fα : fα (W ) −! Uβ and fα ◦ fβ : fβ (W ) −! Uα are differentiable. 3. The family fUα; fαg is maximal relative to 1: and 2:. A pair (Uα; fα) with p 2 fα(Uα) is called a parametrization or coordinate system of M at p. A familiy fUα; fαgα satisfying 1: and 2: is called a differentiable structure of M. Example 1 (The real projective space). We identify the real projective space n with the quotient space n+1 n f0g , where ∼ is the PR R =∼ equivalence relation given by (x1; : : : ; xn+1) ∼ (λx1; : : : ; λxn+1) λ 2 R; λ 6= 0 We denote an element in x 2 n by x = [x ; : : : ; x ]. In order to proof that the real projective PR 1 n+1 space is a differentiable manifold, we need to find a family satisfying 1: and 2: in definition 1. Let n Vi = f[x1; : : : ; xn+1]jxi 6= 0g ⊂ PR n and define the functions fi : R −! Vi by fi(y1; : : : ; yn) = [y1; : : : ; yi−1; 1; yi; : : : ; yn]. We claim that fVi; fig(i2f1;:::;n+1g) is a differential structure on M. n −1 n n First we note that fi : R −! Vi is bijective. Moreover, fi (Vi) = R is open in R and Sn f ( n) = n . i=1 i R PR n n −1 n We have fi(R ) \ fj(R ) = Vi \ Vj 6= ;. We need to show that fi (Vi \ Vj) is open in R and −1 that fj ◦ fi is differentiable. Note that x1 xi−1 xi+1 xn Vi \ Vj = f[x1; : : : ; xn+1jxi 6= 0 ^ xj 6= 0g = f[ ;:::; ; 1; ;:::; ]jxi 6= 0 ^ xj 6= 0g xi xi xi xi With that observation we get −1 x1 xi−1 xi+1 xn fi (Vi \ Vj) = f ;:::; ; ;:::; jxj 6= 0 ^ xi 6= 0g = f(x1; : : : ; xn)jxj 6= 0g xi xi xi xi 2 n which is open in R . Moreover, we have (without loss of generality i > j) −1 −1 fj ◦ fi(x1; : : : ; xn) = fj ([x1; : : : ; xi−1; 1; xi; : : : ; xn+1]) −1 x1 xj−1 xj+1 xi−1 1 xi xn+1 = fj ([ ;:::; ; 1; ;:::; ; ; ;:::; ]) xj xj xj xj xj xj xj x x x x 1 x x = ( 1 ;:::; j−1 ; j+1 ;:::; i−1 ; ; i ;:::; n+1 ) xj xj xj xj xj xj xj −1 which is differentiable on fi (Vi \ Vj). Definition 2 (differentiable map between manifolds). Let M1 and M2 be n resp. m-dimensional manifolds. A map ' : M1 −! M2 is differentiable m at a point p 2 M1 if given a parametrization g : V ⊂ R −! M2 around '(p), there exists a n parametrization f : U ⊂ R −! M1 around p such that '(f(U)) ⊂ g(V ) and the map −1 n m g ◦ ' ◦ f : U ⊂ R −! R −1 is differentiable at f (p). The map ' is differentiable on an open set of M1 if it is differentiable at all points of this set. We call a differentiable map '1 : M1 −! M2 = R a differentiable function on M1 and a differentiable map '2 : I ⊂ R −! M2 a differentiable curve on M2. The differentiability of a map between manifolds is well-defined, i.e. independent of the choice of parametrizations, because the change of parametrizations is by definition differentiable. Motivation 2. 3 3 In R a tangent vector v at a point p to a differentiable curve α : I ⊂ R −! S ⊂ R ; α(0) = p on 0 0 0 0 t a regular surface S is simply defined as v := α (0) = [α1(0); α2(0); α3(0)] . Again, since we do n not necessarily have the ambient space R , we need an alternative characterization. The idea is that a tangent vector is fully characterized by the derivatives of functions along this tangent vector. 3 Let ' : R −! R be differentiable in a neighbourhood of p 2 S. We calculate the directional derivative of ' in direction of the tangent vector v: 3 3 ! d X @' @αi X @ (' ◦ α)j = j = α0 (0) ' dt t=0 @α @t t=0 i @α i=1 i i=1 i Notice that we can get the coordinates of v by applying the operator P3 α0 (0) @ to the i=1 i @αi functions '1(x; y; z) = (x; 0; 0);'2(x; y; z) = (0; y; 0);'3(x; y; z) = (0; 0; z). This observation motivates the following definition: Definition 3 (tangent vector). Let f0g ⊂ I be an open subset of R and α : I −! M a differentiable curve on a differentiable manifold M through p 2 M, α(0) = p. Let D be the set of functions on M, which are differentiable at p. The tangent vector to the curve α at p is the map 0 α (0) : D −! R given by d α0(0)' = (' ◦ α)j ;' 2 D dt t=0 A tangent vector at p 2 M is the tangent vector to some differential curve α : I −! M with α(0) = p. 3 Proposition 1. The set TpM of tangent vectors at a point p 2 M of an n-dimensional differ- entiable manifold M, called the tangent space, is an n-dimensional real vector space. n Proof. Choose a parametrization f : U ⊂ R −! M around p, without loss of generality p = f(~0), and a curve α : I −! M on M through p, α(0) = p. Let ' 2 D. −1 −1 n Then by definition f ◦ α ◦ id ≡ f ◦ α : I −! R is differentiable. Therefore, we can −1 write f ◦ α = (x1(t); : : : ; xn(t)) with differentiable functions xi : I −! R. We also write ' ◦ f(q) = '(x1; : : : ; xn), where q = (x1; : : : ; xn) 2 U. Then we have d d α0(0)' = (' ◦ α)j = (' ◦ f ◦ f −1 ◦ α)j dt t=0 dt t=0 d = '(x (t); : : : ; x (t))j dt 1 n t=0 n X @ = ( x0 (0)( )j )' i @x 0 i=1 i This observation leads us to the assumption that TpM, the tangent space to M at p, is equal to T = h( @ )j ji = 1; : : : ; ni. (Notice that @ depends on the choice of f!). We have already f @xi 0 @xi seen that TpM ⊂ Tf . For the other direction, T ⊂ T M, let v 2 T , i.e. v = Pn λ ( @ )j 2 T . We now have f p f i=1 i @xi 0 f to look for a curve α for which v is the tangent vector at p.
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