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Differential Geometry 2 Andreasˇcap Differential Geometry 2 Fall Term 2011/12 Andreas Capˇ Institut fur¨ Mathematik, Universitat¨ Wien, Nordbergstraße 15, A{ 1090 Wien E-mail address: [email protected] Contents Preface v Chapter 1. Symplectic geometry and contact geometry 1 Initial considerations { distributions and integrability 1 Symplectic geometry 7 Contact geometry 15 Chapter 2. Riemannian metrics and the Levi-Civita connection 23 The homogeneous model 24 The orthonormal frame bundle 27 The Levi{Civita connection 30 Chapter 3. Basic Riemannian geometry 37 The covariant derivative 37 Parallel transport and geodesics 48 The Riemann curvature 59 Bibliography 65 iii Preface v CHAPTER 1 Symplectic geometry and contact geometry In this section, we want to build the bridge between analysis on manifolds and differ- ential geometry. It is also intended to complement our later discussion of Riemannian geometry. Riemannian metrics are the simplest example of a geometric structure of finite type. They have local invariants (e.g. curvature) and the group of automorphisms of such a structure is small. These facts are (at least vaguely) familiar from the case of hypersurfaces in Euclidean spaces discussed in the first course on differential geometry. The structures we consider in this chapter are the simplest examples of the opposite end of the spectrum. They do not admit local invariants, so any two structures are locally isomorphic, and there are many diffeomorphisms preserving the structure. Still they are very interesting in view of their close connections to classical mechanics and the geometric theory of partial differential equations. Initial considerations { distributions and integrability A simple idea how to define a geometric structure on a manifold M would be to choose some distinguished object, e.g. a vector field or a differential form. 1.1. Distinguished vector fields. We first want to show that there is not much about single vector fields, unless they have a zero. Locally around a point in which they are non{zero, all vector fields look the same up to diffeomorphism: Proposition 1.1. Let M be a smooth manifold and let ξ 2 X(M) be a vector field. If x 2 M is a point such that ξ(x) 6= 0, then there is a chart (U; u) for M with x 2 U such @ that ξjU = @u1 . Proof. This is an easy consequence of the existence of the flow. First we take a chart (U;~ u~) with x 2 U~,u ~(x) = 0 andu ~(U~) = Rn. Composingu ~ with a linear isomorphism, we may in addition assume that Txu~ · ξ(x) = e1. Thus it suffices to prove n @ the result in the case M = R , x = 0 and ξ(0) = @x1 (0). Now recall that the flow of ξ is defined on an open neighborhood D(ξ) of Rn × f0g in Rn × R. Hence the set W := f(y1; : : : ; yn) 2 Rn : (0; y2; : : : ; yn; y1) 2 D(ξ) is open and we can define a smooth map ' : W ! Rn by ξ '(y1; : : : ; yn) := Fly1 (0; y2; : : : ; yn): For any y 2 W we have (∗) T ' · e = d j Flξ (~u−1(0; y ; : : : ; y )) = d j Flξ('(y)) = ξ('(y)): y 1 dt t=0 y1+t 2 n dt t=0 t On the other hand, the definition immediately implies that '(0) = 0 and T0'·ei = ei for all i > 1, so T0' = id. Hence shrinking W , we may assume that ' is a diffeomorphism from W onto an open neighborhood U of 0 in Rn. Putting u = '−1, we get a chart n @ (U; u) for R around 0 and (∗) exactly means that ξjU = @u1 . If one studies vector fields, one thus only has to look at neighborhoods of a zero. Of ξ course ξ(x) = 0 is equivalent to Flt (x) = x for all t, so this study is the study of fixed points of dynamical systems, and ideas from dynamics play an important role in this. 1 2 1. SYMPLECTIC GEOMETRY AND CONTACT GEOMETRY 1.2. Distinguished 1{forms. Let us next consider the case of a distinguished one form α 2 Ω1(M), which is already considerably more varied than the case of a vector field. This becomes easily visible if one reinterprets the result for vector fields in Proposition 1.1 slightly: Recalling that a chart map is just a diffeomorphism onto an open subset of Rn, we can simply say that for x 2 M such that ξ(x) 6= 0, there is a diffeomorphism f from an open neighborhood U of x in M onto an open subset of Rn, ∗ @ such that ξjU = f @x1 . Applying this twice we can immediately conclude that for two manifolds M; N of the same dimension, two vector fields ξ 2 X(M) and η 2 X(N) and points x 2 M such that ξ(x) 6= 0 and y 2 N with η(y) 6= 0, there is a diffeomorphism from an open neighborhood U of x in M onto an open neighborhood of y in N such ∗ that ξjU = f η. So locally around a point in which they are non{zero, any two vector fields look the same up to diffeomorphism. The situation can not be as simple for one{forms due to the existence of the exterior derivative. Recall that for a one{form α 2 Ω1(M), the exterior derivative dα 2 Ω2(M) is characterized by dα(ξ; η) = ξ · α(η) − η · α(ξ) − α([ξ; η]) for ξ; η 2 X(M). Naturality of the exterior derivative says that f ∗(dα) = d(f ∗α). Having given two one{forms α 2 Ω1(M) and β 2 Ω1(N) on manifolds of the same dimension and points x 2 M and y 2 N such that both α(x) and β(y) are nonzero, we not only have to take into account the linear maps α(x): TxM ! R and β(y): TyN ! R (which of course look the same up to linear isomorphism). For a locally defined diffeomorphism f with f ∗β = α and f(x) = y to exist, also the skew symmetric bilinear maps dα(x): TxM × TxM ! R and dβ(y): TyN × TyN ! R must be compatible via the linear isomorphism Txf : TxM ! TyM. Before we study skew symmetric bilinear maps in more detail, let us turn to the simplest possible case. An analog of Proposition 1.1 in the realm of one{forms should say when a one{form locally looks like dx1 2 Ω1(Rn) up to diffeomorphism. Since d(dx1) = 0, this can be possible only for closed one{forms, i.e. for α 2 Ω1(M) such that dα = 0. For closed forms, the analog of Proposition 1.1 actually is true. Before we show this, we prove one of the fundamental results on the exterior derivative, the Lemma of Poincar´e: Lemma 1.2 (Lemma of Poincar´e). Any closed differential form on a smooth manifold M is locally exact. More precisely, if ! 2 Ωk(M) a smooth k{form such that d! = 0, then for each point x 2 M, there is an open subset U ⊂ M with x 2 U and a (k−1){form k−1 ' 2 Ω (U) such that !jU = d'. Proof. It suffices to deal with the case M = Rn, since locally around any point we can find a chart which is a diffeomorphism onto Rn. n n Consider the multiplication map α : R × R ! R defined by α(t; x) =: αt(x) = tx. ∗ ∗ Clearly α1 = id, so (α1) ! = ! and (α0) ! = 0, and we can write ! as Z 1 ∗ ∗ d ∗ ! = (α1) ! − (α0) ! = dt (αt) !dt; 0 where we integrate a smooth curve in the vector space Ωk(Rn). For the Euler vector n E t field E 2 X(R ) defined by E(x) = x, the flow is evidently given by Flt (x) = α(e ; x), E so we have αt = Fllog t. Using this, we compute d ∗ d E ∗ 1 E ∗ 1 ∗ dt αt ! = dt (Fllog t) ! = t (Fllog t) LE! = t αt LE!; INITIAL CONSIDERATIONS { DISTRIBUTIONS AND INTEGRABILITY 3 where LE is the Lie derivative along E. Now LE! = diE! + iEd! for the insertion operator iE, and by assumption d! = 0. Since pullbacks commute with the exterior d ∗ 1 ∗ derivative, we end up with dt αt ! = d t αt iE!. Next, we observe that Txαt = t id, so 1 ∗ 1 t αt iE!(x)(X2;:::;Xk) = t !(tx)(tx; tX2; : : : ; tXk) = !(tx)(x; tX2; : : : ; tXk); so this is defined for all t and we can view it as a smooth curve of (k − 1){forms. But since d defines a continuous linear operator Ωk−1(Rn) ! Ωk(Rn), it commutes with the R 1 1 ∗ integral of a smooth curve. Thus, defining ' := 0 t αt iE!dt, we obtain Z 1 Z 1 1 ∗ d ∗ d' = d t αt iE!dt = dt αt ! = !: 0 0 Proposition 1.2. Let M be a smooth manifold and let α 2 Ω1(M) be a closed one{ form. If x 2 M is a point such that α(x) 6= 0, then there is a chart (U; u) for M with 1 x 2 U such that αjU = du . Proof. We may start with an arbitrary chart (U;~ u~) such thatu ~(U~) is a ball in Rn. By the lemma of Poincar´e, dα = 0 implies that there is a smooth function f : U~ ! R n such that αjU~ = df. Since α(x) 6= 0, we can renumber the coordinates in R in such @ ~ n a way that df(x)( @u1 ) 6= 0. Considering the smooth map ' :u ~(U) ! R defined by 1 n −1 2 n @f '(y ; : : : ; y ) = ((f ◦ u~ )(y); y ; : : : ; y ) we see that det(Ty') = @u1 (y) 6= 0.
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