Differential geometry Lecture 10: Dual bundles, 1-forms, and the Whitney sum David Lindemann University of Hamburg Department of Mathematics Analysis and Differential Geometry & RTG 1670 22. May 2020 David Lindemann DG lecture 10 22. May 2020 1 / 23 1 Dual vector bundles and 1-forms 2 Direct sum of vector bundles David Lindemann DG lecture 10 22. May 2020 2 / 23 Recap of lecture 9: defined pushforward & pullback of vector fields locally rectified vector fields proved that [X ; Y ] measures infinitesimal change of Y along local flow of X defined Lie derivative of vector fields erratum: mixed up terms one parameter families and one parameter groups David Lindemann DG lecture 10 22. May 2020 3 / 23 Dual vector bundles and 1-forms Recall the definition of dual vector spaces from linear algebra: A real (for our purposes finite dimensional) vector space V ∗ has dual vector space V := f! : V ! R j ! R-linear mapg. Question: How can we translate this concept to vector bundles? Answer: Use the vector bundle chart lemma! Definition Let πE : E ! M be a vector bundle of rank k. The dual ∗ vector bundle πE ∗ : E ! M is pointwise given by −1 ∗ πE ∗ (p) = Ep := HomR(Ep; R) for all p 2 M. The topology, smooth manifold structure, and bundle struc- ture on E ∗ is obtained as follows: let f( i ; Vi ) j i 2 Ag be a collection of local trivializations of a vector bundle E of rank k, such that there exists an atlas A = f('i ; πE (Vi )) j i 2 Ag of M note: fVi j i 2 Ag is an open covering of E David Lindemann DG lecture 10 22. May 2020 4 / 23 Dual vector bundles and 1-forms (continuation) then B := f((' × id k ) ◦ ; V ) j i 2 Ag is an atlas on E i R i i recall: for any finite dimensional real vector space W , (W ∗)∗ and W are isomorphic via W 3 v 7! (! 7! !(v));! 2 W ∗: the topology on E ∗ is given by pre-images of open images of the dual local trivializations which are defined by −1 k ei : πE ∗ (πE (Vi )) ! πE (Vi ) × R ;!p 7! (p; w); k where w 2 R is the unique vector, such that −1 !p(vp) = hw; pr k (πE (vp))i for all vp 2 π (p) and h·; ·i R E k denotes the Euclidean scalar product on R induced by its canonical coordinates the dual atlas B∗ on E ∗ is then defined by ∗ B := f((' × id k ) ◦ ; V ) j i 2 Ag; i R ei i and it follows that E ∗ ! M is a vector bundle of rank k David Lindemann DG lecture 10 22. May 2020 5 / 23 Dual vector bundles and 1-forms Exercise Show that E ! M and (E ∗)∗ ! M are isomorphic as vector bundles. If we know the transition functions for the local trivializations of E ! M, we also know the transition functions of the dual local trivializations of E ∗ ! M: Lemma The transition functions of E ∗ ! M are given by −1 −1 T ei ◦ ej :(p; w) 7! p; (Ap ) w for all p 2 πE (Vi ), where A : πE (Vi ) ! GL(n) is given by the transition functions of E ! M, −1 i ◦ j :(p; v) 7! (p; Apv): Proof: Exercise! David Lindemann DG lecture 10 22. May 2020 6 / 23 Dual vector bundles and 1-forms The most important example of a dual vector bundle for this course is the dual to the tangent bundle of a smooth manifold: Definition The vector bundle T ∗M := (TM)∗ ! M is called the cotangent bundle of M. Pointwise we denote ∗ ∗ Tp M = (TM)p for all p 2 M. As for the tangent bundle we identify for any U ⊂ M open and p 2 U the vector spaces ∗ ∼ ∗ Tp U = Tp M via the inclusion map. In order to gain a better understanding of the cotangent bundle T ∗M ! M, we must take a deeper look at its local trivializations & charts of the total space T ∗M induced by local coordinates on the base manifold M: (see next page) David Lindemann DG lecture 10 22. May 2020 7 / 23 Dual vector bundles and 1-forms let ' = (x 1;:::; x n) be a local coordinate system on U ⊂ M want to use ' to define a local coordinate system on −1 ∗ πT ∗M (U) ⊂ T M compatible with the bundle structure compatible with bundle structure := charts of the total −1 n space composed with ' × idR from the right must be define local trivializations define candidates for local coordinate systems −1 n e : πT ∗M (U) ! '(U) × R ; ! !! @ @ ∗ e : !p 7! '(πT M (!p));!p 1 ;:::;!p n @x p @x p verify: with := (' ◦ π; d') induced local coordinate −1 system on πTM (U) ⊂ TM [note: interpret codomain of n d'p as R via canonical identification], obtain ∗ !p(vp) = hpr n ( (!p)); pr n ( (vp))i 8!p 2 T M; vp 2 TpM; R e R p h·; ·i = Euclidean scalar product in canonical coordinates David Lindemann DG lecture 10 22. May 2020 8 / 23 Dual vector bundles and 1-forms hence: transition functions of the e are of the form −1 −1 −1 T ei ◦ ej :(p; w) 7! p; (d('i ◦ 'j )p ) w −1 T = p; (d('j ◦ 'i ) −1 ) w ; 'i ◦'j (p) and thus smooth −1 n furthermore, (' × idR ) ◦ e define local trivializations (on level of sets) with smooth matrix part ∗ e define smooth structure on total space T M by vector bundle chart lemma X Remark: The right hand side of !p(vp) = hpr n ( (!p)); pr n ( (vp))i is independent of the chosen R e R local coordinate system ' on M covering p 2 M. David Lindemann DG lecture 10 22. May 2020 9 / 23 Dual vector bundles and 1-forms Understanding sections in the cotangent bundle is, as for vector fields, of critical importance when studying differential geometry. Definition Sections in T ∗M ! M are called 1-forms and denoted by Ω1(M) := Γ(T ∗M): ∗ 1 For U ⊂ M open, sections in Γ(T MjU ) are denoted by Ω (U) and called local 1-forms. We can easily obtain examples of 1-forms by taking the differ- ential of smooth functions: Example Let f 2 C 1(M). Then the differential of f , df 2 Ω1(M), is given by df : p 7! dfp: (continued on next page) David Lindemann DG lecture 10 22. May 2020 10 / 23 Dual vector bundles and 1-forms Example (continuation) 1 n @ @f In local coordinates (x ;:::; x ) we have df @xi = @xi for all 1 ≤ i ≤ n. This implies that df can locally be written as n X @f i df = dx : @x i i=1 In particular it follows for f = x j that the coordinate 1-forms j j @ j dx fulfil dx @xi ≡ δi on the domain of definition of the local coordinates. [ this is the \global" (on chart neighbourhoods) version of dx j @ = δj ] p @xi p i Recall that we have shown that for local coordinates (x 1;:::; x n) n o of M covering p 2 M, the set of vectors @ 1 ≤ i ≤ n @xi p is a basis of TpM. A similar result holds for each covector space ∗ Tp M. David Lindemann DG lecture 10 22. May 2020 11 / 23 Dual vector bundles and 1-forms Lemma Let ' = (x 1;:::; x n) be local coordinates defined on U ⊂ M and let p 2 U be arbitrary but fixed. Then i fdxp j 1 ≤ i ≤ ng ∗ is a basis of Tp M. It is precisely the dual basis to the basis n o @ 1 ≤ i ≤ n of T M. Any local 1-form ! 2 Ω1(U) @xi p p can be written as n X i ! = fi dx i=1 1 with uniquely determined smooth functions fi 2 C (U) for 1 ≤ i ≤ n. Proof: i ∗ fdxp j 1 ≤ i ≤ ng being a basis of Tp M that is dual to n o @ 1 ≤ i ≤ n follows from dx i @ = δi @xi p p @xj p j (continued on next page) David Lindemann DG lecture 10 22. May 2020 12 / 23 Dual vector bundles and 1-forms (continuation of proof) 1 observe: for any fi 2 C (U), 1 ≤ i ≤ n, the right hand n P i ∗ side of ! = fi dx is a local section of T M i=1 this follows from the construction of the smooth manifold structure on the total space T ∗M via charts of the form e = (' ◦ π; d'∗) where ' is a chart on M, d' denotes the vector part of d', and ∗ the pointwise dual linear map, which in particular implies that each dx i is a local 1-form on the other hand: for a given local 1-form ! define @ ! := ! 81 ≤ i ≤ n i @x i 1 it now suffices to show that !i 2 C (U) and, after that, to define fi := !i !i being a local smooth function follows from observing −1 that !i ◦ ' is precisely the i-th entry in the vector part of e ◦ ! ◦ '−1 and thereby by definition a smooth map uniqueness of the fi can be shown as follows: (next page) David Lindemann DG lecture 10 22. May 2020 13 / 23 Dual vector bundles and 1-forms (continuation of proof) suppose that locally n n X i X i ! = fi dx = fei dx i=1 i=1 such that for at least one 1 ≤ j ≤ n, fj 6= fej choose p 2 U, such that fj (p) 6= fej (p) and calculate n ! ! n ! ! X i @ X i @ fi dx = fj (p) 6= fej (p) = fei dx @x j @x j i=1 p i=1 p which is a contradiction We have constructed the dual bundle T ∗M ! M to TM ! M, and we have studied their respective local sections.