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De Rham Cohomology APPENDIX De Rham Cohomology We shall in this appendix explain the main ideas surrounding de Rham co- homology. This is done as a service to the reader who has learned about tensors and algebraic topology but had only sporadic contact with Stokes’ theorem. First we give an introduction to Lie derivatives on manifolds. We then give a digest of forms and important operators on forms. Then we explain how one integrates forms and prove Stokes’ theorem for manifolds without boundary. Finally, we define de Rham cohomology and show how the Poincar´elemma and the Meyer-Vietoris lemma together imply that de Rham cohomology is simply standard cohomology. The cohomology theory that comes closest to de Rham cohomology is Cechˇ coho- mology. As this cohomology theory often is not covered in standard courses on algebraic topology, we define it here and point out that it is easily seen to satisfy the same properties as de Rham cohomology. 1. Lie Derivatives Let X be a vector field and F t the corresponding locally defined flow on a smooth manifold M.ThusF t (p) is defined for small t and the curve t → F t (p) is the integral curve for X that goes through p at t =0. The Lie derivative of a tensor in the direction of X is defined as the first order term in a suitable Taylor expansion of the tensor when it is moved by the flow of X. Let us start with a function f : M → R. Then t f F (p) = f (p)+t (LX f)(p)+o (t) , where the Lie derivative LX f is just the directional derivative DX f = df (X) . We can also write this as t f ◦ F = f + tLX f + o (t) , LX f = DX f = df (X) . When we have a vector field Y things get a little more complicated. We wish to consider Y |F t , but this can’t be directly compared to Y as the vectors live in −t different tangent spaces. Thus we look at the curve t → DF Y |F t(p) that lies in TpM. Then we expand for t near 0 and get −t DF Y |F t(p) = Y |p + t (LX Y ) |p + o (t) for some vector (LX Y ) |p ∈ TpM. This Lie derivative also has an alternate definition. Proposition 49. For vector fields X, Y on M we have LX Y =[X, Y ] . 375 376 DE RHAM COHOMOLOGY Proof. We see that the Lie derivative satisfies −t DF (Y |F t )=Y + tLX Y + o (t) or equivalently t t Y |F t = DF (Y )+tDF (LX Y )+o (t) . It is therefore natural to consider the directional derivative of a function f in the t direction of Y |F t − DF (Y ) . D | − t f = D | f − D t f Y F t DF (Y ) Y F t DF (Y ) t t =(DY f) ◦ F − DY f ◦ F = DY f + tDX DY f + o (t) −DY (f + tDX f + o (t)) = t (DX DY f − DY DX f)+o (t) = tD[X,Y ]f + o (t) . This shows that t Y |F t − DF (Y ) LX Y = lim t→0 t =[X, Y ] . We are now ready to define the Lie derivative of a (0,p)-tensor T and also give an algebraic formula for this derivative. We define t ∗ F T = T + t (LX T )+o (t) or more precisely t ∗ t t F T (Y1, ..., Yp)=T DF (Y1) , ..., DF (Yp) = T (Y1, ..., Yp)+t (LX T )(Y1, ..., Yp)+o (t) . Proposition 50. If X isavectorfieldandT a (0,p)-tensor on M, then p (LX T )(Y1, ..., Yp)=DX (T (Y1, ..., Yp)) − T (Y1, ..., LX Yi, ..., Yp) i=1 Proof. We restrict attention to the case where p =1. The general case is similar but requires more notation. Using that t t Y |F t = DF (Y )+tDF (LX Y )+o (t) we get ∗ F t T (Y )=T DF t (Y ) t = T Y | t − tDF (L Y ) + o (t) F X = T (Y ) ◦ F t − tT DF t (L Y ) + o (t) X t = T (Y )+tDX (T (Y )) − tT DF (LX Y ) + o (t) . 1. LIE DERIVATIVES 377 Thus ∗ (F t) T (Y ) − T (Y ) (LX T )(Y ) = lim t→0 t t = lim DX (T (Y )) − T DF (LX Y ) t→0 = DX (T (Y )) − T (LX Y ) . Finally we have that Lie derivatives satisfy all possible product rules. From the above propositions this is already obvious when multiplying functions with vector fields or (0,p)-tensors. However, it is less clear when multiplying tensors. Proposition 51. Let T1 and T2 be (0,pi)-tensors, then LX (T1 · T2)=(LX T1) · T2 + T1 · (LX T2) . Proof. Recall that for 1-forms and more general (0,p)-tensors we define the product as · · T1 T2 (X1, ..., Xp1 ,Y1, ..., Yp2 )=T1 (X1, ..., Xp1 ) T2 (Y1, ..., Yp2 ) . The proposition is then a simple consequence of the previous proposition and the product rule for derivatives of functions. Proposition 52. Let T be a (0,p)-tensor and f : M → R a function, then p LfXT (Y1, ..., Yp)=fLX T (Y1, ..., Yp)+df (Yi) T (Y1, ..., X, ..., Yp) . i=1 Proof. We have that p LfXT (Y1, ..., Yp)=DfX (T (Y1, ..., Yp)) − T (Y1, ..., LfXYi, ..., Yp) i=1 p = fDX (T (Y1, ..., Yp)) − T (Y1, ..., [fX,Yi] , ..., Yp) i=1 p = fDX (T (Y1, ..., Yp)) − f T (Y1, ..., [X, Yi] , ..., Yp) i=1 p +df (Yi) T (Y1, ..., X, ..., Yp) i=1 The case where X|p = 0 is of special interest when computing Lie derivatives. t t We note that F (p)=p for all t. Thus DF : TpM → TpM and −t DF (Y |p) − Y |p LX Y |p = lim t→0 t d = DF −t | (Y | ) . dt t=0 p d −t | | This shows that LX = dt (DF ) t=0 when X p =0. From this we see that if θ is a 1-form then LX θ = −θ ◦ LX at points p where X|p =0. This is related to the following interesting result. 378 DE RHAM COHOMOLOGY Lemma 68. If a function f : M → R has a critical point at p then the Hessian of f at p does not depend on the metric. i Proof. Assume that X = ∇f and X|p =0. Next select coordinates x around p such that the metric coefficients satisfy gij|p = δij . Then we see that L g dxidxj | = L (g ) | + δ L dxi dxj + δ dxiL dxj X ij p X ij p ij X ij X = δ L dxi dxj + δ dxiL dxj ij X ij X i j = LX δijdx dx |p. Thus Hessf|p is the same if we compute it using g and the Euclidean metric in the fixed coordinate system. Lie derivatives also come in handy when working with Lie groups. For a Lie −1 group G we have the inner automorphism Adh : x → hxh and its differential at x = e denoted by the same letters Adh : g → g. Lemma 69. The differential of h → Adh is given by U → adU (X)=[U, X] Proof. If we write Adh (x)=Rh−1 Lh (x), then its differential at x = e is t t t given by Adh = DRh−1 DLh.NowletF be the flow for U. Then F (g)=gF (e)= t Lg (F (e)) as both curves go through g at t =0andhaveU as tangent everywhere t since U is a left-invariant vector field. This also shows that DF = DRF t(e).Thus d ad (X) | = DR −t DL t (X| ) | U e dt F (e) F (e) e t=0 d = DR −t X| t | dt F (e) F (e) t=0 d −t = DF X| t | dt F (e) t=0 = LU X =[U, X] . This is used in the next Lemma. Lemma 70. Let G = Gl (V ) be the Lie group of invertible matrices on V. The Lie bracket structure on the Lie algebra gl (V ) of left invariant vector fields on Gl (V ) is given by commutation of linear maps. i.e., if X, Y ∈ TI Gl (V ) , then [X, Y ] |I = XY − YX. Proof. Since x → hxh−1 is a linear map on the space hom (V,V ) we see that −1 t Adh (X)=hXh . The flow of U is given by F (g)=g (I + tU + o (t)) so we have d [U, X]= F t (I) XF−t (I) | dt t=0 d = ((I + tU + o (t)) X (I − tU + o (t))) | dt t=0 d = (X + tUX − tXU + o (t)) | dt t=0 = UX − XU. 2. ELEMENTARY PROPERTIES 379 2. Elementary Properties 1 Given p 1-forms ωi ∈ Ω (M) on a manifold M we define (ω1 ∧···∧ωp)(v1, ..., vp)=det([ωi (vj)]) where [ωi (vj)] is the matrix with entries ωi (vj) . We can then extend the wedge product to all forms using linearity and associativity. This gives the wedge product operation Ωp (M) × Ωq (M) → Ωp+q (M) , (ω, ψ) → ω ∧ ψ. This operation is bilinear and antisymmetric in the sense that: ω ∧ ψ =(−1)pq ψ ∧ ω. The wedge product of a function and a form is simply standard multiplication. There are three other important operations defined on forms: the exterior derivative d :Ωp (M) → Ωp+1 (M) , the Lie derivative p p LX :Ω (M) → Ω (M) , and the interior product p p−1 iX :Ω (M) → Ω (M) . The exterior derivative of a function is simply its usual differential, while if we are given a form ω = f0df 1 ∧···∧df p, then we declare that dω = df 0 ∧ df 1 ∧···∧df p.
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