Affine and Riemannian Connections

Transcription

1 Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons on M = x natural bass vector felds X components of the vector feld X = X Motvaton and Outlne The am of ths talk s to ntroduce the concept of parallel transport for Remannan manfolds In general, the result of a parallel transport wll not only depend upon the ntal and fnal pont, but also upon the path between them Therefore, what we are lookng for s a way to locally dentfy vectors at dfferent tangent spaces along a curve (a so-called connecton) Conceptonally, ths s related to havng a way to derve vector felds along curves or, (as we wll see) equvalently, wth respect to vectors: We could then defne a vector feld to be locally constant f and only f ts dervatve s constantly zero However, for a general manfold M there s a-pror no canoncal way to dentfy vectors at dfferent tangent spaces and at the same tme there s no way to derve vector felds along curves In the case of submanfolds of R n we could just propose to derve each component ndvdually The problem s, that the resultng vector mght not le n the approprate tangent space, whereas for general manfolds we do not know what non-tangent vectors shall be We would solve the problem by projectng the resultng vector onto the tangent space, but there are nfntely many ways to project a vector onto a subspace of a vector space However, n the case of the eucldean R n there s one dstngushed projecton: the orthogonal projecton At the very end of ths talk, we wll prove that ntroducng a Remannan structure ndeed gves rse to a dstngushed lnear connecton, once two natural condtons are mposed: One of whch deals wth the symmetry of the yet-to-defne Chrstoffel symbols It can be justfed by the desre that the locally shortest lnes always be straght 1

2 The other one has to do wth the compatblty of the connecton wth the metrc: We expect a par of vectors to keep ts scalar product constant when parallel-transported along a curve In order to fnd a defnton for a lnear connecton, we wll borrow some propertes from a smlar concept the Le dervatve We wll then examne why the Le dervatve s not what we are lookng for and strengthen one condton Remember X, Y X(M) : L X Y (p) [X, Y ](p) = ( X j j Y Y j j X ) p (p),j It subjects to the followng propertes: R-lnearty n X, R-lnearty n Y, Lebnz rule: L X (fy ) = X(f) Y + f L X Y Note the occurrence of the term j X : The Le dervatve apples the dfferental to compare the values of Y : L X Y (p) = lm t 0 dφ t Φ t (p) Y (Φ t (p)) Y (p) t where {Φ t } t s the one-parameter local group of dffeomorphsms generated by X The dfferental turns the vectors Y (Φ t (p)) as the representng curves are turned But we want to carry over the vectors Y (Φ t (p)) unturned by parallel-transport Therefore, the value of the dervatve X Y (p) should only depend upon X(p) We express ths by strengthenng the frst condton to D(M)-lnearty (X(M) s at the same tme an R-vector space and a D(M)-module), Affne connectons Defnton 1 An affne connecton on a smooth manfold M s a mappng : X(M) X(M) X(M) (X, Y ) X Y subject to the propertes: affne connecton fx+gy Z = fl X Z + g Y Z (D(M)-lnearty n the frst argument) X (αy + βz) = αl X Y + β X Z (R-lnearty n the second argument) X (fy ) = f X Y + X(f) Y (Lebnz rule), where X, Y, Z X(M), f, g D(M) 2

3 Remark 2 To an affne connecton there are assocated smooth functons Γ k j := ( j ) k D(M), the so called Chrstoffel symbols By the lnearty propertes, they determne the connecton completely: ( X Y ) k = X(Y k ) + Γ k jx Y j = ( j Y k + Γ k jy j) X,j j Chrstoffel symbols The Chrstoffel symbols are not the components of a tensor feld! It can be shown 1 that under a coordnate change they obey the transformaton law j = x k x x j x k Γ k x x j j +,j,k }{{} l transf law for tensors Γ k x k x l 2 x l x x j In the sequel, let M denote a smooth manfold wth a gven affne connecton Lookng for a way to derve vector felds V along a curve c : t c(t), one would lke to defne := ċv However, globally not every vector feld along c s the restrcton of a vector feld on M Nevertheless, f lnearty s requred, ths defnes D unquely, as by a choce of coordnates every vector feld along c can be wrtten as a lnear combnaton of vector felds on M: Proposton 3 There s a unque way of assocatng to a vector feld V along a dfferentable curve c : I M another vector feld along c such that D (αv + βw ) = α + β DW (R-lnearty) D (fv ) = fv + f (Lebnz rule) f V (t) Y (c(t)): = ċy Proof Unqueness: Introduce coordnates around every pont of c(i) and wrte (c(t)) = x (t), V = j V j j wth V j, j regarded as dependng on the curve parameter t By the above propertes, wrte ths as = j ( V j j + ) ẋ V j j Use ths local expresson to show exstence; by unqueness, ths does not depend upon the choce of coordnates Defnton 4 A vector feld V along a dfferentable curve c : I M s called parallel ff 0 Proposton 5 Let c : I M be a dfferentable curve, V 0 T c(t0 )M Then there exsts a unque parallel vector feld V along c such that V (t 0 ) = V 0 ; V (t) s called the parallel transport of V (t 0 ) along c Lemma 6 (global Pcard Lndelöf theorem) E Banach space, f : [a, b] E E contnuous and globally Lpschtzan n the second varable; Then for each y 0 E there exsts a global soluton to the Cauchy problem and there are no further local solutons ẏ(t) = f(t, y(t)), y(0) = y 0, covarant dervatve parallel transport 1 Yvonne Choquet-Bruhat, Introducton to General Relatvty, Black Holes and Cosmology, Oxford Unversty Press 2015, p 19 3

4 Proof of the proposton By a compactness argument, t suffces to show exstence and unqueness wthn the doman of a chart Adapt the above notaton and wrte V 0 = j (V 0) j j (c(t 0 )) To show unqueness, suppose that there exsts a V wth the desred property It follows that 0 = = ( V j j + ) V j ẋ j = ( V k + Γ k jẋ V j) k j k,j = k : 0 = V k +,j Γ k jẋ V j By the global Pcard Lndelöf theorem, ths system of dfferental equatons possesses a global unque soluton whch satsfes the ntal condtons V k (t 0 ) = (V 0 ) k : f(t, V) :=,j,k Γ k j(c(t)) ẋ (t) V k e k s Lpschtzan n the second argument snce t Γ k j (c(t)) ẋ (t) s bounded Remannan Connectons In the sequel, M s assumed to be Remannan Defnton 7 s sad to be compatble wth, ff P, P = const for any two parallel vector felds P, P along a smooth curve c Proposton 8 s compatble wth, ff for all vector felds V, W along c : I M d V, W, W + V, DW Corollary 9 s compatble wth, ff X, Y, Z X(M) : X Y, Z = X Y, Z + Y, X Z Proof of the proposton =: obvous; = : By t 0 I choosng an orthonormal bass {P (t 0 )} n =1 of T c(t 0 )M and extendng t to an orthonormal bass {P (t)} of T c(t) M for each t, V = V P and / = V P (the same for W ) and thus d V, W = d V W = ( V W + V Ẇ ) =, W + V, DW Defnton 10 s sad to be symmetrc ff X, Y X(M) : X Y Y X [X, Y ] Γ k j Γ k j Theorem 11 (Lev-Cvtà) There exsts a unque lnear connecton on M the Remannan or Lev-Cvtà connecton st s symmetrc and compatble wth, Lev-Cvtà connecton 4

5 Proof Unqueness: X Y, Z = X Y, Z + Y, X Z, Y Z, X = Y Z, X + Z, Y X, Z X, Y = Z X, Y X, Z Y, X Y, Z + Y Z, X Z X, Y = [X, Z], Y + [Y, Z], X + [X, Y ], Z + 2 Z, Y X It follows the Koszul formula: Z, Y X = 2{ 1 X Y, Z + Y Z, X Z X, Y [X, Z], Y [Y, Z], X [X, Y ], Z } Exstence: Use the Koszul formula as defnton From the Koszul formula follows k, j = 1 2 { j, k + j k, k, j ± 0, } and thus Γ k j = 1 g km { g jk + j g k k g j } 2 k 5