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Unformatted text preview: Math 6455 Nov 14, 2006 1 Differential Geometry I Fall 2006, Georgia Tech Lecture Notes 15 Riemannian Geodesics Here we show that every Riemannian manifold admits a unique connection, called the Riemanninan or Levi-Civita connection, which satisfies two properties: symme- try, and compatibility with the metric, as we describe below. This result is known as the fundamental theorem of Rimeannian gemetry. Further we will show that the geodesics which arise from a Riemannian connection are locally minimize distance. 0.1 The bracket For any pair of vector fields X , Y ∈ X ( M ) we may define a new vector field [ X,Y ] ∈ X ( M ) as follows. First recall that T p M is isomorphic to D p M the space of derivations of the germ of functions of M at. Thus we may define [ X,Y ] by desrcribing how it acts on functions at each point: [ X,Y ] p f := X p ( Y f )- Y p ( Xf ) . One may check that this does indeed define a derivation, i.e., [ X,Y ] p ( λf + g ) = λ [ X,Y ] p f + [ X,Y ] p g , and [ X,Y ] p ( fg ) = ([ X,Y ] p f ) g ( p ) + f ( p )([ X,Y ] p g ). Further note that if e i ( p ); = e i denotes the standard basis vector field of R n then [ e i ,e j ] = 0 (since partial derivatives commute). On the other hand it is not difficult to construct examples of vector fields whose bracket does not vanish: Example 0.1. Let X , Y be vector fields on R 2 given by X ( x,y ) = (1 , 0) and Y ( x,y ) = (0 ,x ). Then [ X,Y ] f = X x ∂f ∂y- Y ∂f ∂x = ∂f ∂y + x ∂ 2 f ∂x∂y- x ∂ 2 f ∂y∂x = ∂f ∂y Lemma 0.2. Let f : M → N be a diffeomorphism, and X , Y ∈ X ( M ) . Then df ([ X,Y ]) = [ dfX,dfY ] . Proof. Recall that for any vectorfield Z on M and function g on N , we have ( ( dfZ ) g ) ( f ( p )) = ( dfZ ) f ( p ) g = ( df p Z ) g = Z p ( g ◦ f ) . 1 Last revised: November 21, 2006 1 Thus if we let Z := dfZ , and p := f ( p ), then ( Zg ) ◦ f ( p ) = ( Zg )( p ) = Z p g = Z p ( g ◦ f ) = Z ( g ◦ f ) ( p ) . Using the last set of identities, we may now compute [ X,Y ] g ( p ) = [ X,Y ] p ( g ◦ f ) = X p Y ( g ◦ f )- Y p X ( g ◦ f ) = X p ( Y g ) ◦ f- Y p ( Xg ) ◦ f = X p ( Y g )- Y p ( Xg ) = [ X, Y ] g ( p ) . Corollary 0.3. Let ( U,φ ) be a local chart of M and E i ( p ) := dφ- 1 φ ( p ) ( e i ) be the associated coordinate vector fields on U . Then [ E i ,E j ] = 0 ....
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This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Tech.
- Spring '08