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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 LeviCivita 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
 Staff
 Geometry

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