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LectureNotes14U - Math 497C Curves and Surfaces Fall 2004,...

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Math 497C Nov 11, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 14 2.11 The Induced Lie Bracket on Surfaces; The Self- Adjointness of the Shape Operator Revisited If V , W are tangent vectorFelds on M , then we deFne [ V,W ] M := V W −∇ W V, which is again a tangent vector Feld on M . Note that since, as we had veriFed in an earlier exercise, S is self-adjoint, the Gauss’s formula yields that [ V,W ]= V W W V = W V −∇ V W + ³ - V,S ( W ) ® - W, S ( V ) ® ´ n =[ V,W ] M . In particular if V and W are tangent vectorFelds on M , then [ V,W ] is also a tangent vectorFeld. Let us also recall here, for the sake of completeness, the proof of the self- adjointness of S . To this end it suffices to show that if E i , i =1 ,2 ,i sa basis for T p M , then h E i ,S p ( E j ) i = h S p ( E i ) ,E j i . In particular we may let E i = X i (0 , 0), where X : U M is a regular patch of M centered at p .Now note that - X i ,S p ( X j ) ® = - X i ,dn p ( X j ) ® = - X i , ( n X ) j ® = - X ij , ( n X ) ® . Since the right hand side of the above expression is symmetric with respect to i
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LectureNotes14U - Math 497C Curves and Surfaces Fall 2004,...

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