differential geometry w notes from teacher_Part_22

# differential geometry w notes from teacher_Part_22 - 43 2.3...

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2.3. THE COTANGENT BUNDLE 43 The coordinates p are not functions on M . The Poincar´e form is not a 1-form on M . A general 1-form on T * M is α = n X i = 1 α i ( q , p ) dq i + n X i = 1 v i ( q , p ) dp i . Theorem 2.3.1 The Poincar´e 1 -form is well deﬁned globally on the cotangent bundle of any manifold. Proof : 1. Let ( q α , p α ) and ( q β , p β ) be two overlapping coordinate patches of T * M . 2. Then dq i α = n X j = 1 q i α q j β dq j β and n X i = 1 p α i dq i α = n X j = 1 p β j dq j β . ± We give now an intrinsic deﬁnition of the Poincar´e form. Let ( q , p ) T * M be a point in T * M . We want to deﬁne a 1-form λ T * ( q , p ) ( T * M ) at this point ( q , p ) T * M . Let π : T * M M be the projection deﬁned for any q M , p T * q M by π ( q , p ) = q . Then the pullback is the map π * : T * q M T * ( q , p ) ( T * M ). For each 1-form p T * q M it deﬁnes a 1-form π * ( p )

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differential geometry w notes from teacher_Part_22 - 43 2.3...

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