differential geometry w notes from teacher_Part_35

# differential geometry w notes from teacher_Part_35 - 69 3.2...

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3.2. ORIENTATION AND THE VOLUME FORM 69 3.2.2 Orientation of a Manifold Let M be a manifold and let T p M be the tangent space at a point p M . Let ( U , x ) be a local coordinate patch about a point p M . Then the vectors x i , i = 1 , . . . , n form a basis in T p M . Let ( U 0 , x 0 ) be another local coordinate system about a point p , that is, there is a local di eomorphism x i = x i ( x 0 ). Then the vectors x 0 i = x j x 0 i x j form another basis in T p M . The orientation of the bases { i } and { 0 j } is the same (or consistent ) if det ± x i x 0 j ! > 0 . If it is possible to choose an orientation of all tangent spaces T p M at all points in a continuous fashion, then the orientation of all tangent spaces is consistent. A manifold M is called orientable if there is an atlas such that the ori- entation of all charts of this atlas can be chosen consistently, that is, the Jacobians of all transition functions have positive determinant. Each connected orientable manifold has exactly two possible orientations.

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differential geometry w notes from teacher_Part_35 - 69 3.2...

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