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differential geometry w notes from teacher_Part_36

# differential geometry w notes from teacher_Part_36 - 71 3.2...

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3.2. ORIENTATION AND THE VOLUME FORM 71 That is, N is transverse to V if for any point p V , N p T p V . If N is transverse to V , then N 0 on V . Let W be an n -diemsnional manifold. An ( n - 1)-diemsnional submanifold M of W is called a hypersurface in W . A hypersurface M in W is two-sided in W if there is a continuous vector N in W transverse to M . Examples. Normals to surfaces in R 3 . Theorem 3.2.1 A two-sided hypersurface in an orientable manifold is orientable. Remarks. Orientability of a manifold is an intrinsic property of a manifold. Two-sidedness of a manifold depends on the embedding of the manifold as a hypersurface in a higher-dimensional manifold. Example. Every manifold (even a nonorientable one) M is a two-sided hypersurface in a manifold M × R . 3.2.4 Projective Spaces The real projective space R P 2 is the sphere S 2 with antipodal points identi- fied. The sphere S 2 is two-sided in R 3 , so it is orientable.

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