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Tangent Space

  1. Dec 30, 2016 #1
    1. Let p be an arbitrary point on the unit sphere S2n+1 of Cn+1=R2n+2. Determine the tangent space TpS2n+1 and show that it contains an n-dimensional complex subspace of Cn+1


    2. Relevant equations


    3. It is easy to find tangent space of S1; it is only tangent vector field of S1. But what must do for higher dimension and how can I show it contains an n-dimensional subspace of Cn+1. Thanks for your helps.
     
  2. jcsd
  3. Dec 30, 2016 #2

    Orodruin

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    What is the definition of ##S^{2n+1}##? Given a point on the sphere, what infinitesimal Steps can you take from that point and still be on the sphere?
     
  4. Dec 30, 2016 #3

    Orodruin

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    By the way, this does not really answer the question. You are essentially saying "the tangent space is the tangent space". You should specify exactly what type of vectors are in this space.
     
  5. Dec 30, 2016 #4
    I think that tangent vector field must be
    X=-x2d/dx1+x1d/dx2 for any P(x1,x2) point on the C1. d/dx stand for partial derivative.
     
  6. Dec 30, 2016 #5

    Orodruin

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    And how did you figure this out?

    Edit: Also note that any vector proportional to that one will do.
     
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