Understanding Tangent Space of S2n+1 in Cn+1

[ozlem]

1. Let p be an arbitrary point on the unit sphere S²ⁿ⁺¹ of Cⁿ⁺¹=R²ⁿ⁺². Determine the tangent space T_pS²ⁿ⁺¹ and show that it contains an n-dimensional complex subspace of Cⁿ⁺¹

Homework 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.

 

[Orodruin]

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?

 

[Orodruin]

It is easy to find tangent space of S1; it is only tangent vector field of S1.

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.

 

[ozlem]

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.

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.

 

[Orodruin]

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.

And how did you figure this out?

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