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ozlem
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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
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 said:It is easy to find tangent space of S1; it is only tangent vector field of S1.
I think that tangent vector field must beOrodruin said: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.
And how did you figure this out?ozlem said: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.
In mathematics, the tangent space of a manifold is the vector space that approximates the manifold near a given point. For S2n+1 in Cn+1, the tangent space is a 2n-dimensional vector space that contains all the possible directions in which a curve can pass through a point on the sphere.
The tangent space of S2n+1 in Cn+1 can be calculated by finding the span of the vectors that are tangent to the sphere at a specific point. These vectors can be obtained by taking the partial derivatives of the parametric equations of the sphere with respect to each coordinate.
Understanding the tangent space of S2n+1 in Cn+1 is crucial in studying the local behavior of curves and surfaces on the sphere. It allows for the calculation of derivatives and tangent vectors, which are important in fields such as differential geometry and physics.
The tangent space of S2n+1 in Cn+1 is closely related to the curvature of the sphere. The curvature at a point can be calculated using the vectors in the tangent space, and the shape of the tangent space can provide information about the overall curvature of the sphere.
Yes, there are visual representations of the tangent space of S2n+1 in Cn+1. One way to visualize it is by imagining a plane tangent to the sphere at a specific point. The tangent space at that point would then be the set of all possible directions that a curve can take on that plane.