# Understanding Tangent Space of S2n+1 in Cn+1

• ozlem
In summary, the conversation discusses finding the tangent space of the unit sphere S2n+1 and showing that it contains an n-dimensional complex subspace of Cn+1. The question also asks for clarification on the definition of S2n+1 and the type of vectors in the tangent space. The expert suggests a possible tangent vector field for C1 and notes that any vector proportional to it would suffice.
ozlem
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

## 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.

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?

ozlem said:
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.

Orodruin 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.
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.

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.
And how did you figure this out?

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

## What is the definition of tangent space in S2n+1 in Cn+1?

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.

## How is the tangent space of S2n+1 in Cn+1 calculated?

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.

## What is the significance of understanding tangent space in S2n+1 in Cn+1?

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.

## How does the tangent space of S2n+1 in Cn+1 relate to the concept of curvature?

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.

## Is there a visual representation of the tangent space of S2n+1 in Cn+1?

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.

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