Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Please help, I am just learning about manifolds, I need to know how to solve this problem:

Let M be the image of the application $\phi : \mathbb{R}^n \rightarrow \mathbb{R}^{2n} $

$$\phi(u_1,u_2,...,u_n) = \frac{1}{1+ \displaystyle \sum_{i=1}^{n}u_i^2}(u_1,u_2,...,u_n,u_1^2,u_2^2,...,u_n^2) $$

Show that M is a differentiable manifold and compute its dimension.

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

It is almost a graph.

Choose as atlas the set $\{(M,\psi)\}$ where $\psi \colon M \to \mathbb R^n$ is defined by

$$\psi(x_1,\ldots,x_{2n}) = \left(1 + \sum_{i = 1}^n x_{n + i} \right)(x_1,\ldots,x_n)$$

Then $\psi \circ \phi = \text{Id}_{\mathbb R^n}$ and $\phi \circ \psi = \text{Id}_M$. $\psi$ is obviously continuous, hence it defines a homeomorphism, i.e. a local chart.

With this differentiable structure your manifold $M$ is diffeomorphic to $\mathbb R^n$, hence it has dimension $n$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.