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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.

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1 Answer 1

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 $n$.

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