Is an open Ball a differentiable manifold?

Question

I am studying differentiable manifolds and tangent spaces and I have seen some examples of differential manifolds such as the spherical surface. Now I’m wondering if, in the case of an open ball, do we still have a 3 dimensional differential manifold or simply a manifold? and in such a case how should tangent space be inter-related?

Answer 1

Open ball should be similar to $\mathbb{R}^n$.

For an open interval $I$, for example, you can define you chart as $(I, \phi:x\in I \rightarrow x \in \mathbb{R}^1)$. The chart covers all $I$, and $\phi$ is a homeomorphism to $\mathbb{R}^1$. This is your smooth atlas, hence, $I$ is a smooth manifold.

Moreover, see book Lorint W. Tu: An introduction to manifolds, Example 5.12.: Open subset of a manifold is also a manifold.

The tangent space of an open ball is the same as for $\mathbb{R}^n$.