I'm studying differential geometry using De Carmo. To define the notion of a derivative on a manifold we need to associate with each point $x$ on the manifold $M$ a tangent space $TM_x$. In any examples I've encountered the tangent space has ended up being $TM_x=\mathbb{R}^m$ where $m$ is the dimension of the manifold.
I assume this is not always the case but I cannot think of any examples that give a more exotic or unusually tangent space. Can you give an example of a smooth map between smooth manifold with an unusual tangent spaces?
Best Answer
The vector space structure of $T_pM$ is a $\dim_\mathbb{R} M$-dimensional vector space over $\mathbb{R}$, so it is isomorphic to $\mathbb{R}^{\dim_\mathbb{R} M}$.
However, since you tag your question with
lie-groups, there is another interpretation of your question. The tangent spaces to a Lie group each has a natural Lie algebra structure inherited from the parallelization with left-invariant vector fields. So $T_gG$ is not simply the $\mathbb{R}$-vectorspace $\mathbb{R}^k$, but a Lie algebra $\mathfrak{g}$.