# Tangent spaces that are not simply $\mathbb{R}^n$.

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?

• The tangent space of an $m$-dimensinal manifold is an $m$-dimensional linear space, and hence in general isomorphic to $\mathbb{R}^m$. – MisterRiemann May 31 at 11:44
• Each individual tangent space of a smooth manifold $X^n$ is (more or less by the definition of smoothness) an $n$-dimensional space and thus isomorphic to $\mathbb{R}^n$. The more interesting part is the structure of $TX$ as a vector bundle rather than a series of vector spaces; in general, it isn't just $X\times \mathbb{R}^n$. – anomaly May 31 at 13:02
• Perhaps you were interested in knowing about the Tangent bundle---how each tangent spaces fit together. A simple example of a non-trivial tangent bundle (i.e., $TM \neq M \times \mathbb R^n$) is a sphere $S^2$. – jdoicj Jun 3 at 14:03

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 $$\mathfrak{g}$$.