How to shown that the Tangent Bundle of a vector space is a Vector Bundle

Question

Hello,

I have the following question about the tangent bundle $T_M = \bigcup_{p \in M} \{p\} \times T_p M$ defined on a manifold $M$ of class $C^r$ modeled on a normed space $X$. My problem is showing that the tangent bundle also forms a vector bundle. I found the following definition of a vector bundle

A vector bundle is a tuple $E, B, \pi, F, \mathcal{T}$ where $E, B$ are topological spaces, \ $\pi : E \rightarrow B$ a continuous surjection, $F$ a normed metric space, $\mathcal{T}$ is a family $\{U_i, \varphi_i \}_{i \in I}$ of homeomorphism $\varphi_i : U_i \times F \rightarrow \pi^{- 1} (U_i)$ with $B = \bigcup_{i \in I} U_i$ such that

• $\forall b \in B \succ \pi^{- 1} (\{b\})$ has the structure of a normed vectorspace

• $\forall i \in I$ we have $\forall x \in U_i$ and $\forall v \in F$ that $\pi (\varphi_i (x, v)) = x$

• $\forall i \in I, x \in U_i$ the map $\varphi_i^{(x)} : F \rightarrow \pi^{- 1} (\{x\})$ defined by $\varphi_i^{(x)} (v) = \varphi_i (x, v)$ is a linear function between the vector spaces $F$ and $\pi^{- 1} (\{x\})$

We call

• $E$ the total space of the vector bundle

• $B$ the base space of the vector bundle

• $\pi$ is the projection map of the bundle

• $\mathcal{T}$ is called a trivialization and $(U_i, \varphi_i)$ is called a trivializing neighborhood. \end{itemize}

Now for the tangent bundle it is easy to see that $T_M$ is the total space and $\pi : T_M \rightarrow M : (x, v) \rightarrow x$ is the projection, $M$ is the base space and I think we can equate $F$ with $X$, but how do you go on in finding a trivialization. I thought first about using the induced atlas on $T_M$ (that makes the tangent bundle a differentiable manifold of class $C^{r - 1}$ modelled on $X \times X$ but its mappings has not the correct format.

My problem with using the induced atlas as a trivialization is that it is of the form $\{U_i, \varphi_i \}_{i \in I}$ $\varphi_i : \pi^{- 1} (U_i) \rightarrow \varphi (U_i) \times X$ and using $\varphi_i^{- 1} : \varphi (U_i) \times X \rightarrow \pi^{- 1} (U_i)$ I'm almost there but I have still not found a homeomorphism of the form $U_i \times X \rightarrow \pi^{- 1} (U_i)$. The book I'm reading is talking about a tangent space and says it is \ vector bundle but does not define a vector bundle at all, so I looke up the definition of a vector bundle and failed to

Maybe I'm missing on the definition of a vector bundle (most examples I found on the internet are about finite dimensional spaces ).

Can anybody help me?

Thanks a lot in advance

Marc Mertens

Answer 1

The fiber $F$ should be the vector space of tangent vectors to $M$ at $x$. It sounds like everything else is clear to you except for the local trivializations.

Cover $M$ with coordinate patches. Suppose $x\in M$ and $x$ is contained in two coordinate patches $\mathcal{U}_1$ and $\mathcal{U}_2$ with coordinate functions $x^1, \dots, x^m$ and $y^1, \dots y^m$, respectively. Then over $\mathcal{U}_1$ the tangent space has the basis $\{\partial/\partial x^1, \dots \partial/\partial x^m\}$. Similarly, over $\mathcal{U}_2$ the tangent space has the basis $\{\partial/\partial y^1, \dots \partial/\partial y^m\}$. The transition function is then given by the Jacobian matrix $[\partial x^i/\partial y^j]\in GL(\mathbb{R}^m)$.