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
Best Answer
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 $[\partial x^i/\partial y^j]\in GL(\mathbb{R}^m)$.