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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

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What is wrong with the induced atlas? –  Sergei Ivanov Apr 13 '10 at 18:36
3  
I think you need to consult at least a few more references. This is really basic standard stuff that is explained very carefully in many places. One book that should have this and that many have recommended on MathOverflow is John Lee's Introduction to Smooth Manifolds. –  Deane Yang Apr 13 '10 at 20:04
12  
Marc, you should mention right at the top that you're interested in the infinite dimensional case. For many people, a manifold is by definition finite dimensional. –  HJRW Apr 13 '10 at 20:21
2  
You already have a homeomorphism between $\pi^{-1}(U)$ and $\varphi(U)\times X$ . And you have a homeomorphim between $U\times X$ and $\varphi(U)\times X$, it is just $\varphi\times id_X$. Compose them and you get a trivialization. –  Sergei Ivanov Apr 13 '10 at 21:07
1  
How do you define $T_p(M)$ if $M$ is modeled on arbitrary normed vector spaces and how is the norm on $T_p(M)$ defined? –  Martin Brandenburg Apr 13 '10 at 22:09
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1 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)$.

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5  
He is apparently interested in the case where $M$ is infinite dimensional, modeled on a normed space. –  Mariano Suárez-Alvarez Apr 13 '10 at 18:53
    
Ah, I see. Is it MO protocol that I delete my answer? –  Bill Kronholm Apr 13 '10 at 19:05
    
This is indeed the case, I found some texts about the finite dimensional case. BTW for the definition of a vector bundle I found also only the finite dimensional case so I extended this to the infinite case (hope my definition is correct) –  Marc Mertens Apr 13 '10 at 19:05
    
Don't know what MO protocol is but I made a error in my comment –  Marc Mertens Apr 13 '10 at 19:07
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