Simple question about definition of tangent bundleby pellman Tags: bundle, definition, simple, tangent 

#1
Nov1608, 02:20 PM

P: 565

So I'm trying to learn about fibre bundles and I am looking at the example of a tangent bundle.
Given a differentiable manifold M. Denote the tangent space at [tex]p \in M[/tex] by [tex]T_p M[/tex]. Is the definition of the tangent bundle [tex]TM = \lbrace (p, T_p M)p \in M \rbrace[/tex] or is it [tex]TM = \lbrace (p, V)p \in M , V \in T_p M\rbrace[/tex]? Maybe I'm splitting hairs but there should be standard definition of one or the other, right? I can discuss further why I think it matters but first let's just see if anyone is certain about the answer. 



#2
Nov1608, 03:03 PM

Emeritus
Sci Advisor
PF Gold
P: 16,101

1. A topological space E, which consists of 1a. A set of points E 1b. A topology on E 2. A continuous function E > M (often called the 'projection map', or the 'structure map') 



#3
Nov1608, 03:24 PM

P: 565

Hurkyl, you da man. Thanks for the quick response.
So bijective is bad? That's part of what I don't get. I'm following Nakahara. You can see the page I am on here http://books.google.com/books?id=cH...um=1&ct=result So when he says [tex]\pi^{1}(p)=T_p M[/tex] he's being very loose with the inverse notation, right? [tex]\pi^{1}[/tex] doesn't really exist, since [tex]\pi((p,V))=p[/tex] for every [tex]V \in T_p M[/tex]? 



#4
Nov1608, 04:11 PM

Emeritus
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PF Gold
P: 16,101

Simple question about definition of tangent bundle 



#5
Nov1608, 05:07 PM

P: 565

Ok. That gives me enough to press on. I'm sure I will get it when I see other examples. Thanks again.



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