Parallelizable Manifold


by amd939
Tags: manifold, parallelizable
amd939
amd939 is offline
#1
Oct31-07, 04:00 PM
P: 3
Hi, I am new to manifold and having a hard time on it. Could anyone please help me on the following problem. Please write down your thoughts. Thanks alot.

Prove that (S^n) X R is parallelizable for all n.
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shoehorn
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#2
Nov1-07, 12:11 AM
P: 448
Show us what you've done so far and tell us precisely where you're stuck.
amd939
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#3
Nov1-07, 01:04 PM
P: 3
Hi,I tried to show S^nXR is parallelizable by showing S^nXR is diffeomorphic to R^n+1\{0} so if R^n+1\{0} is parallelizable then the problem solved. But I just don't know how to show R^n+1\{0} is parallelizable.

Reverie
Reverie is offline
#4
Nov6-07, 07:56 PM
P: 28

Parallelizable Manifold


Quote Quote by amd939 View Post
Hi,I tried to show S^nXR is parallelizable by showing S^nXR is diffeomorphic to R^n+1\{0} so if R^n+1\{0} is parallelizable then the problem solved. But I just don't know how to show R^n+1\{0} is parallelizable.
R^(n+1)\{0} is a subset of R^(n+1). You can explicitly write down a global trivialization for R^(n+1) that restricts to a global frame for R^(n+1)\{0}.

A global frame for R^(n+1) is...

(1,0,...,0)
(0,1,0,...,0)
.
.
.
(0,...,0,1)


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