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The problem is "Find an explicit trivialization of the tangent bundle of $\mathbb{S}^{3}$", but I am confused because aren't all trivialization's of a tangent bundle just $\Phi(v^{i}\frac{\partial}{\partial u^{i}}|_{p})=(p,v^{1},...,v^{n})$? How can you get any more explicit?

EDIT: How hard is it to find a global trivialization of $\mathbb{S}^{3}$?

SihOASHoihd
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1 Answers1

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I cannot tell what your notation means.

But, what is meant by a trivialization of the tangent bundle $T\mathbb S^3$ is a diffeomorphism $F: T \mathbb{S}^3 \to \mathbb{S}^3 \times \mathbb R^3$ with the property that for each $p \in \mathbb S^3$, the function $F$ maps $T_p \mathbb{S}^3$ to $\{p\} \times \mathbb R^3$, and the composition $$T_p \mathbb{S}^3 \xrightarrow{F} \{p\} \times \mathbb R^3 \mapsto \mathbb R^3 $$ is a linear isomorphism.

The notation you have written does not have the appearance of such a function.

Alekos Robotis
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Lee Mosher
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  • You are asked to find a global trivialization. Since $S^3\not\cong \mathbb{R}^3$, you should not expect to find a global set of coordinates. So, such a global trivialization should not depend on coordinate representations. – Alekos Robotis May 17 '20 at 02:50
  • Is it hard to find an explicit global trivialization? – SihOASHoihd May 17 '20 at 03:23