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How can one show that the tangent bundle TS² of the 2-sphere is not trivial ? I know we can use the tools of algebraic topology, but I'm looking for a way to show it only with elementary tools of differential geometry.

More precisely, I constructed an atlas for TS² by using the stereographic projection on S². I also computed the expression of the change of coordinates for the tangent vectors in TS², following the equator; it is the application defined by the matrix :

[tex]\left(\begin{array}{cc}-1 & 0 \\ 0 & 1 \end{array}\right)\left(\begin{array}{cc}\cos 2\theta & \sin2\theta \\ & \\ -\sin2\theta & \cos2\theta\end{array}\right)[/tex]

where [tex]\theta[/tex] is the polar angle coordinate of the point on the equator.

From that point on, how can I show that TS² is non trivial ? (--> How can I show that there is no non-vanishing section of TS² ?)

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# Tangent bundle on the sphere

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