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February 2nd, 2019, 09:24 AM   #1
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Exercise on Tangent bundle

Hi everybody,
I have to verify that $S'\times S^2$ doesn't have a frame. That's what I would do: $S'$ has a frame so $T_S'= S'\times \mathbb R$. $S^2$ doesn't have a frame so $T_{S^2}\ne S^2 \times \mathbb {R}^2$. $S'\times S^2$ has a frame if and only if $T_{S'\times S^2}= S'\times S^2 \times \mathbb {R}^3$. But since $T_{S^2}\ne S^2 \times \mathbb {R}^2$ that's impossibile. Am I right? thank you all
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bundle, diffeomorphism, exercise, frame, tangent, tangent bundle



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