If we have a set $X=A\cup B$ and $Y= A\cap B$. What is the difference between $TY$ (the tangent bundle of $Y$) and $T_{Y}X$ (the tangent bundle of $X$ at $Y$)? That is, $Y$ is subset of $X$, so I can not understand what we mean by tangent bundle of $X$ at $Y$ and how it is different from tangent bundle of $Y$.
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1Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Sep 04 '21 at 09:22
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I'm not sure what you mean by $T_YX$. Did you see this in a book? – Michael Albanese Sep 04 '21 at 10:47
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I saw this in an article for Nils Dencker named "the propagation of polarization in Double Refraction " in the third page of the article, but he did not gave the definition of it. – Ray Sep 04 '21 at 11:25
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Isn't it possible that $Y=A\cap B$ is a singleton ${y}$? Then I would interpret $T_YX$ as $T_yX$. Maybe more generally, it could be the vector bundle of fibers $T_yX$ over $Y$... – Berci Sep 04 '21 at 13:13
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No the intersection is not a singelton. And in this case how it differs from TY – Ray Sep 04 '21 at 13:30
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If for example $A,B$ are disks sharing the boundary circle $S^1=Y=A\cap B$, then $X\cong S^2$, and $T_yX$ is 2 dimensional whereas $T_yY$ is a 1 dimensional subspace of it. – Berci Sep 04 '21 at 18:41
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Thank you for this explanation. Now I understood it. Yes there a difference. – Ray Sep 05 '21 at 11:31