Replacing each fiber by the (co)tangent space of the fiber

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Let $Fto Xto B$ be a fiber bundle (of smooth manifolds) with fiber $F$ and base $B$. As a set, we may write
$$
X=bigcup_bin BF_b
$$
where $F_b$ denote the fiber at $bin B$.




Is there a canonical way to describe
$$
bigcup_bin B T^*F_b ~textor~ bigcup_bin B TF_b
$$
as some new fiber bundle over $B$?




I attempt to consider $T^*X$ or $TX$ but we have also horizontal parts; the above sets only consists of vertical parts. I guess the fiber bundle structures on the above sets depends on some choices and may not be canonically constructed; but I have no idea how to understand it.




differential-geometry differential-topology riemannian-geometry smooth-manifolds fiber-bundles




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    Let $Fto Xto B$ be a fiber bundle (of smooth manifolds) with fiber $F$ and base $B$. As a set, we may write
    $$
    X=bigcup_bin BF_b
    $$
    where $F_b$ denote the fiber at $bin B$.




    Is there a canonical way to describe
    $$
    bigcup_bin B T^*F_b ~textor~ bigcup_bin B TF_b
    $$
    as some new fiber bundle over $B$?




    I attempt to consider $T^*X$ or $TX$ but we have also horizontal parts; the above sets only consists of vertical parts. I guess the fiber bundle structures on the above sets depends on some choices and may not be canonically constructed; but I have no idea how to understand it.




    differential-geometry differential-topology riemannian-geometry smooth-manifolds fiber-bundles




    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $Fto Xto B$ be a fiber bundle (of smooth manifolds) with fiber $F$ and base $B$. As a set, we may write
      $$
      X=bigcup_bin BF_b
      $$
      where $F_b$ denote the fiber at $bin B$.




      Is there a canonical way to describe
      $$
      bigcup_bin B T^*F_b ~textor~ bigcup_bin B TF_b
      $$
      as some new fiber bundle over $B$?




      I attempt to consider $T^*X$ or $TX$ but we have also horizontal parts; the above sets only consists of vertical parts. I guess the fiber bundle structures on the above sets depends on some choices and may not be canonically constructed; but I have no idea how to understand it.




      differential-geometry differential-topology riemannian-geometry smooth-manifolds fiber-bundles




      share|cite|improve this question













      Let $Fto Xto B$ be a fiber bundle (of smooth manifolds) with fiber $F$ and base $B$. As a set, we may write
      $$
      X=bigcup_bin BF_b
      $$
      where $F_b$ denote the fiber at $bin B$.




      Is there a canonical way to describe
      $$
      bigcup_bin B T^*F_b ~textor~ bigcup_bin B TF_b
      $$
      as some new fiber bundle over $B$?




      I attempt to consider $T^*X$ or $TX$ but we have also horizontal parts; the above sets only consists of vertical parts. I guess the fiber bundle structures on the above sets depends on some choices and may not be canonically constructed; but I have no idea how to understand it.





      differential-geometry differential-topology riemannian-geometry smooth-manifolds fiber-bundles





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Aug 2 at 14:58
























      asked Jul 29 at 23:50









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