Line bundles associated to principal circle bundles

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Let $pi: P rightarrow B$ be a principal circle bundle over $B$ and $rho: S^1 times mathbbC rightarrow mathbbC$ an effective left action. Then, one can associate to the bundle $pi$ a complex line bundle $pi_rho:P times_rho mathbbC rightarrow B$ by the canonical projection, where $$ P times_rho mathbbC := [p,z]in P times mathbbC, . $$
My question is the following: Define two left circle actions
$rho_1, rho_2: S^1 times mathbbC rightarrow mathbbC$ by

$$rho_1(theta, z)=e^ithetaz, rho_2(theta,z)=e^-ithetaz.$$
Then,



1) Are two associated bundle $pi_rho_j:P times_rho_jmathbbC rightarrow B$ ($j=1,2$) isomorphic as vector bundles?;



2) Are the two total spaces $P times_rho_jmathbbC$ ($j=1,2$) mutually diffeomorphic?



I am happy to get to know the answer to each question. Thank you in advance.




lie-groups complex-geometry vector-bundles group-actions principal-bundles




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    up vote
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    Let $pi: P rightarrow B$ be a principal circle bundle over $B$ and $rho: S^1 times mathbbC rightarrow mathbbC$ an effective left action. Then, one can associate to the bundle $pi$ a complex line bundle $pi_rho:P times_rho mathbbC rightarrow B$ by the canonical projection, where $$ P times_rho mathbbC := [p,z]in P times mathbbC, . $$
    My question is the following: Define two left circle actions
    $rho_1, rho_2: S^1 times mathbbC rightarrow mathbbC$ by

    $$rho_1(theta, z)=e^ithetaz, rho_2(theta,z)=e^-ithetaz.$$
    Then,



    1) Are two associated bundle $pi_rho_j:P times_rho_jmathbbC rightarrow B$ ($j=1,2$) isomorphic as vector bundles?;



    2) Are the two total spaces $P times_rho_jmathbbC$ ($j=1,2$) mutually diffeomorphic?



    I am happy to get to know the answer to each question. Thank you in advance.




    lie-groups complex-geometry vector-bundles group-actions principal-bundles




    share|cite|improve this question





















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      Let $pi: P rightarrow B$ be a principal circle bundle over $B$ and $rho: S^1 times mathbbC rightarrow mathbbC$ an effective left action. Then, one can associate to the bundle $pi$ a complex line bundle $pi_rho:P times_rho mathbbC rightarrow B$ by the canonical projection, where $$ P times_rho mathbbC := [p,z]in P times mathbbC, . $$
      My question is the following: Define two left circle actions
      $rho_1, rho_2: S^1 times mathbbC rightarrow mathbbC$ by

      $$rho_1(theta, z)=e^ithetaz, rho_2(theta,z)=e^-ithetaz.$$
      Then,



      1) Are two associated bundle $pi_rho_j:P times_rho_jmathbbC rightarrow B$ ($j=1,2$) isomorphic as vector bundles?;



      2) Are the two total spaces $P times_rho_jmathbbC$ ($j=1,2$) mutually diffeomorphic?



      I am happy to get to know the answer to each question. Thank you in advance.




      lie-groups complex-geometry vector-bundles group-actions principal-bundles




      share|cite|improve this question











      Let $pi: P rightarrow B$ be a principal circle bundle over $B$ and $rho: S^1 times mathbbC rightarrow mathbbC$ an effective left action. Then, one can associate to the bundle $pi$ a complex line bundle $pi_rho:P times_rho mathbbC rightarrow B$ by the canonical projection, where $$ P times_rho mathbbC := [p,z]in P times mathbbC, . $$
      My question is the following: Define two left circle actions
      $rho_1, rho_2: S^1 times mathbbC rightarrow mathbbC$ by

      $$rho_1(theta, z)=e^ithetaz, rho_2(theta,z)=e^-ithetaz.$$
      Then,



      1) Are two associated bundle $pi_rho_j:P times_rho_jmathbbC rightarrow B$ ($j=1,2$) isomorphic as vector bundles?;



      2) Are the two total spaces $P times_rho_jmathbbC$ ($j=1,2$) mutually diffeomorphic?



      I am happy to get to know the answer to each question. Thank you in advance.





      lie-groups complex-geometry vector-bundles group-actions principal-bundles





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      asked Jul 21 at 18:07









      Takao

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