If the tangent bundle is trivial, then the cotangent bundle is trivial. To see this, consider (X_i) a global frame for TM. Then define a global frame (\alpha^i) for T*M by setting [itex]\alpha^i(X_j)=\delta_{ij}[/itex] and extend by linearity.(adsbygoogle = window.adsbygoogle || []).push({});

Does trivial cotangent bundle implies trivial tangent bundle? A similar argument based on global frames does not seem to work in this direction: given a global frame (\alpha^i) for T*M, how do you define a global frame for TM? It does not makse sense to say "Let X_i be the vector field such that [itex]\alpha^i(X_j)=\delta_{ij}[/itex]" because such a vector field might not exist. And if locally, [itex]\alpha^i=\sum_j\alpha^i_jdx^j[/itex], then defining a (global) vector field by setting [itex]X_i:=\sum_j\alpha^i_j\partial_j[/itex] is inconsistent because the coefficients [itex]\alpha^i_j[/itex] do no transform correctly.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Does trivial cotangent bundle implies trivial tangent bundle?

Loading...

Similar Threads - Does trivial cotangent | Date |
---|---|

I Does a covariant version of Euler-Lagrange exist? | Feb 20, 2018 |

A Principal bundle triviality, groups and connections | Feb 10, 2017 |

How does regularity of curves prevent "cusps"? | Feb 16, 2015 |

Does spherical symmetry imply spherical submanifolds? | Jun 4, 2014 |

How does metric give complete information about its space? | Apr 15, 2014 |

**Physics Forums - The Fusion of Science and Community**