In trying to understand why not all tangent bundles are trivial, I've attempted to prove that they are all trivial and see where things go wrong. Unfortunately, I finished the proof and cannot find my mistake. Here it is:(adsbygoogle = window.adsbygoogle || []).push({});

Let M be an n-manifold with coordinate charts [tex](U_\alpha, \phi_\alpha)[/tex]. Therefore [tex](\pi^{-1}(U_\alpha), \tilde{\phi_\alpha})[/tex] are charts for TM where [tex]\pi[/tex] is the projection map and [tex]\tilde{\phi_\alpha}(p, v^i \frac{\partial}{\partial x^i}\vert_p) = (\phi(p),v^1, \ldots, v^n)[/tex]. I claim that the map F from TM to M x R^n given by [tex]F(p, v^i \frac{\partial}{\partial x^i}\vert_p) = (p, v^1, \ldots, v^n)[/tex] is a diffeomorphism. Clearly F is bijective so it is sufficient to check that F is a local diffeomorphism. Thus let [tex](p, v^i \frac{\partial}{\partial x^i}\vert_p)[/tex] be an arbitrary point in TM. [tex]p \in U_\alpha[/tex] for some [tex]\alpha[/tex] so [tex]\pi^{-1}(U_\alpha)[/tex] is an open set (indeed a chart) of TM containing [tex](p, v^i \frac{\partial}{\partial x^i}\vert_p)[/tex] and [tex]F(\pi^{-1}(U_\alpha)) = U_\alpha \times R^n[/tex] is a chart of M x R^n. But the restriction of F to [tex]\pi^{-1}(U_\alpha)[/tex] is [tex](\phi_\alpha^{-1} \times Id_{R^n}) \circ \tilde{\phi_\alpha}[/tex], which is a diffeomorphism (being a composition of diffeomorphisms).

Where did I go wrong?

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

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

# Trivial Tangent Bundles

Loading...

Similar Threads - Trivial Tangent Bundles | Date |
---|---|

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

Trivial Isometry Group for the Reals | Sep 4, 2014 |

How to prove a vector bundle is non-trivial using transition functions | Apr 5, 2014 |

Existence of Hodge Dual: obvious or non-trivial? | Feb 18, 2014 |

Does trivial cotangent bundle implies trivial tangent bundle? | Mar 2, 2010 |

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