Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I'm reading the book Geometrical methods of mathematial physics by Brian Schutz. In chapter 3, on Lie groups, he states and proves that the vector fields on a manifold over which a particular tensor is invariant (i.e. has 0 Lie derivative over) form a Lie algebra. And associated with every Lie algebra is a Lie group.

Does this have any implications on the manifold somehow "containing" a Lie group in it (as some sort of submanifold maybe?), since the exponentiation of the Lie algebra gives you the identity component of the Lie group it's associated with? This seems not right since, for example, the Lie group of symmetries of a manifold may be of higher dimension than the manifold itself (e.g. Minkowski spacetime has dimension 4, while its group of symmetries, the Poincare group is of dimension 10).

Nevertheless, it seems that there has to be some sort of association going on? Or am I just taking crazy pills?

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

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!

# Manifolds with symmetries

Loading...

Similar Threads - Manifolds symmetries | Date |
---|---|

I Parallelizable Manifold | Nov 8, 2017 |

A On the dependence of the curvature tensor on the metric | Nov 2, 2017 |

A Can you give an example of a non-Levi Civita connection? | Oct 30, 2017 |

A Is tangent bundle TM the product manifold of M and T_pM? | Oct 20, 2017 |

Jet Prolongation Formulas for Lie Group Symmetries | Dec 13, 2015 |

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