Hi, I'm trying to attack a problem where the Riemannian metric depends explicitly on time, and is therefore a time-dependent assignment of an inner product to the tangent space of each point on the manifold.(adsbygoogle = window.adsbygoogle || []).push({});

Specifically, in coordinates I encounter a term which looks like

[tex]v^iv^j\frac{\partial g_{ij}(t,\gamma(t))}{\partial t}[/tex]

where gamma is a smooth curve on the manifold and v is an arbitrary element of the appropriate tangent space. I'd like to be able to write this object in a nice coordinate free way, but I can't quite see how to, since writing something like

[tex] \langle v_x,v_x \rangle_{\partial g/\partial t} [/tex]

doesn't make any sense, since the derivative of g does not necessairly satisfy the requirements of being an appropriate inner product....

Any insight on this, or on time dependent riemannian metrics in general, would help

**Physics Forums | Science Articles, Homework Help, Discussion**

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!

# Time-dependent Riemannian metric

Loading...

Similar Threads for dependent Riemannian metric |
---|

I Surface Metric Computation |

A Is the Berry connection a Levi-Civita connection? |

I Lie derivative of a metric determinant |

A On the dependence of the curvature tensor on the metric |

A Pushforward map |

**Physics Forums | Science Articles, Homework Help, Discussion**