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**

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

# Time-dependent Riemannian metric

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