Suppose we have a foliation of leaves (hypersurfaces) with codimension one of some Riemannian manifold ##M## with metric ##g##. For any point ##p## in ##M## we can then find some flat coordinate chart ##(U,\phi) = (U, (x^\mu, y))## such that setting ##y## to a constant locally labels each leaf (hypersurface) uniquely.(adsbygoogle = window.adsbygoogle || []).push({});

Let the dimension of the manifold ##M## be ##n## and let latin indicies run from ##1## to ##n## and greek run from ##1## to ##n-1##.

In the chart we can introduce a coordinate basis and express the metric as ##g=g_{ab}dx^a \otimes dx^b##. Now suppose that

$$\frac{\partial}{\partial y} g_{ab} = 0$$

in _any_ flat chart for the foliation.

Since the ##y##-curves defined by ##x^\mu## to constant are only defined within each flat chart for the foliation they are not globally defined (they might be disconnected and be at completely different angles from their respective leaves). And hence ##frac{\partial}{\partial y}## is not a globally defined killing vector.

But what are this kind of isometry then kalled? Somehow the metric ##g## respects the foliation. Are there any equivalent definitions of this situation?

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

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

# What kind of isometry? A metric tensor "respects" the foliation?

Loading...

Similar Threads for kind isometry metric |
---|

I Surface Metric Computation |

I Metrics and topologies |

I Lie derivative of a metric determinant |

I Conformal Related metrics |

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

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