- #1

center o bass

- 560

- 2

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?