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I just need to verify that I understand this correctly.

For some four dimensional manifold and group of isometries:

the dimension of the isometry group is given by the number of Killing vectors, while the dimension of the orbit group is given by the number of linearly independent Killing vectors. Right?

And in terms of understanding the orbit group: I think I understand what the orbit is, it's the points that are "traced out" from the function -- so given a point on a sphere and a rotation, we'd trace out the 2-sphere (by rotating around the polar and azimuthal angles)? So the dim of the orbit group for the rotations of a point on the sphere would have to be 2?

I'm still trying to make all the connections, but still just barely learning what Killing vectors and isometries and orbits are. :tongue2:

And are the only types of orbits Riemannian and Lorentzian? Is there such a thing as a "Null" orbit, and what does this mean exactly?

Thanks,

quasar