Killing vector Definition and 53 Threads

Can a killing vector field generate a diffeomorphism that only shifts points inside a small part of the manifold and preserves points outside of it? Rotation preserves the metric of a sphere but shifts every points on the sphere, I'd to find out if there is a killing vector field that...

Let (U,V,\theta, \phi) be Kruskal coordinates on the Kruskal manifold, where -UV=\left(\frac{r}{2m}-1\right)e^{r/2m},\hspace{1cm} t=2m\ln\left(\frac{-V}{U}\right) and \theta and \phi are the usual polar angles. The metric is ds^2=\frac{-32m^3}{r}e^{\frac{-r}{2m}}dUdV+r^2d\Omega^2. The vector...

What is a time-like killing vector?