  We investigate the gravitational collapse of rapidly rotating relativistic
supermassive stars by means of a 3+1 hydrodynamical simulations in conformally
flat spacetime of general relativity. We study the evolution of differentially
rotating supermassive stars of $q \equiv J/M^{2} \sim 1$ ($J$ is the angular
momentum and $M$ is the gravitational mass of the star) from the onset of
radial instability at $R/M \sim 65$ ($R$ is the circumferential radius of the
star) to the point where the conformally flat approximation breaks down. We
find that the collapse of the star of $q \gtrsim 1$, a radially unstable
differentially rotating star form a black hole of $q \lesssim 1$. The main
reason to prevent formation of a black hole of $q \gtrsim 1$ is that quite a
large amount of angular momentum stays at the surface. We also find that most
of the mass density collapses coherently to form a supermassive black hole with
no appreciable disk nor bar. In the absence of nonaxisymmetric deformation, the
collapse of differentially rotating supermassive stars from the onset of radial
instability are the promising sources of burst and quasinormal ringing waves in
the Laser Interferometer Space Antenna.
