The general world model for homogeneous and isotropic universe has been proposed. For this purpose, we introduce a global and fiducial system of reference (world reference frame) constructed on a (4+1)-dimensional space-time, and assume that the universe is spatially a 3-dimensional hypersurface embedded in the 4-dimensional space. The simultaneity for the entire universe has been specified by the global time coordinate. We define the line element as the separation between two neighboring events on the expanding universe that are distinct in space and time, as viewed in the world reference frame. The information that determines the kinematics of the geometry of the universe such as size and expansion rate has been included in the new metric. The Einstein’s field equations with the new metric imply that closed, flat, and open universes are filled with positive, zero, and negative energy, respectively. The curvature of the universe is determined by the sign of mean energy density. We have demonstrated that the flat universe is empty and stationary, equivalent to the Minkowski space-time, and that the universe with positive energy density is always spatially closed and finite. In the closed universe, the proper time of a comoving observer does not elapse uniformly as judged in the world reference frame, in which both cosmic expansion and time-varying light speeds cannot exceed the limiting speed of the special relativity. We have also reconstructed cosmic evolution histories of the closed world models that are consistent with recent astronomical observations, and derived useful formulas such as energy-momentum relation of particles, redshift, total energy in the universe, cosmic distance and time scales, and so forth. The notable feature of the spatially closed universe is that the universe started from a non-singular point in the sense that physical quantities have finite values at the initial time as judged in the world reference frame. It has also been shown that the inflation with positive acceleration at the earliest epoch is improbable.

Abstract
1. Introduction
2. How to define metric forhomogeneous and isotropic universes?
    (a) Friedmann-Robertson-Walker metric
    (b) Global reference frame and new metric
3. Metric and evolution equations ofexpanding universes
    (a) Flat universe
    (b) Closed universe
    (c) Open universe
    (d) Our Universe is spatially closed
    (e) The Friedmann equations
4. Physical and astronomical aspectsof the expanding closed universe
    (a) Time-varying light speed and cosmic expansionspeed
    (b) Cosmic evolution history
    (c) Energy-momentum relation of particles
    (d) Doppler shift and cosmological redshift ofphotons
    (e) Total energy in the Universe
    (f) Energy density, pressure, and temperaturein thermal equilibrium
    (g) Cosmic distance and time scales
5. Inflation
6. Conclusion
References

• Chan-Gyung Park(Division of Science Education and Institute of Fusion Science, Chonbuk National University) Corresponding author