Homogeneous literally means "to be the same throughout," no matter where you are in the universe. If you look at the universe from Earth or from a galaxy a million light-years away, it will look the same.uniform with respect to position (Homogeneous), and uniform with respect to viewing angle (Isotropic). But Manifold the concept itself uniform structure. Then why we need Isotropic and Homogeneous?
The manifold refers to space and time while homogeneous and isotropic refers to the large scale distribution of matter within the universe.
Thank you for all your answers. I did PhD in pure maths. I would like to work on Quantum gravity. Can I apply for another PhD for physics?
Isotropy and homogeneity are two key cosmological concepts which are an idealization of reality. In practice, we know that we do see stars and galaxies where matter is clumped. However in a large enough scale (more than 300 mega parsec) these clumps are smoothed out, universe then is approximately homogeneous. Similarly in the CMBR one finds anisotropy in the temperature T (mean temperature is 2.7 K) of the order of one part in 105. That is isoptropy of the universe is also only an approximate reality.
These two key approximations follow from the cosmological principle according to which there is no preferred observer. That is if you observe natural phenomenon on earth and I do it in a different place then both of us will completely agree on the underlying laws of Physics. In reality me and you will mostly agree, but there also remains a chance of a slight disagreement.