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I am looking for (at least) one example of the following manifolds:

  1. Flat, homogeneous and isotropic
  2. Curved, homogeneous and isotropic
  3. Flat, non-homogeneous and isotropic
  4. Flat, homogeneous and non-isotropic
  5. Curved, non-homogeneous and isotropic
  6. Curved, homogeneous and non-isotropic
  7. Flat, non-homogeneous and non-isotropic
  8. Curved, non-homogeneous and non-isotropic

By isotropic, I mean isotropic at least about some specific point.

This is what comes to (my) mind:

  1. Euclidean space
  2. (2-)Sphere
  3. Euclidean space with one point removed / Vector space
  4. Minkowski space
  5. Sphere with one point removed?
  6. ?
  7. Euclidean space with two point removed?
  8. A generic manifold

I hope you can provide some better examples.

  • 1
    An example of 6 would be a rotating cosmological spacetime: http://physics.stackexchange.com/q/1048/ –  Aug 05 '14 at 19:01
  • 2
    If a space is flat, then it is isomorphic to a space with all $\pm 1$ along the diagonals. Under usual assumptions, your only choices are Minkowski and Euclidean space. So 4 and 7 are the empty set, unless you play games global topology and use that as an excuse for anisotropy. – Zo the Relativist Aug 05 '14 at 19:05
  • I sincerely hope that if anyone answers this comprehensively, then she/he does so in the form of a table as opposed to a list. – joshphysics Aug 05 '14 at 19:25
  • @JerrySchirmer: Shouldn't Minkowski space be considered anisotropic since it has a light-cone structure and therefore a preferred (time-like) direction? –  Aug 06 '14 at 11:40
  • @LittleBrownOne: usually people don't talk about spacetimes as being isotropic or not, just spaces. – Zo the Relativist Aug 06 '14 at 14:48
  • @JerrySchirmer: ok, fair enough. But, do you agree with my observation that if we were to talk about isotropy of spacetime, then the Minkowski space time would be anisotropic? –  Aug 06 '14 at 15:00
  • @LittleBrownOne: yes, but it's kind of boringly so. – Zo the Relativist Aug 06 '14 at 15:28
  • @Jerry Perhaps to you :-) –  Aug 06 '14 at 15:29

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