List all the circle bundles over the circle, sphere, projective plane, torus, and Klein bottle.
State the orientability of the total space, the base and the bundle (orientability of a circle bundle is equivalent to whether it is $ U_1 $ principal see https://mathoverflow.net/questions/144092/is-every-orientable-circle-bundle-principal ).
For circle bundles over a surface state which Thurston geometry the total space admits, if any.
Also state whether or not the total space of the bundle is homogeneous (transitive action of a Lie group)
This question is of interest since $ U_1 $ principal bundles are a topic of independent mathematical interest as are Thurston geometries on 3 manifolds. Also circle bundles over surfaces are related to Seifert fibrations which are an important topic in the theory of 3 manifolds.
Also based on structure theory of homogeneous spaces due to Mostow https://math.stackexchange.com/a/4374850/758507 nearly all homogeneous 3 manifold should be fiber bundles of the sort asked about in this question.
Base $ S^1 $:
All three: $ S^1 \to T^2 \to S^1 $ (in general any trivial bundle $ S^1 \times M $ for any orientable manifold $ M $ has base, bundle and total space all orientable). Total space is Riemannian homogeneous.
Only base orientable: $ S^1 \to K^2 \to S^1 $ where $ K^2 $ is the Klein bottle. Total space is a solvmanifold.
Base $ S^2 $:
Base $ T^2 $:
All three: The circle bundles $ S^1 \to MT(\begin{bmatrix} 1 & r \\ 0 & 1 \end{bmatrix}) \to T^2 $ where $ MT(\begin{bmatrix} 1 & r \\ 0 & 1 \end{bmatrix}) $ denotes the mapping torus of $ T^2 $ corresponding to the mapping class $ \begin{bmatrix} 1 & r \\ 0 & 1 \end{bmatrix} $, which is the $ r $th power of the Dehn twist $ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $. For $ r=0 $ this is $ T^3 $ and admits $ E^3 $ (flat) geometry while for $ r \neq 0 $ these are the nilmanifolds $ N_r $ described in Is every Nil manifold a nilmanifold? and they admit Nil geometry. $ E^3 $ geometry for $ r=0 $ otherwise Nil geometry. Total space is a nilmanifold.
Only base orientable: $ S^1 \rtimes_b T^2 $ two of the four flat compact non orientable three manifolds. For $ b=0 $ this is $ S^1 \times K^2 $ with first homology $ \mathbb{Z}^2 \times C_2 $, for $ b=1 $ this is the mapping torus of the Dehn twist diffeomorphism of $ K^2 $ with first homology $ \mathbb{Z}^2 $. The total space is not orientable. These coincide with the two $ U_1 $ principal bundles over $ K^2 $. $ E^3 $ geometry. Total space of any circle bundle over a torus is solvmanifold by https://mathoverflow.net/questions/416611/torus-bundles-and-compact-solvmanifolds .
Base $ \mathbb{R}P^2 $:
Only bundle orientable: $ S^1 \to S^1 \times \mathbb{R}P^2 \to \mathbb{R}P^2 $. (in general any trivial bundle $ S^1 \times M $ for any non orientable manifold $ M $ has only the bundle orientable). $ E^1 \times S^2 $ geometry. Total space is Riemannian homogeneous.
Only bundle orientable: $ S^1 \to (S^2 \times S^1)/(-1,-1) \to \mathbb{R}P^2 $. This is the mapping torus of the antipodal map of $ S^2 $. It is the unique nontrivial $ U_1 $ principal bundle over $ \mathbb{R}P^2 $. $ E^1 \times S^2 $ geometry. Total space is Riemannian homogenous see Mapping torus of the antipodal map of $ S^2 $ .
Only total space orientable: $ S^1 \to P_{4n,1} \to \mathbb{R}P^2 $ where $ P_{4n,1} $ is the standard prism manifold with $ 4n $ element dicyclic fundamental group. $ S^3 $ geometry. Total space is Riemannian homogeneous.
Only total space orientable: $ S^1 \to UT(\mathbb{R}P^2) \cong L_{4,1} \to \mathbb{R}P^2 $, the unit tangent bundle of $ \mathbb{R}P^2 $. $ S^3 $ geometry. Total space is Riemannian homogeneous.
Only total space orientable: $ S^1 \to \mathbb{R}P^3 \# \mathbb{R}P^3 \to \mathbb{R}P^2 $. $ E^1 \times S^2 $ geometry. Total space has transitive action by $ SE(3,\mathbb{R}) $ see Connected sum of two copies of $ RP^3 $
Base $ K^2 $:
Only the bundle is orientable: The two principal $ U_1 $ bundles over $ K^2 $ coincide with the two non principal $ S^1 $ bundles over $ T^2 $. These are two of the four non orientable compact flat three manifolds they can also be viewed as two of the four mapping tori of $ K^2 $. $ E^3 $ geometry. Again the total space of the bundle is always a solvmanifold since the total space of any circle bundle over a torus is a solvmanifold by https://mathoverflow.net/questions/416611/torus-bundles-and-compact-solvmanifolds .
none of 3: The other two of the four non orientable compact flat three manifolds, they can also be viewed as the other two of the four mapping tori of $ K^2 $. They are non principal bundles $ S^1 \to S^1 \rtimes_b K^2 \to K^2 $ both with non orientable total space. For $ b=0 $ this is the mapping torus of the Y homoeomorphism of $ K^2 $, it has first homology $ \mathbb{Z}^2 \times C_2 \times C_2 $. For $ b=1 $ this is the mapping torus of $ K^2 $ for the mapping class corresponding to the combination of a Dehn twist and a Y homoemorphism. It has first homology $ \mathbb{Z}^2 \times C_4 $. $ E^3 $ geometry. Both have total space which are not homogeneous see the argument given here Is a mapping torus of a solvmanifold always a solvmanifold? and the arguments given here https://math.stackexchange.com/a/4374850/758507
Only total space orientable: The circle bundles $ S^1 \to MT(\begin{bmatrix} -1 & -r \\ 0 & -1 \end{bmatrix}) \to K^2 $ where $ MT(\begin{bmatrix} -1 & -r \\ 0 & -1 \end{bmatrix}) $ denotes the mapping torus of $ T^2 $ corresponding to the mapping class $ \begin{bmatrix} -1 & -r \\ 0 & -1 \end{bmatrix} $. These manifolds are double covered by $ MT(\begin{bmatrix} 1 & 2r \\ 0 & 1 \end{bmatrix}) $. For $ r=0 $ this is the unit tangent bundle of the Klein $ UT(K^2) $, which admits $ E^3 $(flat) geometry, while for $ r \neq 0 $ these admit Nil geometry. $ E^3 $ geometry for $ r=0 $, Nil geometry otherwise. The mapping torus of $ T^2 $ is the total space of a bundle $ T^2 \rtimes S^1 $ and thus must be a solvmanifold by https://mathoverflow.net/questions/416611/torus-bundles-and-compact-solvmanifolds .