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According to Scott's paper "The geometries of 3-manifolds", the only closed manifolds that admit an $S^2 \times \mathbb R$ geometry are the two $S^2$ bundles over $S^1$, $P^2 \times S^1$ and $P^3 \# P^3$. It seems like the last one is the only candidate (unless I'm missing a way in which $S^2 \times S^1$ can be viewed as such a bundle).

Edit. As pointed out below, I spoke too soon. Of course such bundles could have spherical geometry too. Scott's paper is a nice reference for that too!

According to Scott's paper "The geometries of 3-manifolds", the only closed manifolds that admit an $S^2 \times \mathbb{R}$ geometry are the two $S^2$ bundles over $S^1$, $P^2 \times S^1$ and $P^3 \# P^3$. It seems like the last one is the only candidate (unless I'm missing a way in which $S^2 \times S^1$ can be viewed as such a bundle).

Edit. As pointed out below, I spoke too soon. Of course such bundles could have spherical geometry too. Scott's paper is a nice reference for that too!

According to Scott's paper "The geometries of 3-manifolds", the only closed manifolds that admit an $S^2\times R$ geometry are the two $S^2$ bundles over $S^1$, $P^2\times S^1$ and $P^3\sharp P^3$. It seems like the last one is the only candidate (unless I'm missing a way in which $S^2\times S^1$ can be viewed as such a bundle).

(How do I get a connect sum symbol in maths mode!?)

EDIT: Used \sharp as Richard suggested.

FURTHER EDIT: As pointed out below, I spoke too soon. Of course such bundles could have spherical geometry too. Scott's paper is a nice reference for that too!

According to Scott's paper "The geometries of 3-manifolds", the only closed manifolds that admit an $S^2 \times \mathbb R$ geometry are the two $S^2$ bundles over $S^1$, $P^2 \times S^1$ and $P^3 \# P^3$. It seems like the last one is the only candidate (unless I'm missing a way in which $S^2 \times S^1$ can be viewed as such a bundle).

Edit. As pointed out below, I spoke too soon. Of course such bundles could have spherical geometry too. Scott's paper is a nice reference for that too!

According to Scott's paper "The geometries of 3-manifolds", the only closed manifolds that admit an $S^2\times R$ geometry are the two $S^2$ bundles over $S^1$, $P^2\times S^1$ and $P^3\sharp P^3$. It seems like the last one is the only candidate (unless I'm missing a way in which $S^2\times S^1$ can be viewed as such a bundle).

(How do I get a connect sum symbol in maths mode!?)

EDIT: Used \sharp as Richard suggested.

According to Scott's paper "The geometries of 3-manifolds", the only closed manifolds that admit an $S^2\times R$ geometry are the two $S^2$ bundles over $S^1$, $P^2\times S^1$ and $P^3\sharp P^3$. It seems like the last one is the only candidate (unless I'm missing a way in which $S^2\times S^1$ can be viewed as such a bundle).

(How do I get a connect sum symbol in maths mode!?)

EDIT: Used \sharp as Richard suggested.

FURTHER EDIT: As pointed out below, I spoke too soon. Of course such bundles could have spherical geometry too. Scott's paper is a nice reference for that too!