foliations with diagonal leaves
The torus has a foliation by Villarceau circles which are "diagonal" in the sense that the projection from the torus onto either factor, when restricted to a circular leaf of the foliation, is a homeomorphism.
(Specifically, we can take the circles to be translates of the diagonal $Delta$ under the $S^1$ group operation: for $x in S^1$, the leaf $L_x$ is ${(a, ax): a in S^1}$. The way that I understand the Hopf fibration is to look at the three-sphere as the join of two circles: we can foliate each torus slice as above, and then show that at each end, when the torus is collapsed to a circle, that circle is still compatible with the foliation.)
For what other manifolds $M$ is there a foliation of $M times M$ with this property?
It seems clear that we can generalize the construction above if $M$ is a Lie group: the map $M times M to M$, $(x, y) mapsto xy^{-1}$ is a fiber bundle of manifolds, its fibers therefore form a foliation, and the leaves are clearly diagonal. Can we do it in any other cases? For example, can we do it when $M = S^2$?
Bonus question: are there any other interesting examples for general topological spaces $X$ of fiber bundles $X times X to X$ whose fibers are diagonal?
differential-topology lie-groups smooth-manifolds
asked Nov 25 at 20:15
Hew Wolff
2,173716
add a comment |
The torus has a foliation by Villarceau circles which are "diagonal" in the sense that the projection from the torus onto either factor, when restricted to a circular leaf of the foliation, is a homeomorphism.
(Specifically, we can take the circles to be translates of the diagonal $Delta$ under the $S^1$ group operation: for $x in S^1$, the leaf $L_x$ is ${(a, ax): a in S^1}$. The way that I understand the Hopf fibration is to look at the three-sphere as the join of two circles: we can foliate each torus slice as above, and then show that at each end, when the torus is collapsed to a circle, that circle is still compatible with the foliation.)
For what other manifolds $M$ is there a foliation of $M times M$ with this property?
It seems clear that we can generalize the construction above if $M$ is a Lie group: the map $M times M to M$, $(x, y) mapsto xy^{-1}$ is a fiber bundle of manifolds, its fibers therefore form a foliation, and the leaves are clearly diagonal. Can we do it in any other cases? For example, can we do it when $M = S^2$?
Bonus question: are there any other interesting examples for general topological spaces $X$ of fiber bundles $X times X to X$ whose fibers are diagonal?
differential-topology lie-groups smooth-manifolds
asked Nov 25 at 20:15
Hew Wolff
2,173716
add a comment |
The torus has a foliation by Villarceau circles which are "diagonal" in the sense that the projection from the torus onto either factor, when restricted to a circular leaf of the foliation, is a homeomorphism.
(Specifically, we can take the circles to be translates of the diagonal $Delta$ under the $S^1$ group operation: for $x in S^1$, the leaf $L_x$ is ${(a, ax): a in S^1}$. The way that I understand the Hopf fibration is to look at the three-sphere as the join of two circles: we can foliate each torus slice as above, and then show that at each end, when the torus is collapsed to a circle, that circle is still compatible with the foliation.)
For what other manifolds $M$ is there a foliation of $M times M$ with this property?
It seems clear that we can generalize the construction above if $M$ is a Lie group: the map $M times M to M$, $(x, y) mapsto xy^{-1}$ is a fiber bundle of manifolds, its fibers therefore form a foliation, and the leaves are clearly diagonal. Can we do it in any other cases? For example, can we do it when $M = S^2$?
Bonus question: are there any other interesting examples for general topological spaces $X$ of fiber bundles $X times X to X$ whose fibers are diagonal?
differential-topology lie-groups smooth-manifolds
asked Nov 25 at 20:15
Hew Wolff
2,173716
The torus has a foliation by Villarceau circles which are "diagonal" in the sense that the projection from the torus onto either factor, when restricted to a circular leaf of the foliation, is a homeomorphism.
(Specifically, we can take the circles to be translates of the diagonal $Delta$ under the $S^1$ group operation: for $x in S^1$, the leaf $L_x$ is ${(a, ax): a in S^1}$. The way that I understand the Hopf fibration is to look at the three-sphere as the join of two circles: we can foliate each torus slice as above, and then show that at each end, when the torus is collapsed to a circle, that circle is still compatible with the foliation.)
For what other manifolds $M$ is there a foliation of $M times M$ with this property?
It seems clear that we can generalize the construction above if $M$ is a Lie group: the map $M times M to M$, $(x, y) mapsto xy^{-1}$ is a fiber bundle of manifolds, its fibers therefore form a foliation, and the leaves are clearly diagonal. Can we do it in any other cases? For example, can we do it when $M = S^2$?
Bonus question: are there any other interesting examples for general topological spaces $X$ of fiber bundles $X times X to X$ whose fibers are diagonal?
differential-topology lie-groups smooth-manifolds
differential-topology lie-groups smooth-manifolds
asked Nov 25 at 20:15
Hew Wolff
2,173716
asked Nov 25 at 20:15
Hew Wolff
2,173716
asked Nov 25 at 20:15
Hew Wolff
2,173716
asked Nov 25 at 20:15
Hew Wolff
2,173716
asked Nov 25 at 20:15
Hew Wolff
2,173716
2,173716
add a comment |
add a comment |
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