Skip to main content
MathOverflow

Return to Question

add link to Bott's paper, minor spelling / grammar changes
Source Link
j.c.
j.c.
  • 13.6k
  • 3
  • 52
  • 90

A Bott's theoremA theorem of Bott states that if a manifold admits a metric with all geodesics closed, then its homologies arehomology is isomorphic to homologiesthe homology of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \mathbb{CP}^n, \mathbb{HP}^n$ or $\mathbb{C}a\mathbb{P}^2$.

The problem of constructing such metrics is known to be extremely hard.

So my particular interest is: what else is known about this problem in the case of dimension $3$?

For example, are all homologicalhomology spheres known to admit such metrics? If a homologicalhomology sphere (or a homologicalhomology $\mathbb{RP}^3$ --- btwby the way, are there any examples, which are not $\mathbb{RP}^3$?) hashas finite $\pi_1$, its universal coveringcover is $S^3$, and I believe that all geodesics willwill be closed in the metric induced by the standard metric on $S^3$.

But I've just learned that there are examples of homologicalhomology spheres with infinite fundamental group. TheirTheir Thurston geometry is modeled byon the universal coveringcover of $\operatorname{SL}(2, \mathbb{R})$. It would be very interesting to know if such homologicalhomology spheres admit metrics with all geodesics closed.

A Bott's theorem states that if a manifold admits a metric with all geodesics closed, then its homologies are isomorphic to homologies of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \mathbb{CP}^n, \mathbb{HP}^n$ or $\mathbb{C}a\mathbb{P}^2$.

The problem of constructing such metrics is known to be extremely hard.

So my particular interest is: what else is known about this problem in the case of dimension $3$?

For example, are all homological spheres known to admit such metrics? If a homological sphere (or a homological $\mathbb{RP}^3$ --- btw are there any examples, which are not $\mathbb{RP}^3$?) has finite $\pi_1$, its universal covering is $S^3$, and I believe that all geodesics will be closed in the metric induced by the standard metric on $S^3$.

But I've just learned that there are examples of homological spheres with infinite fundamental group. Their Thurston geometry is modeled by the universal covering of $\operatorname{SL}(2, \mathbb{R})$. It would be very interesting to know if such homological spheres admit metrics with all geodesics closed.

A theorem of Bott states that if a manifold admits a metric with all geodesics closed, then its homology is isomorphic to the homology of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \mathbb{CP}^n, \mathbb{HP}^n$ or $\mathbb{C}a\mathbb{P}^2$.

The problem of constructing such metrics is known to be extremely hard.

So my particular interest is: what else is known about this problem in the case of dimension $3$?

For example, are all homology spheres known to admit such metrics? If a homology sphere (or a homology $\mathbb{RP}^3$ --- by the way, are there any examples which are not $\mathbb{RP}^3$?) has finite $\pi_1$, its universal cover is $S^3$, and I believe that all geodesics will be closed in the metric induced by the standard metric on $S^3$.

But I've just learned that there are examples of homology spheres with infinite fundamental group. Their Thurston geometry is modeled on the universal cover of $\operatorname{SL}(2, \mathbb{R})$. It would be very interesting to know if such homology spheres admit metrics with all geodesics closed.

Source Link
V. Rogov
V. Rogov
  • 1.2k
  • 5
  • 13

3-manifolds with all geodesics closed

A Bott's theorem states that if a manifold admits a metric with all geodesics closed, then its homologies are isomorphic to homologies of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \mathbb{CP}^n, \mathbb{HP}^n$ or $\mathbb{C}a\mathbb{P}^2$.

The problem of constructing such metrics is known to be extremely hard.

So my particular interest is: what else is known about this problem in the case of dimension $3$?

For example, are all homological spheres known to admit such metrics? If a homological sphere (or a homological $\mathbb{RP}^3$ --- btw are there any examples, which are not $\mathbb{RP}^3$?) has finite $\pi_1$, its universal covering is $S^3$, and I believe that all geodesics will be closed in the metric induced by the standard metric on $S^3$.

But I've just learned that there are examples of homological spheres with infinite fundamental group. Their Thurston geometry is modeled by the universal covering of $\operatorname{SL}(2, \mathbb{R})$. It would be very interesting to know if such homological spheres admit metrics with all geodesics closed.