## Bi-invariant metrics on the group of symplectomorphisms

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- by Zhigang Han PDF
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**361**(2009), 3343-3357 Request permission

## Abstract:

This paper studies the extension of the Hofer metric and general Finsler metrics on the Hamiltonian symplectomorphism group $\textrm {Ham}(M,\omega )$ to the identity component $\textrm {Symp}_0(M,\omega )$ of the symplectomorphism group. In particular, we prove that the Hofer metric on $\textrm {Ham}(M,\omega )$ does not extend to a bi-invariant metric on $\textrm {Symp}_0(M,\omega )$ for many symplectic manifolds. We also show that for the torus $\mathbb T^{2n}$ with the standard symplectic form $\omega$, no Finsler metric on $\textrm {Ham}(\mathbb T^{2n},\omega )$ that satisfies a strong form of the invariance condition can extend to a bi-invariant metric on $\textrm {Symp}_0(\mathbb T^{2n},\omega )$. Another interesting result is that there exists no $C^1$-continuous bi-invariant metric on $\textrm {Symp}_0(\mathbb T^{2n},\omega )$.## References

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## Additional Information

**Zhigang Han**- Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Amherst, Massachusetts 01003-9305
- Email: han@math.umass.edu
- Received by editor(s): October 1, 2007
- Published electronically: December 31, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**361**(2009), 3343-3357 - MSC (2000): Primary 53D35; Secondary 57R17
- DOI: https://doi.org/10.1090/S0002-9947-08-04713-2
- MathSciNet review: 2485430