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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Bi-invariant metrics on the group of symplectomorphisms


Author: Zhigang Han
Journal: Trans. Amer. Math. Soc. 361 (2009), 3343-3357
MSC (2000): Primary 53D35; Secondary 57R17
Published electronically: December 31, 2008
MathSciNet review: 2485430
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Abstract: This paper studies the extension of the Hofer metric and general Finsler metrics on the Hamiltonian symplectomorphism group $ {\rm Ham}(M,\omega)$ to the identity component $ {\rm Symp}_0(M,\omega)$ of the symplectomorphism group. In particular, we prove that the Hofer metric on $ {\rm Ham}(M,\omega)$ does not extend to a bi-invariant metric on $ {\rm 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 $ {\rm Ham}(\mathbb{T}^{2n},\omega)$ that satisfies a strong form of the invariance condition can extend to a bi-invariant metric on $ {\rm Symp}_0(\mathbb{T}^{2n},\omega)$. Another interesting result is that there exists no $ C^1$-continuous bi-invariant metric on $ {\rm Symp}_0(\mathbb{T}^{2n},\omega)$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04713-2
PII: S 0002-9947(08)04713-2
Keywords: Hofer metric, Finsler metric, bi-invariant extension, admissible lift
Received by editor(s): October 1, 2007
Published electronically: December 31, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.