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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the Weil-Petersson metric on Teichmüller space

Authors: A. E. Fischer and A. J. Tromba
Journal: Trans. Amer. Math. Soc. 284 (1984), 319-335
MSC: Primary 32G15; Secondary 58B20
MathSciNet review: 742427
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Abstract: Teichmüller space for a compact oriented surface $M$ without boundary is described as the quotient $\mathcal {A}/{\mathcal {D}_0}$, where $\mathcal {A}$ is the space of almost complex structures on $M$ (compatible with a given orientation) and ${\mathcal {D}_0}$ are those ${C^\infty }$ diffeomorphisms homotopic to the identity. There is a natural ${\mathcal {D}_0}$ invariant ${L_2}$ Riemannian structure on $\mathcal {A}$ which induces a Riemannian structure on $\mathcal {A}/{\mathcal {D}_0}$. Infinitesimally this is the bilinear pairing suggested by Andre Weil—the Weil-Petersson Riemannian structure. The structure is shown to be Kähler with respect to a naturally induced complex structure on $\mathcal {A}/{\mathcal {D}_0}$.

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Article copyright: © Copyright 1984 American Mathematical Society