## On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil-Petersson metric.(English)Zbl 0606.32014

Using the methods and original approach to Teichmüller theory developed by the author and A. Fischer [Math. Ann. 267, 311-345 (1984; Zbl 0518.32015); Trans. Am. Math. Soc. 284, 319-335 (1984; Zbl 0578.58006); J. Reine Angew. Math. 352, 151-160 (1984; Zbl 0573.32021)] the author obtains a formula for sectional curvature of Teichmüller space T(p) with respect to its Weil-Petersson metric in terms of the Laplace- Beltrami operator on functions (Scott Wolpert has derived independently a similar result). He shows that the sectional curvature as well as the holomorphic sectional curvature and Ricci curvature are negative [see L. V. Ahlfors, J. Anal. Math. 9, 161-176 (1961; Zbl 0148.312)]. Bounds on the holomorphic and the Ricci curvature are given; in particular, the holomorphic sectional curvature of T(p) (p is the genus of the surface, $$p>1)$$ is strictly negative and bounded by $$-1/4\pi (p- 1).$$
Reviewer: B.N.Apanasov

### MSC:

 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Full Text:

### References:

 [1] Ahlfors, L.V.: On quasiconformal mappings, J. Analyse Math. 4, 1-58 (1954) · Zbl 0065.30502 [2] Ahlfors, L.V.: Curvature properties of Teichmüller’s space, J. Analyse Math. 9, 161-176 (1961) · Zbl 0148.31201 [3] Ahlfors, L.V.: The complex analytic structure on the space of closed Riemann surfaces. In Analytic functions, Princeton Univ. Press (1960) · Zbl 0100.28903 [4] Bers, L.: Quasiconformal mappings and Teichmüller theory? In Analytic functions, 89-120, Princeton Univ. Press (1960) · Zbl 0100.28904 [5] Cheeger, J., Ebin, D.: Comparison Theorems in Riemannian Geometry, North Holland, Amsterdam (1975) · Zbl 0309.53035 [6] Earle, C., Eells, J.: A fibre bundle discription of Teichmüller theory. J. Differential Geometry 3, 19-43 (1969) · Zbl 0185.32901 [7] Eisenhart, L. P.: Riemannian geometry, Princeton Univ. Press (1966) · Zbl 0174.53303 [8] Fischer, A.E., Tromba, A.J.: On a purely Riemannian proof of the structure and dimension of the unramified moduli space of a compact Riemann surface, Math. Ann. 267, 311-345 (1984) · Zbl 0532.32008 [9] Fischer, A. E., Tromba, A. J.: On the Weil-Petersson metric on Teichmüller space, Trans. AMS Vol. 284, No. 1, July 1984 · Zbl 0578.58006 [10] Fischer, A. E., Tromba, A. J.: Almost complex principle fibre bundles and the complex structure on Teichmüller space, Crelles J., Band 352 (1984) · Zbl 0573.32021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.