Metric topology on the moduli space




Gromov-Hausdorff topology, ε-topology, Lipschitz-topology, smooth Lipschitz-topology


We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to Tuschmann’s question.


Download data is not yet available.

Author Biography

Jialong Deng, Georg-August-Universität

Mathematisches Institut


N. A'Campo, L. Ji and A. Papadopoulos, On the early history of moduli and Teichmüller spaces, arXiv e-prints, page arXiv:1602.07208, Feb 2016.

D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, volume 33, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001.

D. A. Edwards, The structure of superspace, in: Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte, N. C., 1974 dedicated to Math. Sect. Polish Acad. Sci.), pages 121-133, 1975.

A. E. Fischer, The theory of superspace, in: Relativity (Proc. Conf. Midwest, Cincinnati, Ohio, 1969), pages 303-357, 1970.

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53-73.

M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA (2007).

C.-H. Li, Quantum fluctuations, conformal deformations, and Gromov's topology - Wheeler, DeWitt, and Wilson meeting Gromov, arXiv e-prints, page arXiv:1509.03895, Sep 2015.

W. Tuschmann, Spaces and moduli spaces of Riemannian metrics, Front. Math. China 11, no. 5 (2016), 1335-1343.

W. Tuschmann and D. J. Wraith, Moduli spaces of Riemannian metrics, volume 46, Oberwolfach Seminars, Birkhäuser Verlag, Basel, 2015.

J. A. Wheeler, Superspace, in: Analytic methods in mathematical physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), pages 335-378, 1970.




How to Cite

J. Deng, “Metric topology on the moduli space”, Appl. Gen. Topol., vol. 22, no. 1, pp. 11–15, Apr. 2021.