Asian Journal of Mathematics

Volume 20 (2016)

Number 1

Teichmüller space is totally geodesic in Goldman space

Pages: 21 – 46



Qiongling Li (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)


We construct a new Riemannian metric on Goldman space $\mathcal{B}(S)$, the space of the equivalence classes of convex projective structures on the surface $S$, and then prove the new metric, as well as the metric of Darvishzadeh and Goldman, restricts to be the Weil–Petersson metric on Teichmüller space, embedded as a submanifold of Goldman space $\mathcal{B}(S)$. Moreover, Teichmüller space endowed with the Weil–Petersson metric then is totally geodesic in the Riemannian manifold $\mathcal{B}(S)$.


Weil–Petersson metric, real projective structure

2010 Mathematics Subject Classification

57N16, 58B20

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