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Let $S$ be a closed oriented surface of genus $g\geq 2$. Consider the Teichmuller space $T(S)$. Let $d_t$ be the Teichmuller metric and $d_{WP}$ be the Weil-Petersson metric on $T(S)$. Let $P:S_1\rightarrow S$ be a covering map. It is a well known result that if $P$ is finite then the embedding $(T(S),d_t)\rightarrow (T(S_1),d_t)$ is an isometric embedding (see Section 7). My question is

Q) Is the map $((T(S),d_{WP})\rightarrow (T(S_1),d_{WP})$ an isometric embedding?

My idea is the following. As the WP pairing and the (co)metric on $T(S)$ is defined by $\pi_1(S)$ invariant quadratic differentials, they are naturally $\pi_1(S_1)$ invariant. Hence the above map is an isometry. But for this argument, I don't need $P$ to be finite. I am not sure whether this argument is correct or I am making some silly mistakes.

Any kind of suggestion/reference will be extremely helpful. Thanks in advance.

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    $\begingroup$ See the remark on p. 2 of this paper: arxiv.org/abs/0806.2304 For a covering, the embedding of Teichmuller space induced by a covering increases distance by the square root of the degree of the cover. $\endgroup$
    – Ian Agol
    Commented Aug 28, 2020 at 16:30
  • $\begingroup$ @IanAgol Thank you very much for the reference. I forgot about the normalization. $\endgroup$
    – Cusp
    Commented Aug 28, 2020 at 17:31

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