@ Appl. Gen. Topol. 22, no. 1 (2021), 11-15doi:10.4995/agt.2021.13066 © AGT, UPV, 2021 Metric topology on the moduli space Jialong Deng Mathematisches Institut, Georg-August-Universität, Göttingen, Germany (jialong.deng@mathematik.uni-goettingen.de) Communicated by S. Romaguera Abstract 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. 2010 MSC: 54E35; 54F65; 51F99. Keywords: Gromov-Hausdorff topology; ε-topology; Lipschitz-topology; smooth Lipschitz-topology. 1. Introduction Any smooth closed manifold M can be given a smooth Riemannian metric, and then one can ask: How many Riemannian metrics are there, and how many different geometries of this kind does the manifold actually allow? That means one wants to understand the space of Riemannian metrics on M, which is denoted by R(M), and the moduli space which is denoted by M(M). Here the moduli space is the quotient space of R(M) by the action of diffeomorphism group of M. Those two questions originated from Riemann when he set up Riemannian geometry in the nineteenth century ([1]). Especially, the moduli space M(M) is the superspace in physics (see [10], [4], [3]). The Cn,α-compact-open topology (n ∈ N+ and α ∈ R+) is the most com- mon consideration [9]. Tuschmann asked the following question in [8, Sec- tion 3, (8)]: What can one say about the topology of moduli spaces under the Gromov-Hausdorff metric? What if one uses the Lipschitz topology? Received 28 January 2020 – Accepted 28 December 2020 http://dx.doi.org/10.4995/agt.2021.13066 J. Deng 2. Metric topology Inspired by Tuschmann’s questions, we will introduce four kinds of metric topology on the moduli space, and then discuss the relationship among them. Let X and Y be metric spaces of finite diameter, then the Gromov-Hausdorff distance is defined as ρGH(X, Y ) := inf Z {dZH(f(X), g(Y ))}, where dH is Haus- dorff metric and Z takes all metric spaces such that f (resp. g) are isomet- ric embeddings X (resp. Y ) into Z (see [5]). The Gromov-Hausdorff dis- tance ρGH is a pseudo-metric in the collection of all compact metric spaces. Furthermore, ρGH(X, Y ) = 0 if and only if X is isometric to Y . For g1 and g2 in R(M), the Gromov-Hausdorff distance can be defined on it by ρGH(g1, g2) = ρGH((M, d1), (M, d2)), where d1 and d2 are induced metrics on M by g1 and g2. Since M is closed, the Gromov-Hausdorff distance is well-defined on R(M). Moreover, ρGH(f ∗ 1 g1, f ∗ 2 g2) = ρGH(g1, g2), where f1 and f2 are diffeomorphism of M and f ∗ 1 g1, f ∗ 2 g2 are push-back metrics on M. Then one can define ρGH on M(M) as above and then ρGH is a metric on M(M). Therefore, M(M) can be endowed with the metric topology called GH-topology by the Gromov-Hausdorff metric ρGH. Definition 2.1 (Edwards [3]). A map f : X → Y is called an ε-isometry between compact metric spaces X and Y , if |dX(a, b) − dY (f(a), f(b))| ≤ ε for all a, b ∈ X. Definition 2.2 (ε-distance). Assume g1 and g2 are in R(M), then we define the ε-distance by ρε(g1, g2) := ρε(d1, d2) = inf {ε | Iε(d1, d2) 6= φ 6= Iε(d2, d1)} , where Iε(d1, d2) are the set of ε-isometries from (M, d1) to (M, d2). Remark 2.3. Note that |Diam(d1) − Diam(d2)| ≤ ρε(d1, d2) ≤ max{Diam(d1), Diam(d2)}, where Diam(di) is the diameter of (M, gi), i = 1, 2. Thus, ρε is well-defined on R(M). Moreover, ρε is the pseudo-metric and ρε(g1, g2) = 0 if and only if g1 is isometric to g2 on R(M). Then ε-metric, which is also denoted by ρε, can be defined on M(M) as ρGH. Thus, it induces a metric topology on M(M) called ε-topology. The conformal class dense theorem of the ε-topology on M(M) was proved by Liu in [7, Corollary 2.2]. Theorem 2.4 (Liu [7]). Each conformal class is dense in M(M) that is en- dowed with ε-topology. Lemma 2.5. If ρGH(X, Y ) ≤ ε, then there is a 2ε-isometric map f : X → Y . If there is an ε-isometric map f : X → Y , then ρGH(X, Y ) ≤ 3 2 ε. Remark 2.6. The lemma can be proved by using another definition of Gromov- Hausdorff metric, i.e. ρGH(X, Y ) = 1 2 inf R {dis (R)}, where the infimum is taken © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 12 Metric topology on the moduli space over all correspondences R ⊆ X × Y . A correspondence between two metric spaces X and Y is a subset R of X ×Y such that the projections πX : X ×Y → X and πY : X × Y → Y remain surjective when they are restricted to R. Corollary 2.7. ε-topology is equivalent to GH-topology on M(M). Thus, the conformal class dense theorem is also true on GH-topology. That means the GH-topology is coarse and a finer topology is needed to define on M(M). Let X and Y be two compact metric spaces, the dilation of a Lipschitz map f : X → Y is defined by Dil(f) := sup a,b∈X,a 6=b dY (f(a), f(b)) dX(a, b) . If f−1 is also a Lipschitz map then it is called the bi-Lipschitz homeomor- phism. The Lipschitz-distance ρL between X and Y is defined by ρL(X, Y ) := inf f:X→Y log{max{Dil(f), Dil(f−1)}}, where the infinum is taken over all bi-Lipschitz homeomorphisms between X and Y . Then the Lipschitz-distance ρL can be defined on R(M) as the defini- tion of Gromov-Hausdorff distance on R(M). Moreover, ρL is pseudo-metric on R(M) and ρL(g1, g2) = 0 if and only if g1 is isometric to g2 (see [2, Theorem 7.2.4]). Thus, it can induce a Lipschitz-metric ρL on M(M), and then ρL induces the Lipschitz-topology on M(M) called L- topology. Furthermore, Lipschitz convergence implies Gromov-Hausdorff Con- vergence, where the convergence means Cauchy sequence convergence related to their metrics (see [6, Proposition 3.6]). Proposition 2.8. L-topology is finer than GH-topology on M(M). The GH-topology and L-topology on M(M) only catch the metric informa- tion of the basic manifold and lose much essential information of the smooth structure. So it may be useful to modify the definition of L-topology on M(M) to a finer topology on M(M). For any homorphism of metric space f : (X, dX) → (Y, dY ), the Lipschitz constant of f is defined by L(f) := inf{k ≥ 1 | dX(x, y) k ≤ dY (f(x), f(y)) ≤ kdX(x, y), x, y ∈ X}. If the set is empty, then let L(f) be infinity. Lemma 2.9. Suppose that M and N are smooth closed Riemannian manifolds, then any diffeomorphism of M and N has bounded Lipschitz constant. Remark 2.10. The normal of tangent maps of diffeomorphism on the unit tan- gent bundle over closed manifold are uniform bounded, since the tangent maps are continuous and the total spaces of unit tangent bundle over compact man- ifold are compact. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 13 J. Deng For the composition of diffeomorphism f ◦ g : M → N → W , we have L(f ◦ g) ≤ L(f) · L(g) by direct computation. Definition 2.11. Assume g1 and g2 are in R(M), we define ρSL(g1, g2) = ρSL((M, d1), (M, d2)) := inf{log L(f) | f ∈ Diff}, where Diff is the diffeomorphism group of M. Lemma 2.12. ρSL is a pseudo-metric on R(M) and ρSL(g1, g2) = 0 if and only if g1 is isometric to g2 on M. Remark 2.13. If ρSL(d1, d2) = 0, then the isometry map between (M, d1) and (M, d2) can be constructed by using the closeness of M and the Arzela-Ascoli theorem. Continuing the game, one can define the metric topology on M(M) called SL-topology by the metric ρSL. Theorem 2.14. SL-topology � L-topology � GH-topology ∼= ε-topology. Usually those four metrics are not complete metrics on M(M), so M(M) is local compact topology spaces endowed with their induced metric topology in general. But if we restrict it to the subset of M(M), it may have some precompact propositions. For example, Gromov precompactness theorem and other convergence theorems on the moduli space [6, Chapter 5]. For the non-compact case, one can ask what is the right topology on R≥0(V ) and M≥0(V ), where V is a non-compact manifold, R≥0(V ) is the Riemannian metric with non-negative sectional curvature, and M≥0(V ) is the moduli space of V with non-negative sectional curvature? Acknowledgements. I thank Xuchao Yao for useful discussions. References [1] 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. [2] D. Burago, Y. Burago and S. 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Tuschmann, Spaces and moduli spaces of Riemannian metrics, Front. Math. China 11, no. 5 (2016), 1335–1343. [9] W. Tuschmann and D. J. Wraith, Moduli spaces of Riemannian metrics, volume 46, Oberwolfach Seminars, Birkhäuser Verlag, Basel, 2015. [10] J. A. Wheeler, Superspace, in: Analytic methods in mathematical physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), pages 335–378, 1970. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 15