no. 2
A geometric study of Wasserstein spaces: Euclidean spaces
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 297-323.

In this article we consider Wasserstein spaces (with quadratic transportation cost) as intrinsic metric spaces. We are interested in usual geometric properties: curvature, rank and isometry group, mostly in the case of Euclidean spaces. Our most striking result is that the Wasserstein space of the line admits “exotic” isometries, which do not preserve the shape of measures.

@article{ASNSP_2010_5_9_2_297_0,
     author = {Kloeckner, Beno\^\i t},
     title = {A geometric study of Wasserstein spaces: Euclidean spaces},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {297--323},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {2},
     year = {2010},
     zbl = {1218.53079},
     mrnumber = {2731158},
     language = {en},
     url = {archive.numdam.org/item/ASNSP_2010_5_9_2_297_0/}
}
Kloeckner, Benoît. A geometric study of Wasserstein spaces: Euclidean spaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 297-323. http://archive.numdam.org/item/ASNSP_2010_5_9_2_297_0/

[1] L. Ambrosio and N. Gigli, Construction of the parallel transport in the Wasserstein space, Methods Appl. Anal. 15 (2008), 1–30. | MR 2482206 | Zbl 1179.28009

[2] L. Ambrosio, N. Gigli and G. Savaré, “Gradient Flows in Metric Spaces and in the Space of Probability Measures”, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. | MR 2129498 | Zbl 1090.35002

[3] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 805–808. | MR 923203 | Zbl 0652.26017

[4] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375–417. | MR 1100809 | Zbl 0738.46011

[5] M. R. Bridson and A. Haefliger, “Metric Spaces of Non-positive Curvature”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 319, Springer-Verlag, Berlin, 1999. | MR 1744486 | Zbl 0988.53001

[6] D. Burago, Y. Burago and S. Ivanov, “A Course in Metric Geometry”, Graduate Studies in Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 2001. | MR 1835418 | Zbl 0981.51016

[7] J.-M. Gambaudo and É. Ghys, Braids and signatures, Bull. Soc. Math. France 133 (2005), 541–579. | EuDML 272312 | Numdam | MR 2233695 | Zbl 1103.57001

[8] S. Helgason, “The Radon Transform”, Progress in Mathematics, Vol. 5, Birkhäuser, Boston, Mass., 1980. | MR 573446 | Zbl 0453.43011

[9] J. Jost, “Nonpositive Curvature: Geometric and Analytic Aspects”, Lectures in Mathematics ETH Zürich, Birkhäuser, Verlag, Basel, 1997. | MR 1451625 | Zbl 0896.53002

[10] N. Juillet, Optimal transport and geometric analysis in Heisenberg groups, Université Grenoble 1 et Rheinische Friedrich-Wilhelms-Universität Bonn, December 2008. | Zbl 1246.53049

[11] M. Knott and C. S. Smith, On the optimal mapping of distributions, J. Optim. Theory Appl. 43 (1984), 39–49. | MR 745785 | Zbl 0519.60010

[12] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), 903–991. | MR 2480619 | Zbl 1178.53038

[13] J. Lott, Some geometric calculations on Wasserstein space, Comm. Math. Phys. 277 (2008), 423–437. | MR 2358290 | Zbl 1144.58007

[14] J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Königl. Sächs. Ges. Wiss. Leipzig 69 (1917), 262–277. English translation: Determination of a function from the values of its integrals along certain manifolds, In: “Computed Tomography” (Cincinnati, Ohio, 1982), Proc. Sympos. Appl. Math., Vol. 27, Amer. Math. Soc., Providence, R.I., 1982, 71–86. | JFM 46.0436.02

[15] M.-K. Von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), 923–940. | MR 2142879 | Zbl 1078.53028

[16] C. S. Smith and M. Knott, Note on the optimal transportation of distributions, J. Optim. Theory Appl. 52 (1987), 323–329. | MR 879207 | Zbl 0586.49005

[17] K.-T. Sturm, On the geometry of metric measure spaces, I, II, Acta Math. 196 (2006), 65–131, 133–177. | MR 2237207 | Zbl 1105.53035

[18] A. Takatsu, On Wasserstein geometry of the space of Gaussian measures, arXiv: 0801.2250, 2008. | MR 2648273

[19] A. Takatsu and T. Yokota, Cone structure of L 2 -Wasserstein spaces, arXiv:0812.2752. | MR 2949241 | Zbl 1253.28001

[20] C. Villani, “Topics in Optimal Transportation”, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. | MR 1964483 | Zbl 1106.90001

[21] C. Villani, “Optimal Transport Old and New”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 338, Springer-Verlag, Berlin, 2009. | MR 2459454 | Zbl 1156.53003