References

[1] A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10, Springer-Verlag, Berlin, 1987. MR zbM

[2] R. L. Bryant, M. Dunajski, M. Eastwood, Metrisability of two-dimensional projective structures, J. Differential Geom. 83 (2009), 465–499. MR zbM

[3] E. Calabi, Complete affine hyperspheres. I, Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), (1972), 19–38. MR zbM

[4] C. B. Croke, V. A. Sharafutdinov, Spectral rigidity of a compact negatively curved manifold, Topology 37 (1998), 1265–1273. MR zbM

[5] M. Dubois-Violette, Structures complexes au-dessus des variétés, applications, in Mathematics and physics (Paris, 1979/1982), Progr. Math. 37, Birkhäuser Boston, Boston, MA, 1983, pp. 1–42. MR

[6] D. Dumas, M. Wolf, Polynomial cubic differentials and convex polygons in the projective plane, Geom. Funct. Anal. 25 (2015), 1734–1798. MR zbM

[7] V. Guillemin, D. Kazhdan, Some inverse spectral results for negatively curved \(2\)-manifolds, Topology 19 (1980), 301–312. MR zbM

[8] N. Hitchin, Complex manifolds and Einstein’s equations, in Twistor geometry and nonlinear systems, Lecture Notes in Math. 970, Springer, Berlin, 1982, pp. 73–99. MR zbM

[9] N. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), 449–473. MR zbM

[10] D. Jane, G. P. Paternain, On the injectivity of the X-ray transform for Anosov thermostats, Discrete Contin. Dyn. Syst. 24 (2009), 471–487. MR zbM

[11] F. Labourie, Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007), 1057–1099. MR zbM

[12] J. C. Loftin, Affine spheres and convex \(\mathbb{RP}^n\)-manifolds, Amer. J. Math. 123 (2001), 255–274. MR zbM

[13] V. S. Matveev, P. J. Topalov, Metric with ergodic geodesic flow is completely determined by unparameterized geodesics, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 98–104. MR zbM

[14] T. Mettler, Weyl metrisability of two-dimensional projective structures, Math. Proc. Cambridge Philos. Soc. 156 (2014), 99–113. MR zbM

[15] T. Mettler, Geodesic rigidity of conformal connections on surfaces, Math. Z. 281 (2015), 379–393. MR

[16] T. Mettler, Minimal Lagrangian connections on compact surfaces, Adv. Math. 354 (2019), Article ID 106747, 36 p. MR zbM

[17] T. Mettler, Extremal conformal structures on projective surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) XX (2020), 1621–1663.

[18] T. Mettler, G. P. Paternain, Holomorphic differentials, thermostats and Anosov flows, Math. Ann. 373 (2019), 553–580. MR zbM

[19] N. R. O’Brian, J. H. Rawnsley, Twistor spaces, Ann. Global Anal. Geom. 3 (1985), 29–58. MR zbM

[20] G. P. Paternain, M. Salo, G. Uhlmann, Tensor tomography on surfaces, Invent. Math. 193 (2013), 229–247. MR zbM

[21] G. P. Paternain, M. Salo, G. Uhlmann, Spectral rigidity and invariant distributions on Anosov surfaces, J. Differential Geom. 98 (2014), 147–181. MR zbM

[22] V. Sharafutdinov, G. Uhlmann, On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points, J. Differential Geom. 56 (2000), 93–110. MR zbM

[23] M. Spivak, A comprehensive introduction to differential geometry. Vol. II, third ed., Publish or Perish, Inc., Wilmington, Del., 1999.

[24] C. P. Wang, Some examples of complete hyperbolic affine \(2\)-spheres in \({\bf R}^3\), in Global differential geometry and global analysis (Berlin, 1990), Lecture Notes in Math. 1481, Springer, Berlin, 1991, pp. 271–280. MR zbM