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Weyl metrisability of two-dimensional projective structures

Published online by Cambridge University Press:  19 September 2013

THOMAS METTLER
THOMAS METTLER*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA. e-mail: t.mettler@damtp.cam.ac.uk
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Abstract

We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the ‘twistor’ bundle of conformal inner products having holomorphic image. The second solution allows to use standard results in algebraic geometry to show that the Weyl connections on the two-sphere whose geodesics are the great circles are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 1 , January 2014 , pp. 99 - 113
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Alvarez–Paiva, J. C. and Berck, G. Finsler surfaces with prescribed geodesics. http://arxiv.org/abs/1002.0243arXiv:math/1002.0242v1 (2010).Google Scholar
[2]Bryant, R. L., Chern, S. S., Gardner, R. B., Goldschmidt, H. L. and Griffiths, P. A.Exterior differential systems. Math. Sci. Res. Inst. Publ. vol. 18 (Springer-Verlag, New York, 1991). MR 1083148Google Scholar
[3]Bryant, R., Dunajski, M. and Eastwood, M.Metrisability of two-dimensional projective structures. J. Differential Geom. 83 (2009), no. 3, 465–499. MR 2581355CrossRefGoogle Scholar
[4]Bryant, R. L.Projectively flat Finsler 2-spheres of constant curvature. Selecta Math. (N.S.) 3 (1997), no. 2, 161–203. MR 1466165CrossRefGoogle Scholar
[5]Čap, A. and Slovák, J.Parabolic geometries. I. Math. Surv. Monog. vol. 154, (American Mathematical Society, Providence, RI, 2009), Background and general theory. MR 2532439Google Scholar
[6]Cartan, E.Sur les variétés à connexion projective. Bull. Soc. Math. France 52 (1924), 205–241. MR 1504846CrossRefGoogle Scholar
[7]Dubois–Violette, M.Structures complexes au-dessus des variétés, applications. Mathematics and physics (Paris, 1979/1982). Progr. Math., vol. 37 (Birkhäuser Boston, Boston, MA, 1983), pp. 1–42. MR 728412Google Scholar
[8]Dunajski, M., Mason, L. J. and Tod, P.Einstein–Weyl geometry, the dKP equation and twistor theory. J. Geom. Phys. 37 (2001), no. 1-2, 63–93. MR 1807082 (2002c:53081)CrossRefGoogle Scholar
[9]Dunajski, M. and Tod, P.Four-dimensional metrics conformal to Kähler. Math. Proc. Camb. Phils. Soc. 148 (2010), no. 3, 485–503. MR 2609304CrossRefGoogle Scholar
[10]Eastwood, M. and Matveev, V.Metric connections in projective differential geometry. Symmetries and overdetermined systems of partial differential equations, IMA Vol. Math. Appl. vol. 144 (Springer, New York, 2008), pp. 339–350. MR 2384718CrossRefGoogle Scholar
[11]Eisenhart, L. P. and Veblen, O.The Riemann geometry and its generalization. Proc. Nat. Acad. Sci. 8 (1922), 19–23.CrossRefGoogle ScholarPubMed
[12]Gallavotti, G. and Ruelle, D.SRB states and nonequilibrium statistical mechanics close to equilibrium. Comm. Math. Phys. 190 (1997), no. 2, 279–285. MR 1489572 (99a:82057)CrossRefGoogle Scholar
[13]Hoover, W. G.Molecular dynamics. Lecture Notes in Phys, vol. 258 (Springer, Berlin, 1986).Google Scholar
[14]Ivey, T. A. and Landsberg, J. M.Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems. Graduate Studies in Mathematics, vol. 61, (Amer. Math. Soc. Providence, RI, 2003). MR 2003610Google Scholar
[15]Kobayashi, S. and Nagano, T.On projective connections J. Math. Mech. 13 (1964), 215–235. MR 0159284Google Scholar
[16]Lebrun, C. and Mason, L. J.Zoll manifolds and complex surfaces. J. Differential Geom. 61 (2002), no. 3, 453–535. MR 1979367CrossRefGoogle Scholar
[17]Lebrun, C. and Mason, L. J.Zoll metrics, branched covers and holomorphic disks. Comm. Anal. Geom. 18 (2010), no. 3, 475–502.CrossRefGoogle Scholar
[18]Liouville, R.Sur les invariants de certaines équations différentielles et sur leurs applications. J. l'Ecole Polytechnique 59 (1889), 7–76.Google Scholar
[19]Matveev, V. S.Geodesically equivalent metrics in general relativity. J. Geom. Phys. 62 (2012), no. 3, 675–691.CrossRefGoogle Scholar
[20]Mettler, T.Reduction of β-integrable 2-Segre structures. Comm. Anal. Geom. 21 (2013), no. 2, 331–353.CrossRefGoogle Scholar
[21]Newlander, A. and Nirenberg, L.Complex analytic coordinates in almost complex manifolds Ann. of Math. (2) 65 (1957), 391–404. MR 0088770CrossRefGoogle Scholar
[22]Nirenberg, L.Lectures on linear partial differential equations. (Amer. Math. Soc., Providence, R.I., 1973). MR 0450755CrossRefGoogle Scholar
[23]Nurowski, P.Projective vs metric structures. J. Geom. Phys. 62 (2012), no. 3, 657–674.CrossRefGoogle Scholar
[24]O'Brian, N. R. and Rawnsley, J. H.Twistor spaces. Ann. Global Anal. Geom. 3 (1985), no. 1, 29–58. MR 812312CrossRefGoogle Scholar
[25]Pedersen, H. and Swann, A.Riemannian submersions, four-manifolds and Einstein–Weyl geometry. Proc. London Math. Soc. (3) 66 (1993), no. 2, 381–399. MR 1199072 (94c:53061)CrossRefGoogle Scholar
[26]Thomas, T. Y.On the projective and equi-projective geometries of paths. Proc. Nat. Acad. Sci. 11 (1925), 199–203.CrossRefGoogle ScholarPubMed
[27]Weyl, H. Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung. Göttingen Nachrichten (1921), 99–112.Google Scholar
[28]Wojtkowski, M. P.W-flows on Weyl manifolds and Gaussian thermostats. J. Math. Pures Appl. (9) 79 (2000), no. 10, 953–974. MR 1801870 (2001k:37044)CrossRefGoogle Scholar