On Liouville integrability of ℎ-projectively equivalent Kähler metrics

Kiyohara, Kazuyoshi; Topalov, Peter

doi:10.1090/S0002-9939-2010-10576-2

On Liouville integrability of $h$-projectively equivalent Kähler metrics
HTML articles powered by AMS MathViewer

by Kazuyoshi Kiyohara and Peter Topalov PDF

Proc. Amer. Math. Soc. 139 (2011), 231-242 Request permission

Abstract:

Under a nondegeneracy condition we classify the compact connected Kähler manifolds admitting pairs of $h$-projectively equivalent metrics. Any such manifold is biholomorphically equivalent to $\mathbb CP^n$ and has integrable geodesic flow.

References

Kazuyoshi Kiyohara, Two classes of Riemannian manifolds whose geodesic flows are integrable, Mem. Amer. Math. Soc. 130 (1997), no. 619, viii+143. MR 1396959, DOI 10.1090/memo/0619
M. Igarashi and K. Kiyohara: On Hermite-Liouville manifolds, to appear in J. Math. Soc. Japan.
Kazuyoshi Kiyohara, On Kähler-Liouville manifolds, Differential geometry and integrable systems (Tokyo, 2000) Contemp. Math., vol. 308, Amer. Math. Soc., Providence, RI, 2002, pp. 211–222. MR 1955637, DOI 10.1090/conm/308/05320
Tominosuke Ōtsuki and Yoshihiro Tashiro, On curves in Kaehlerian spaces, Math. J. Okayama Univ. 4 (1954), 57–78. MR 66024
V. S. Matveev and P. Ĭ. Topalov, Trajectory equivalence and corresponding integrals, Regul. Chaotic Dyn. 3 (1998), no. 2, 30–45 (English, with English and Russian summaries). MR 1693470, DOI 10.1070/rd1998v003n02ABEH000069
Vladimir S. Matveev and Peter J. Topalov, Quantum integrability of Beltrami-Laplace operator as geodesic equivalence, Math. Z. 238 (2001), no. 4, 833–866. MR 1872577, DOI 10.1007/s002090100280
Yoshihiro Tashiro, On a holomorphically projective correspondence in an almost complex space, Math. J. Okayama Univ. 6 (1957), 147–152. MR 87181
Peter Topalov, Hierarchy of integrable geodesic flows, Publ. Mat. 44 (2000), no. 1, 257–276. MR 1775764, DOI 10.5565/PUBLMAT_{4}4100_{1}0
Peter Topalov, Geodesic hierarchies and involutivity, J. Math. Phys. 42 (2001), no. 8, 3898–3914. MR 1845225, DOI 10.1063/1.1379068
Peter Topalov, Geodesic compatibility and integrability of geodesic flows, J. Math. Phys. 44 (2003), no. 2, 913–929. MR 1953103, DOI 10.1063/1.1526939
Kentaro Yano and Hitosi Hiramatu, Kaehlerian manifolds with constant scalar curvature admitting a holomorphically projective vector field, J. Differential Geometry 14 (1979), no. 1, 81–92 (1980). MR 577880
Kentaro Yano and Hitosi Hiramatu, Isometry of Kaehlerian manifolds to complex projective spaces, J. Math. Soc. Japan 33 (1981), no. 1, 67–78. MR 597481, DOI 10.2969/jmsj/03310067