Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T14:21:53.348Z Has data issue: false hasContentIssue false
  • English
  • Français

TWO REMARKS ON PQε-PROJECTIVITY OF RIEMANNIAN METRICS

Published online by Cambridge University Press:  02 August 2012

VLADIMIR S. MATVEEV  and
STEFAN ROSEMANN
VLADIMIR S. MATVEEV
Affiliation:
Institute of Mathematics, FSU Jena, Jena 07737, Germany e-mails: vladimir.matveev@uni-jena.de, stefan.rosemann@uni-jena.de
STEFAN ROSEMANN
Affiliation:
Institute of Mathematics, FSU Jena, Jena 07737, Germany e-mails: vladimir.matveev@uni-jena.de, stefan.rosemann@uni-jena.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that PQε-projectivity of two Riemannian metrics introduced in [15] (P. J. Topalov, Geodesic compatibility and integrability of geodesic flows, J. Math. Phys.44(2) (2003), 913–929.) implies affine equivalence of the metrics unless ε ∈ {0,−1,−3,−5,−7,. . .}. Moreover, we show that for ε=0, PQε-projectivity implies projective equivalence.

Keywords

53B2053B3553C2153C2253C5537J3570H06
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 1 , January 2013 , pp. 131 - 138
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Apostolov, V., Calderbank, D. and Gauduchon, P., Hamiltonian 2-forms in Kähler geometry. I. General theory, J. Differ. Geom. 73 (3) (2006), 359–412.Google Scholar
2.Fedorova, A., Kiosak, V., Matveev, V. and Rosemann, S., The only Kähler manifold with degree of mobility ≥ 3 is (CP(n), g Fubini-Study), Proc. Lond. Math. Soc., doi: 10.1112/plms/pdr053 (2012).Google Scholar
3.Kiyohara, K. and Topalov, P. J., On Liouville integrability of h-projectively equivalent Kähler metrics, Proc. Am. Math. Soc. 139 (2011), 231–242.Google Scholar
4.Levi-Civita, T., Sulle trasformazioni delle equazioni dinamiche, Ann. Di Math. 24 (1896), 255–300.Google Scholar
5.Matveev, V. S., Hyperbolic manifolds are geodesically rigid, Invent. Math. 151 (3) (2003), 579–609.Google Scholar
6.Matveev, V. S., Proof of the projective Lichnerowicz–Obata conjecture, J. Differ. Geom. 75 (3) (2007), 459–502.Google Scholar
7.Matveev, V. S. and Rosemann, S., Proof of the Yano-Obata conjecture for holomorph-projective transformations, arXiv:1103.5613 [math.DG], 2011.Google Scholar
8.Matveev, V. S. and Topalov, P. J., Integrability in the theory of geodesically equivalent metrics, J. Phys. A 34 (11) (2001), 2415–2433.CrossRefGoogle Scholar
9.Mikes, J., Special F-planar mappings of affinely connected spaces onto Riemannian spaces, Mosc. Univ. Math. Bull. 49 (3) (1994), 15–21 (English; original in Russian; translated from Vestn. Mosk. Univ., Ser. I (3) (1994), 18–24).Google Scholar
10.Mikes, J., Geodesic mappings of affine-connected and Riemannian spaces, J. Math. Sci. 78 (3) (1996), 311–333.CrossRefGoogle Scholar
11.Mikes, J., Holomorphically projective mappings and their generalizations, J. Math. Sci. (New York) 89 (3) (1998), 1334–1353.CrossRefGoogle Scholar
12.Otsuki, T. and Tashiro, Y., On curves in Kaehlerian spaces, Math. J. Okayama Univ. 4 (1954), 57–78.Google Scholar
13.Sinjukov, N. S., Geodesic mappings of Riemannian spaces (in Russian) (Nauka, Moscow, Russia, 1979, MR0552022, Zbl 0637.53020).Google Scholar
14.Tashiro, Y., On a holomorphically projective correspondence in an almost complex space, Math. J. Okayama Univ. 6 (1956), 147–152.Google Scholar
15.Topalov, P. J., Geodesic compatibility and integrability of geodesic flows, J. Math. Phys. 44 (2) (2003), 913–929.Google Scholar