Abstract
Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, G-invariant metrics on homogenous space G/H implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metrics of any signature on sphere S3 are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics.
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References
N. Bokan, T. Sukilovic, S. Vukmirovic: Lorentz geometry of 4-dimensional nilpotent Lie groups. Geom. Dedicata 177 (2015), 83–102.
A. V. Bolsinov, V. Kiosak, V. S. Matveev: A Fubini theorem for pseudo-Riemannian geodesically equivalent metrics. J. Lond. Math. Soc., II. Ser. 80 (2009), 341–356.
L. P. Eisenhart: Symmetric tensors of the second order whose first covariant derivatives are zero. Trans. Amer. Math. Soc. 25 (1923), 297–306.
G. S. Hall, D. P. Lonie: Holonomy groups and spacetimes. Classical Quantum Gravity 17 (2000), 1369–1382.
G. S. Hall, D. P. Lonie: Projective structure and holonomy in four-dimensional Lorentz manifolds. J. Geom. Phys. 61 (2011), 381–399.
V. Kiosak, V. S. Matveev: Complete Einstein metrics are geodesically rigid. Commun. Math. Phys. 289 (2009), 383–400.
V. Kiosak, V. S. Matveev: Proof of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two. Commun. Math. Phys. 297 (2010), 401–426.
T. Levi-Civita: Sulle trasformazioni dello equazioni dinamiche. Annali di Mat. 24 (1896), 255–300. (In Italian.)
N. S. Sinyukov: On geodesic mappings of Riemannian spaces onto symmetric Rieman-nian spaces. Dokl. Akad. Nauk SSSR, n. Ser. 98 (1954), 21–23. (In Russian.)
P. Topalov: Integrability criterion of geodesical equivalence. Hierarchies. Acta Appl. Math. 59 (1999), 271–298.
Z. Wang, G. Hall: Projective structure in 4-dimensional manifolds with metric signature (+,+,-,-). J. Geom. Phys. 66 (2013), 37–49.
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The research of the second and the third authors has been partially supported within the project 174012 through the Ministry of Education, Science and Technological Development of the Republic of Serbia.
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Bokan, N., Šukilović, T. & Vukmirović, S. Geodesically equivalent metrics on homogenous spaces. Czech Math J 69, 945–954 (2019). https://doi.org/10.21136/CMJ.2018.0557-17
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DOI: https://doi.org/10.21136/CMJ.2018.0557-17