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Keywords:
invariant metric; geodesically equivalent metric; affinely equivalent metric
invariant metric; geodesically equivalent metric; affinely equivalent metric
Summary:
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 $S^3$ are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics.
References:
[1] Bokan, N., Šukilović, T., Vukmirović, S.: Lorentz geometry of 4-dimensional nilpotent Lie groups. Geom. Dedicata 177 (2015), 83-102. DOI 10.1007/s10711-014-9980-4 | MR 3370025 | Zbl 1326.53065
[2] Bolsinov, A. V., Kiosak, V., Matveev, V. S.: A Fubini theorem for pseudo-Riemannian geodesically equivalent metrics. J. Lond. Math. Soc., II. Ser. 80 (2009), 341-356. DOI 10.1112/jlms/jdp032 | MR 2545256 | Zbl 1175.53022
[3] Eisenhart, L. P.: Symmetric tensors of the second order whose first covariant derivatives are zero. Trans. Amer. Math. Soc. 25 (1923), 297-306 \99999JFM99999 49.0539.01. DOI 10.1090/S0002-9947-1923-1501245-6 | MR 1501245
[4] Hall, G. S., Lonie, D. P.: Holonomy groups and spacetimes. Classical Quantum Gravity 17 (2000), 1369-1382. DOI 10.1088/0264-9381/17/6/304 | MR 1752641 | Zbl 0957.83014
[5] Hall, G. S., Lonie, D. P.: Projective structure and holonomy in four-dimensional Lorentz manifolds. J. Geom. Phys. 61 (2011), 381-399. DOI 10.1016/j.geomphys.2010.10.007 | MR 2746125 | Zbl 1208.83035
[6] Kiosak, V., Matveev, V. S.: Complete Einstein metrics are geodesically rigid. Commun. Math. Phys. 289 (2009), 383-400. DOI 10.1007/s00220-008-0719-7 | MR 2504854 | Zbl 1170.53025
[7] Kiosak, V., Matveev, V. S.: Proof of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two. Commun. Math. Phys. 297 (2010), 401-426. DOI 10.1007/s00220-010-1037-4 | MR 2651904 | Zbl 1197.53055
[8] Levi-Civita, T.: Sulle trasformazioni dello equazioni dinamiche. Annali di Mat. 24 Italian (1896), 255-300 \99999JFM99999 27.0603.04. DOI 10.1007/BF02419530 | MR 2551879
[9] Sinyukov, N. S.: On geodesic mappings of Riemannian spaces onto symmetric Riemannian spaces. Dokl. Akad. Nauk SSSR, n. Ser. 98 (1954), 21-23 Russian. MR 0065994 | Zbl 0056.15301
[10] Topalov, P.: Integrability criterion of geodesical equivalence. Hierarchies. Acta Appl. Math. 59 (1999), 271-298. DOI 10.1023/A:1006369525091 | MR 1744754 | Zbl 0972.53048
[11] Wang, Z., Hall, G.: Projective structure in 4-dimensional manifolds with metric signature $(+,+,-,-)$. J. Geom. Phys. 66 (2013), 37-49. DOI 10.1016/j.geomphys.2012.12.004 | MR 3019271 | Zbl 1285.53014
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