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31
A solution of a problem of Sophus Lie: Normal forms of 2dim metrics admitting two projective vector fields.
, 2007
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Complete Einstein metrics are geodesically rigid
 Comm. Math. Phys
"... We prove that every complete Einstein (Riemannian or pseudoRiemannian) metric g is geodesically rigid: if any other complete metric ¯g has the same (unparametrized) geodesics with g, then the LeviCivita connections of g and ¯g coincide. ..."
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Cited by 16 (6 self)
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We prove that every complete Einstein (Riemannian or pseudoRiemannian) metric g is geodesically rigid: if any other complete metric ¯g has the same (unparametrized) geodesics with g, then the LeviCivita connections of g and ¯g coincide.
SPLITTING AND GLUING LEMMAS FOR GEODESICALLY EQUIVALENT PSEUDORIEMANNIAN METRICS
, 2011
"... Two metrics g and ¯g are geodesically equivalent if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the d ..."
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Cited by 7 (3 self)
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Two metrics g and ¯g are geodesically equivalent if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the description of projective vector fields), to the case when the (1, 1)−tensor Gi j: = gik¯gkj has one real eigenvalue, or two complex conjugate eigenvalues, and give first applications. As a part of the proof of the main result, we generalise the TopalovSinjukov (hierarchy) Theorem for pseudoRiemannian metrics.
Normal forms for pseudoRiemannian 2dimensional metrics whose geodesic flows admit integrals quadratic
"... We discuss pseudoRiemannian metrics on 2dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct (Theorem 1) local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liou ..."
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Cited by 6 (5 self)
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We discuss pseudoRiemannian metrics on 2dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct (Theorem 1) local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely: • they admit geodesically equivalent metrics (Theorem 2); • one can use them to construct a big family of natural systems admitting integrals quadratic in momenta (Theorem 4); • the integrability of such systems can be generalized to the quantum setting (Theorem 5); • these natural systems are integrable by quadratures (Section 2.2.2). 1
The only Kähler manifold with degree of mobility at least 3 is (CP(n),gFubini−Study
 Proc. Lond. Math. Soc
, 2012
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Strictly nonproportional geodesically equivalent metrics have htop(g) = 0
, 2004
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Fubini Theorem for pseudoRiemannian metrics
"... We generalize the following classical result of Fubini [12] for pseudoRiemannian metrics: if three essentially different metrics on M n≥3 share the same unparametrized geodesics, and two of them (say, g and ¯g) are strictly nonproportional (i.e., the minimal polynomial of g iα ¯gαj coincides with t ..."
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Cited by 3 (2 self)
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We generalize the following classical result of Fubini [12] for pseudoRiemannian metrics: if three essentially different metrics on M n≥3 share the same unparametrized geodesics, and two of them (say, g and ¯g) are strictly nonproportional (i.e., the minimal polynomial of g iα ¯gαj coincides with the characteristic polynomial) at least at one point, then they have constant curvature. 1