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 Mat. Zametki, 2005, Volume 77, Issue 3, Pages 412–423 (Mi mz2502)

The eigenvalues of the Sinyukov mapping for geodesically equivalent metrics are globally ordered

V. S. Matveev

Chelyabinsk State University

Abstract: Suppose all geodesics of two Riemannian metrics $g$ and $\overline g$ defined on a (connected, geodesically complete) manifold $M^n$ coincide. At each point $x\in M^n$, consider the common eigenvalues $\rho_1,\rho_2,…,\rho_n$ of the two metrics (we assume that $\rho_1\geqslant\rho_2\geqslant…\geqslant\rho_n$)) and the numbers
$$\lambda_i=(\rho_1\rho_2\dotsb\rho_n)^{1/(n+1)}\frac1{\rho_i}.$$
. We show that the numbers $\lambda_i$ are ordered over the entire manifold: for any two points $x$ and $y$ in M the number $\lambda_k(x)$ is not greater than $\lambda_{k+1}(y)$. If $\lambda_k(x)=\lambda_{k+1}(y)$, then there is a point $z\in M^n$ such that $\lambda_k(z)=\lambda_{k+1}(z)$. If the manifold is closed and all the common eigenvalues of the metrics are pairwise distinct at each point, then the manifold can be covered by the torus.

DOI: https://doi.org/10.4213/mzm2502

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English version:
Mathematical Notes, 2005, 77:3, 380–390

Bibliographic databases:

UDC: 517.9+514.17
Revised: 21.04.2003

Citation: V. S. Matveev, “The eigenvalues of the Sinyukov mapping for geodesically equivalent metrics are globally ordered”, Mat. Zametki, 77:3 (2005), 412–423; Math. Notes, 77:3 (2005), 380–390

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz2502
• https://doi.org/10.4213/mzm2502
• http://mi.mathnet.ru/eng/mz/v77/i3/p412

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This publication is cited in the following articles:
1. Kiosak V., Matveev V.S., “Complete Einstein metrics are geodesically rigid”, Comm. Math. Phys., 289:1 (2009), 383–400
2. Kiosak V., Matveev V.S., “Proof of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two”, Comm. Math. Phys., 297:2 (2010), 401–426
3. Matveev V.S., Rosemann S., “Proof of the Yano-Obata Conjecture for H-Projective Transformations”, J. Differ. Geom., 92:2 (2012), 221–261
4. Mikes J. Stepanova E. Vanzurova A., “Differential Geometry of Special Mappings”, Differential Geometry of Special Mappings, Palacky Univ, 2015, 1–566
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