Geodesic
Metrization of connections with regular curvature
Archivum mathematicum, Tome 45 (2009) no. 4, pp. 325-333 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively existence of positive definite metrics compatible with a real analytic connection with regular curvature tensor on an analytic connected and simply connected manifold, and to construct the family of compatible metrics (determined up to a scalar multiple) in the affirmative case. We also breafly touch related problems concerning geodesic mappings and projective structures.
We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively existence of positive definite metrics compatible with a real analytic connection with regular curvature tensor on an analytic connected and simply connected manifold, and to construct the family of compatible metrics (determined up to a scalar multiple) in the affirmative case. We also breafly touch related problems concerning geodesic mappings and projective structures.
Classification : 53B05, 53B20, 53C05, 53C20
Keywords: manifold; linear connection; metric; pseudo-Riemannian geometry 
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     author = {Van\v{z}urov\'a, Alena},
     title = {Metrization of connections with regular curvature},
     journal = {Archivum mathematicum},
     pages = {325--333},
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     zbl = {1212.53020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2009_45_4_a7/}
}
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Vanžurová, Alena. Metrization of connections with regular curvature. Archivum mathematicum, Tome 45 (2009) no. 4, pp. 325-333. http://geodesic.mathdoc.fr/item/ARM_2009_45_4_a7/

[1] Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg, New York, 2003. | MR | Zbl

[2] Borel, A., Lichnerowicz, A.: Groupes d’holonomie des variétés riemanniennes. C. R. Acad. Sci. Paris 234 (1952), 1835–1837. | MR | Zbl

[3] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I, II. Wiley-Intersc. Publ., New York, Chichester, Brisbane, Toronto, Singapore, 1991.

[4] Kowalski, O.: On regular curvature structures. Math. Z. 125 (1972), 129–138. | DOI | MR | Zbl

[5] Kowalski, O.: Metrizability of affine connections on analytic manifolds. Note di Matematica 8 (1) (1988), 1–11. | MR | Zbl

[6] Mikeš, J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. 78 (1996), 311–333. | DOI | MR

[7] Mikeš, J., Kiosak, V., Vanžurová, A.: Geodesic mappings of manifolds with affine connection. Palacký University, Olomouc (2008). | MR | Zbl

[8] Schmidt, B. G.: Conditions on a connection to be a metric connection. Commun. Math. Phys. 29 (1973), 55–59. | DOI | MR

[9] Vanžurová, A.: Metrization problem for linear connections and holonomy algebras. Arch. Math. (Brno) 44 (2008), 339–349. | MR | Zbl

[10] Vilimová, Z.: The problem of metrizability of linear connections. Master's thesis, 2004, (supervisor: O. Krupková).