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457:{\displaystyle R_{ijkl}=(\partial _{i}\rho _{j}-\partial _{j}\rho _{i})g_{kl}+g_{jl}(\partial _{i}\rho _{k}-\rho _{i}\rho _{k})-g_{il}(\partial _{j}\rho _{k}-\rho _{j}\rho _{k}).}
108:
described
Beltrami's original proof (done in the two-dimensional Riemannian case) as being much more complicated. Relative to a projectively flat chart, there are functions
100:
Beltrami's theorem asserts the converse: any connected pseudo-Riemannian manifold which is locally projectively flat must have constant curvature. With the use of
129:
750:
is constant. Substituting the above identity into the
Riemann tensor as given above, it follows that the chart domain has constant sectional curvature
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1040:
948:
1023:. Die Grundlehren der mathematischen Wissenschaften. Vol. 10 (Second edition of 1923 original ed.). Berlin–Göttingen–Heidelberg:
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813:: if given a geodesic map between pseudo-Riemannian manifolds, one manifold has constant curvature if and only if the other does.
86:
690:{\displaystyle \partial _{j}\rho _{k}-\rho _{j}\rho _{k}=g_{jk}{\frac {g^{il}(\partial _{i}\rho _{l}-\rho _{i}\rho _{l})}{n}}}
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28:
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61:, there are smooth coordinates relative to which all nonconstant geodesics appear as straight lines. In the
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is the dimension of the manifold. It is direct to verify that the left-hand side is a (locally defined)
24:
50:, is a result on the inverse problem of determining a (pseudo-)Riemannian metric from its geodesics.
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Nachrichten von der
Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
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1112:"Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung"
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Ricci-calculus. An introduction to tensor analysis and its geometrical applications
810:
806:. By connectedness of the manifold, this local constancy implies global constancy.
213:{\displaystyle \Gamma _{ij}^{k}=\rho _{i}\delta _{j}^{k}+\rho _{j}\delta _{i}^{k}.}
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940:(Revised & updated second edition of 1976 original ed.). Mineola, NY:
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89:, these charts show that any Riemannian manifold of constant curvature is
1073:. Reprinted in 1988. (Second edition of 1950 original ed.). London:
36:
708:, using only the given form of the Christoffel symbols. It follows from
909:
891:
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Geodesic maps preserve the property of having constant curvature
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892:"Teoria fondamentale degli spazii di curvature costante"
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Beltrami's theorem may be phrased in the language of
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of constant curvature is locally projectively flat.
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938:Differential geometry of curves & surfaces
35:determines a certain class of paths known as
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53:It is nontrivial to see that, on any
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981:. Reprinted in 1997. Princeton:
104:, the proof is straightforward.
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519:. The other curvature symmetry
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1075:Addison-Wesley Publishing Co.
69:, this is justified by the
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95:pseudo-Riemannian manifold
91:locally projectively flat.
1033:10.1007/978-3-662-12927-2
975:Eisenhart, Luther Pfahler
81:, it is justified by the
942:Dover Publications, Inc.
225:Riemann curvature tensor
467:The curvature symmetry
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991:10.1515/9781400884216
934:do Carmo, Manfredo P.
872:, Footnote on p. 110.
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85:. In the language of
25:differential geometry
1144:"Beltrami's theorem"
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234:
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93:More generally, any
71:Beltrami–Klein model
43:, named for Italian
979:Riemannian geometry
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121:Christoffel symbols
83:gnomonic projection
67:hyperbolic geometry
55:Riemannian manifold
1141:Weisstein, Eric W.
910:10.1007/BF02419615
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547:, then says that
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79:spherical geometry
75:positive curvature
63:negative curvature
59:constant curvature
41:Beltrami's theorem
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888:Beltrami, Eugenio
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33:Riemannian metric
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1118:: 99–112.
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967:1352.53002
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817:References
1149:MathWorld
926:120773141
870:Weyl 1921
670:ρ
660:ρ
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637:∂
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375:ρ
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297:ρ
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283:−
274:ρ
264:∂
194:δ
184:ρ
166:δ
156:ρ
135:Γ
73:. In the
37:geodesics
23:field of
1162:Category
1110:(1921).
1108:Weyl, H.
1069:(1961).
1019:(1954).
977:(1926).
936:(2016).
890:(1868).
880:Sources.
77:case of
65:case of
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700:where
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543:and
535:klij
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483:jikl
474:ijkl
1120:JFM
1096:Zbl
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1029:doi
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