Beltrami's theorem (original) (raw)
In de Riemann-meetkunde, een deelgebied van de wiskunde, is de stelling van Beltrami een resultaat dat vernoemd is naar de Italiaanse wiskundige Eugenio Beltrami. De stelling van Beltrami bewijst dat geodetische afbeeldingen de eigenschap van constante kromming behouden.
| Property | Value |
|---|---|
| dbo:abstract | In the mathematical field of differential geometry, any (pseudo-)Riemannian metric determines a certain class of paths known as geodesics. Beltrami's theorem, named for Italian mathematician Eugenio Beltrami, is a result on the inverse problem of determining a (pseudo-)Riemannian metric from its geodesics. It is nontrivial to see that, on any Riemannian manifold of constant curvature, there are smooth coordinates relative to which all nonconstant geodesics appear as straight lines. In the negative curvature case of hyperbolic geometry, this is justified by the Beltrami–Klein model. In the positive curvature case of spherical geometry, it is justified by the gnomonic projection. In the language of projective differential geometry, these charts show that any Riemannian manifold of constant curvature is locally projectively flat. More generally, any pseudo-Riemannian manifold of constant curvature is locally projectively flat. Beltrami's theorem asserts the converse: any connected pseudo-Riemannian manifold which is locally projectively flat must have constant curvature. With the use of tensor calculus, the proof is straightforward. Hermann Weyl 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 ρi such that the Christoffel symbols take the form Direct calculation then shows that the Riemann curvature tensor is given by The curvature symmetry Rijkl + Rjikl = 0 implies that ∂i ρj = ∂j ρi. The other curvature symmetry Rijkl = Rklij, traced over i and l, then says that where n is the dimension of the manifold. It is direct to verify that the left-hand side is a (locally defined) Codazzi tensor, using only the given form of the Christoffel symbols. It follows from Schur's lemma that gil(∂i ρl − ρi ρl) is constant. Substituting the above identity into the Riemann tensor as given above, it follows that the chart domain has constant sectional curvature −1/ngil(∂i ρl − ρi ρl). By connectedness of the manifold, this local constancy implies global constancy. Beltrami's theorem may be phrased in the language of geodesic maps: if given a geodesic map between pseudo-Riemannian manifolds, one manifold has constant curvature if and only if the other does. (en) In de Riemann-meetkunde, een deelgebied van de wiskunde, is de stelling van Beltrami een resultaat dat vernoemd is naar de Italiaanse wiskundige Eugenio Beltrami. De stelling van Beltrami bewijst dat geodetische afbeeldingen de eigenschap van constante kromming behouden. (nl) |
| dbo:wikiPageExternalLink | https://eudml.org/doc/59096 |
| dbo:wikiPageID | 13636654 (xsd:integer) |
| dbo:wikiPageLength | 5719 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1073712457 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Beltrami–Klein_model dbc:Theorems_in_Riemannian_geometry dbr:Princeton_University_Press dbr:Annali_di_Matematica_Pura_ed_Applicata dbr:Geodesic_map dbr:Geodesic dbr:Gnomonic_projection dbr:Constant_curvature dbr:Mathematician dbr:Eugenio_Beltrami dbr:Dover_Publications,_Inc. dbr:Riemann_curvature_tensor dbr:Pseudo-Riemannian_manifold dbr:Riemannian_metric dbr:Hermann_Weyl dbr:Addison-Wesley_Publishing_Co. dbr:Hyperbolic_geometry dbr:Projective_differential_geometry dbr:Codazzi_tensor dbr:Differential_geometry dbr:Tensor_calculus dbr:Schur's_lemma_(Riemannian_geometry) dbr:Spherical_geometry dbr:Mathematical dbr:Springer-Verlag dbr:Christoffel_symbol |
| dbp:1a | Schouten (en) do Carmo (en) Beltrami (en) |
| dbp:1p | 292 (xsd:integer) 301 (xsd:integer) |
| dbp:1y | 1868 (xsd:integer) 1954 (xsd:integer) 2016 (xsd:integer) |
| dbp:2a | Eisenhart (en) Weyl (en) |
| dbp:2loc | Footnote on p. 110 (en) Section 40 (en) |
| dbp:2y | 1921 (xsd:integer) 1926 (xsd:integer) |
| dbp:3a | Schouten (en) |
| dbp:3loc | Section VI.2 (en) |
| dbp:3y | 1954 (xsd:integer) |
| dbp:4a | Struik (en) |
| dbp:4loc | Section 5-3 (en) |
| dbp:4y | 1961 (xsd:integer) |
| dbp:title | Beltrami's theorem (en) |
| dbp:urlname | BeltramisTheorem (en) |
| dbp:wikiPageUsesTemplate | dbt:= dbt:Cite_book dbt:Cite_journal dbt:Math dbt:MathWorld dbt:Mvar dbt:Refbegin dbt:Refend dbt:Reflist dbt:Sfnm dbt:Sfrac dbt:Short_description |
| dcterms:subject | dbc:Theorems_in_Riemannian_geometry |
| gold:hypernym | dbr:Result |
| rdf:type | yago:WikicatTheoremsInGeometry yago:WikicatTheoremsInRiemannianGeometry yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293 |
| rdfs:comment | In de Riemann-meetkunde, een deelgebied van de wiskunde, is de stelling van Beltrami een resultaat dat vernoemd is naar de Italiaanse wiskundige Eugenio Beltrami. De stelling van Beltrami bewijst dat geodetische afbeeldingen de eigenschap van constante kromming behouden. (nl) In the mathematical field of differential geometry, any (pseudo-)Riemannian metric determines a certain class of paths known as geodesics. Beltrami's theorem, named for Italian mathematician Eugenio Beltrami, is a result on the inverse problem of determining a (pseudo-)Riemannian metric from its geodesics. Direct calculation then shows that the Riemann curvature tensor is given by The curvature symmetry Rijkl + Rjikl = 0 implies that ∂i ρj = ∂j ρi. The other curvature symmetry Rijkl = Rklij, traced over i and l, then says that (en) |
| rdfs:label | Beltrami's theorem (en) Stelling van Beltrami (nl) |
| owl:sameAs | freebase:Beltrami's theorem yago-res:Beltrami's theorem wikidata:Beltrami's theorem dbpedia-nl:Beltrami's theorem https://global.dbpedia.org/id/4XU8S |
| prov:wasDerivedFrom | wikipedia-en:Beltrami's_theorem?oldid=1073712457&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Beltrami's_theorem |
| is dbo:knownFor of | dbr:Eugenio_Beltrami |
| is dbo:wikiPageRedirects of | dbr:Beltrami's_Theorem dbr:Beltrami_theorem |
| is dbo:wikiPageWikiLink of | dbr:Beltrami's_Theorem dbr:Eugenio_Beltrami dbr:List_of_theorems dbr:Beltrami_theorem |
| is dbp:knownFor of | dbr:Eugenio_Beltrami |
| is foaf:primaryTopic of | wikipedia-en:Beltrami's_theorem |