On The Existence of Levi Foliations
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| Formato: | Artículo científico |
| Lenguaje: | en |
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Academia Brasileira de Ciências
2001
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| _version_ | 1876462270312087552 |
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| author | Renata N. Ostwald |
| author_facet | Renata N. Ostwald |
| contents | On The Existence of Levi Foliations Renata N. Ostwald Multidisciplinaria (Ciencias Naturales y Exactas) singularities Levi varieties Levi foliations holomorphic foliations Let L C2 be a real 3 dimensional analytic variety. For each regular point p - L there exists aunique complex line lp on the space tangent to L at p. When the field of complex linep → lpis completely integrable, we say that L is Levi variety. More generally; let L - M be a realsubvariety in an holomorphic complex variety M. If there exists a real 2 dimensional integrabledistribution on L which is invariant by the holomorphic structure J induced by M, we say that Lis a Levi variety. We shall prove:Theorem. Let L be a Levi foliation and let F be the induced holomorphic foliation. Then, Fadmits a Liouvillian first integral.In other words, if L is a 3 dimensional analytic foliation such that the induced complex distributiondefines an holomorphic foliationF; that is, if L is a Levi foliation; thenF admits a Liouvillian firstintegrala function which can be constructed by the composition of rational functions, exponentiation,integration, and algebraic functions (Singer 1992). For example, if f is an holomorphicfunction and if θ is real a 1-form on R2; then the pull-back of θ by f defines a Levi foliationL : f -θ = 0 which is tangent to the holomorphic foliation F : df = 0.This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997). 2001 artículo científico 0001-3765 https://www.redalyc.org/articulo.oa?id=32773102 en http://www.redalyc.org/revista.oa?id=327 Anais da Academia Brasileira de Ciências application/pdf Academia Brasileira de Ciências Anais da Academia Brasileira de Ciências (Brasil) Num.1 Vol.73 |
| format | Artículo científico |
| id | redalyc_32773102 |
| institution | Redalyc |
| language | en |
| publishDate | 2001 |
| publisher | Academia Brasileira de Ciências |
| spellingShingle | On The Existence of Levi Foliations Renata N. Ostwald Multidisciplinaria (Ciencias Naturales y Exactas) singularities Levi varieties Levi foliations holomorphic foliations On The Existence of Levi Foliations Renata N. Ostwald Multidisciplinaria (Ciencias Naturales y Exactas) singularities Levi varieties Levi foliations holomorphic foliations Let L C2 be a real 3 dimensional analytic variety. For each regular point p - L there exists aunique complex line lp on the space tangent to L at p. When the field of complex linep → lpis completely integrable, we say that L is Levi variety. More generally; let L - M be a realsubvariety in an holomorphic complex variety M. If there exists a real 2 dimensional integrabledistribution on L which is invariant by the holomorphic structure J induced by M, we say that Lis a Levi variety. We shall prove:Theorem. Let L be a Levi foliation and let F be the induced holomorphic foliation. Then, Fadmits a Liouvillian first integral.In other words, if L is a 3 dimensional analytic foliation such that the induced complex distributiondefines an holomorphic foliationF; that is, if L is a Levi foliation; thenF admits a Liouvillian firstintegrala function which can be constructed by the composition of rational functions, exponentiation,integration, and algebraic functions (Singer 1992). For example, if f is an holomorphicfunction and if θ is real a 1-form on R2; then the pull-back of θ by f defines a Levi foliationL : f -θ = 0 which is tangent to the holomorphic foliation F : df = 0.This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997). 2001 artículo científico 0001-3765 https://www.redalyc.org/articulo.oa?id=32773102 en http://www.redalyc.org/revista.oa?id=327 Anais da Academia Brasileira de Ciências application/pdf Academia Brasileira de Ciências Anais da Academia Brasileira de Ciências (Brasil) Num.1 Vol.73 |
| title | On The Existence of Levi Foliations |
| topic | Multidisciplinaria (Ciencias Naturales y Exactas) singularities Levi varieties Levi foliations holomorphic foliations |
| url | https://www.redalyc.org/articulo.oa?id=32773102 |