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Manifolds of Maps in Riemannian Foliations

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Abstract

Let (M′,F′) and (M,F) be two (compact or not) foliated manifolds, C ∞ F (M′, M) the space of smooth maps which send leaves into leaves. In this paper we prove that C ∞ F (M′, M) admits a structure of an infinite-dimensional manifold modeled on LF-spaces, provided that F is a Riemannian foliation or, more generally, when it admits an adapted local addition.

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References

  1. Boothby, W. M.: An introduction to Differentiable Manifolds and Riemannian Geometry. Pure Appl. Math. 63, Academic Press, New York, 1975.

    Google Scholar 

  2. Gràcia, X. and Pons, J. M.: On a evolution operator connecting Lagrangian and Hamiltonian formalisms. Lett. Math. Phys. 17 (1989).

  3. Hermann, R.: A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc. 11 (1960), 236–242.

    Google Scholar 

  4. Leslie, J.: On a differentiable structure for the group of diffeomorphisms, Topology 6 (1967), 263–271.

    Google Scholar 

  5. Leslie, J.: Two classes of classical subgroups of Diff(M), J. Diff. Geom. 5 (1971), 427–435.

    Google Scholar 

  6. Leslie, J.: A remark on the group of automorphisms of a foliation having a dense leaf, J. Differential Geom. 7 (1972), 597–601.

    Google Scholar 

  7. Margalef, J. and Outerelo, E.: Differential Topology. North-Holland Math. Stud. 123, 1992.

  8. Martínez, E., Cariñena, J. F. and Sarlet, W.: Derivations of differential forms along the tangent bundle projection, Differential Geom. Appl. Part I: 2 (1992), 17–43; Part II: 3 (1993), 1–29.

    Google Scholar 

  9. Michor, P. W.: Manifolds of smooth maps, Cahiers Topologie Geom. Diff. I: 19 (1978), 47–78; II: 21 (1980), 63–86.

    Google Scholar 

  10. Michor, P. W.: Manifolds of Differentiable mappings, Shiva Math. Series, 1980.

  11. Molino, P.: Riemannian Foliations, Progr. in Math. 73, Birkhäuser, Boston, 1988.

    Google Scholar 

  12. Omori, H.: On the group of diffeomorphisms on a compact manifold, In: Proc. Sympos. Pure Math. 15, Amer. Math. Soc. Providence, (1970), 167–183.

    Google Scholar 

  13. Omori, H.: Infinite Dimensional Lie Transformation Groups, Lecture Notes in Math. 427, Springer-Verlag, New York, 1976.

    Google Scholar 

  14. Palais, R. S.: Foundations of Global Non-linear Analysis, W. A. Benjamin, New York, 1968.

    Google Scholar 

  15. Reinhart, B. L.: Foliated manifolds with bundle-like metrics, Ann. of Math. 69(1) (1959). 119–131.

    Google Scholar 

  16. Tondeur, Ph.: Survey of Riemannian foliations, In: Proc. XXIV National Conf. Geom. Topol. A.C. Albu and M. Craioveanu (eds,) Univ. of Timisoara, Romania 1996.

    Google Scholar 

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Authors and Affiliations

  1. Dpto. Xeometria e Topoloxia, Fac. Matemáticas, Univ. de Santiago de Compostela, 15706, Galicia, Spain

    E. Macias-Virgós

  2. Dpto. Matemáticas, Fac. CC. Económicas e Empresariais, Univ. de Vigo, 36207, Galicia, Spain

    E. Sanmartín Carbón

Authors
  1. E. Macias-Virgós
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  2. E. Sanmartín Carbón
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Macias-Virgós, E., Sanmartín Carbón, E. Manifolds of Maps in Riemannian Foliations. Geometriae Dedicata 79, 143–156 (2000). https://doi.org/10.1023/A:1005217109018

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