Abstract
The objects under study, in this article, are Riemannian manifoldsfoliated by hypersurfaces. Looking at the transverse direction as time,we construct the generalised heat operator and, in the spirit of atime-space extension of harmonic morphisms, we introduce the conformal heat morphisms. Concrete examples of these concepts arepresented, as well as, a characterisation of conformal heat morphisms.Lastly, we calculate the heat Lie algebra of the generalised heatoperator.
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Almeida, L. and Ge, Y.: Symmetry results for positive solutions of some elliptic equations on manifolds, Ann. Global Anal. Geom. 18 (2000), 153-170.
Aubin, T.: Non-Linear Analysis on Manifolds. Monge-Ampère Equations, Ser. Comprehensive Stud. Math. 252, Springer-Verlag, Berlin, 1982.
Baird, P. and Gudmundsson, S.: p-Harmonic maps and minimal submanifolds, Math. Ann. 294 (1992), 611-624.
Baird, P. and Wood, J. C.: Harmonic Morphisms between Riemannian Manifolds, London Math. Soc. Monogr. (N.S.), Oxford Univ. Press, Oxford, to appear.
Djehiche, B. and Kolsrud, T.: Canonical transformations for diffusions, C. R. Acad. Sciences Paris 321 (1995), 339-344.
Eells, J. and Lemaire, L.: A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68.
Eells, J. and Lemaire, L.: Selected Topics in Harmonic Maps, CBMS Regional Conf. Ser. in Math. 50, Amer. Math. Soc., Providence, RI, 1983.
Eells, J. and Lemaire, L.: Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385-524.
Eells, J. and Sampson, J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160.
Fuglede, B.: Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), 107-144.
Godbillon, C.: Feuilletages, Progr. Math. 98, Birkhäuser, Boston, 1991.
Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions, J.Math. Kyoto Univ. 19 (1979), 215-229.
Kolsrud, T.: Quantum constants of motion and the heat Lie algebra in a Riemannian manifold, Ann. Inst. H. Poincaré, to appear.
Ladyženskaja, O. A., Solonnikov, V. A., and Ural'ceva, N. N.: Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs 23, Amer. Math. Soc., Providence, RI, 1968.
Loubeau, E.: Morphisms of the heat equation, Ann. Global Anal. Geom. 15 (1997), 487-496.
Sullivan, D.: A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv. 54 (1979), 218-223.
Tischler, D.: Totally parallelizable 3-manifolds, in: L. Auslander and W. H. Gottschalk (eds), Topological Dynamics, Benjamin, New York, 1968, pp. 471-492.
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Kolsrud, T., Loubeau, E. Foliated Manifolds and Conformal Heat Morphisms. Annals of Global Analysis and Geometry 21, 241–267 (2002). https://doi.org/10.1023/A:1014929718946
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DOI: https://doi.org/10.1023/A:1014929718946