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A geometric theory of harmonic and semi-conformal maps. (English) Zbl 1146.51012
This paper has some overlap with the author’s [Theory Appl. Categ. 9, 1–16, electronic only (2001; Zbl 1026.18007)].
The author, using a subscheme \(M_{L}\subseteq M\times M\) between the first and second neighborhood of the diagonal of any Riemannian manifold \(M\), characterizes semi-conformal maps and harmonic maps geometrically. The present paper is more refined in the construction of \(M_{L}\) and the proofs than the preceding paper. The new results in the present paper are the characterization of semi-conformality and a version of the theorem of B. Fuglede [Ann. Inst. Fourier 28, No. 2, 107–144 (1978; Zbl 0339.53026)] and T. Ishihara [J. Math. Kyoto Univ. 19, 215–229 (1979; Zbl 0421.31006)] talking about harmonic \(2\)-jets in place of harmonic germs. The latter is almost immediate from the geometric descriptions of harmonic and semi-conformal maps.

MSC:
51K10 Synthetic differential geometry
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References:
[1] P. Baird and J.C. Wood: Harmonic Morphisms Between Riemannian Manifolds, Oxford University Press, 2003. · Zbl 1055.53049
[2] E. Dubuc: “C ∞ schemes”, Am. J. Math, Vol. 103, (1981), pp. 683-690. http://dx.doi.org/10.2307/2374046 · Zbl 0483.58003
[3] A. Grothendieck: Techniques de construction en géometrie algébrique, Sem. H. Cartan, Paris, 1960-61, pp. 7-17.
[4] A. Kock: Synthetic Differential Geometry, Cambridge University Press, 1981. · Zbl 0466.51008
[5] A. Kock: “A combinatorial theory of connections”, Contemporary Mathematics, Vol. 30, (1984), pp. 132-144. · Zbl 0542.18007
[6] A. Kock: “Geometric construction of the Levi-Civita parallelism”, Theory and Applications of Categories, Vol. 4(9), (1998). · Zbl 0923.51016
[7] A. Kock: “Infinitesimal aspects of the Laplace operator”, Theory and Applications of Categories Vol. 9(1), (2001). · Zbl 1026.18007
[8] A. Kock: “First neighbourhood of the diagonal, and geometric distributions”, Universitatis Iagellonicae Acta Math., Vol. 41. (2003), pp. 307-318. · Zbl 1073.53020
[9] A. Kock and R. Lavendhomme: “Strong infinitesimal linearity, with applications to strong difference and affine connections”, Cahiers de Top. et Géom. Diff., Vol. 25, (1984), pp. 311-324. · Zbl 0564.18009
[10] A. Kumpera and D. Spencer: “Lie Equations”, Annals of Math. Studies, Vol. 73, Princeton, 1972. · Zbl 0258.58015
[11] F.W. Lawvere: “Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body”, Cahiers de Top. et Géom. Diff., Vol. 21, (1980), pp. 377-392. · Zbl 0472.18009
[12] B. Malgrange: “Equations de Lie”, I, J. Diff. Geom., Vol. 6, (1972), pp. 503-522.
[13] D. Mumford: The Red Book of varieties and schemes, Springer L.N.M. 1358, 1988. · Zbl 0658.14001
[14] A. Weil: “Théorie des points proches sur les varietés différentiables”, Colloque Top. et Géom. Diff., Stasbourg 1953.
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