zbMATH — the first resource for mathematics

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.

51K10 Synthetic differential geometry
Full Text: DOI arXiv
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.