A geometric theory of harmonic and semi-conformal maps

Anders Kock

Open Mathematics (2004)

  • Volume: 2, Issue: 5, page 708-724
  • ISSN: 2391-5455

Abstract

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We describe for any Riemannian manifold M a certain scheme M L, lying in between the first and second neighbourhood of the diagonal of M. Semi-conformal maps between Riemannian manifolds are then analyzed as those maps that preserve M L; harmonic maps are analyzed as those that preserve the (Levi-Civita-) mirror image formation inside M L.

How to cite

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Anders Kock. "A geometric theory of harmonic and semi-conformal maps." Open Mathematics 2.5 (2004): 708-724. <http://eudml.org/doc/268808>.

@article{AndersKock2004,
abstract = {We describe for any Riemannian manifold M a certain scheme M L, lying in between the first and second neighbourhood of the diagonal of M. Semi-conformal maps between Riemannian manifolds are then analyzed as those maps that preserve M L; harmonic maps are analyzed as those that preserve the (Levi-Civita-) mirror image formation inside M L.},
author = {Anders Kock},
journal = {Open Mathematics},
keywords = {51K10; 53A30; 53C43},
language = {eng},
number = {5},
pages = {708-724},
title = {A geometric theory of harmonic and semi-conformal maps},
url = {http://eudml.org/doc/268808},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Anders Kock
TI - A geometric theory of harmonic and semi-conformal maps
JO - Open Mathematics
PY - 2004
VL - 2
IS - 5
SP - 708
EP - 724
AB - We describe for any Riemannian manifold M a certain scheme M L, lying in between the first and second neighbourhood of the diagonal of M. Semi-conformal maps between Riemannian manifolds are then analyzed as those maps that preserve M L; harmonic maps are analyzed as those that preserve the (Levi-Civita-) mirror image formation inside M L.
LA - eng
KW - 51K10; 53A30; 53C43
UR - http://eudml.org/doc/268808
ER -

References

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  1. [1] P. Baird and J.C. Wood: Harmonic Morphisms Between Riemannian Manifolds, Oxford University Press, 2003. Zbl1055.53049
  2. [2] E. Dubuc: “C ∞ schemes”, Am. J. Math, Vol. 103, (1981), pp. 683–690. http://dx.doi.org/10.2307/2374046 Zbl0483.58003
  3. [3] A. Grothendieck: Techniques de construction en géometrie algébrique, Sem. H. Cartan, Paris, 1960–61, pp. 7–17. 
  4. [4] A. Kock: Synthetic Differential Geometry, Cambridge University Press, 1981. 
  5. [5] A. Kock: “A combinatorial theory of connections”, Contemporary Mathematics, Vol. 30, (1984), pp. 132–144. 
  6. [6] A. Kock: “Geometric construction of the Levi-Civita parallelism”, Theory and Applications of Categories, Vol. 4(9), (1998). Zbl0923.51016
  7. [7] A. Kock: “Infinitesimal aspects of the Laplace operator”, Theory and Applications of Categories Vol. 9(1), (2001). Zbl1026.18007
  8. [8] A. Kock: “First neighbourhood of the diagonal, and geometric distributions”, Universitatis Iagellonicae Acta Math., Vol. 41. (2003), pp. 307–318. Zbl1073.53020
  9. [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. Zbl0564.18009
  10. [10] A. Kumpera and D. Spencer: “Lie Equations”, Annals of Math. Studies, Vol. 73, Princeton, 1972. Zbl0258.58015
  11. [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. Zbl0472.18009
  12. [12] B. Malgrange: “Equations de Lie”, I, J. Diff. Geom., Vol. 6, (1972), pp. 503–522. 
  13. [13] D. Mumford: The Red Book of varieties and schemes, Springer L.N.M. 1358, 1988. Zbl0658.14001
  14. [14] A. Weil: “Théorie des points proches sur les varietés différentiables”, Colloque Top. et Géom. Diff., Stasbourg 1953. Zbl0053.24903

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