Harmonie reflections

Lieven Vanhecke; Maria-Elena Vazquez-Abal

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1988)

  • Volume: 82, Issue: 2, page 229-236
  • ISSN: 1120-6330

Abstract

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We study local reflections \phi_{\sigma} with respect to a curve \sigma in a Riemannian manifold and prove that \sigma is a geodesic if \phi_{\sigma} is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if \phi_{\sigma} is harmonic for all geodesies \sigma.

How to cite

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Vanhecke, Lieven, and Vazquez-Abal, Maria-Elena. "Harmonie reflections." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 82.2 (1988): 229-236. <http://eudml.org/doc/287366>.

@article{Vanhecke1988,
abstract = {We study local reflections $\phi_\{\sigma\}$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if $\phi_\{\sigma\}$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if $\phi_\{\sigma\}$ is harmonic for all geodesies $\sigma$.},
author = {Vanhecke, Lieven, Vazquez-Abal, Maria-Elena},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Harmonic maps; Reflections with respect to a curve; Harmonic reflections; local reflection; harmonic map; geodesic; constant curvature},
language = {eng},
month = {6},
number = {2},
pages = {229-236},
publisher = {Accademia Nazionale dei Lincei},
title = {Harmonie reflections},
url = {http://eudml.org/doc/287366},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Vanhecke, Lieven
AU - Vazquez-Abal, Maria-Elena
TI - Harmonie reflections
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/6//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 2
SP - 229
EP - 236
AB - We study local reflections $\phi_{\sigma}$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if $\phi_{\sigma}$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if $\phi_{\sigma}$ is harmonic for all geodesies $\sigma$.
LA - eng
KW - Harmonic maps; Reflections with respect to a curve; Harmonic reflections; local reflection; harmonic map; geodesic; constant curvature
UR - http://eudml.org/doc/287366
ER -

References

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  2. CHEN, B.Y., Geometry of submanifolds, Pure and Applied Mathematics, 22, Marcel Dekker, New York, 1973. Zbl0262.53036MR353212
  3. DODSON, C.T.J., VANHECKE, L. and VAZQUEZ-ABAL, M.E., Harmonic geodesic symmetries, «C.R. Math Rep. Acad. Sci. Canada» 9 (1987), 231-235. Zbl0631.53013MR910160
  4. EELLS, J. and SAMPSON, J.H., Harmonic mappings of Riemannian manifolds, «Amer. J. Math.» 86 (1984), 109-160. Zbl0122.40102MR164306
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  6. GHEYSENS, L., Riemannse differentiaalmeetkunde van buisvormige omgevingen, doctoral dissertation, Katholieke Universiteit Leuven, 1981. 
  7. GRAY, A. and VANHECKE, L., The volumes of tubes about curves in a Riemannian manifold, «Proc. London Math. Soc.» 44 (1982), 215-243. Zbl0491.53035MR647431DOI10.1112/plms/s3-44.2.215
  8. TONDEUR, P.H. and VANHECKE, L., Reflections in submanifolds, to appear in «Geometricae Dedicata». Zbl0656.53055MR965832DOI10.1007/BF00147801
  9. VANHECKE, L. and WILLMORE, T.J., Interaction of tubes ad spheres, «Math. Ann.» 263 (1983), 31-42. Zbl0491.53034MR697328DOI10.1007/BF01457081
  10. VANHECKE, L., Geometry and symmetry, Proc. Workshop on Advances in Differential Geometry and Topology, Torino1987, to appear. Zbl0766.53043

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