A characterization of harmonic sections and a Liouville theorem

Simão Stelmastchuk

Archivum Mathematicum (2012)

  • Volume: 048, Issue: 2, page 149-162
  • ISSN: 0044-8753

Abstract

top
Let P ( M , G ) be a principal fiber bundle and E ( M , N , G , P ) an associated fiber bundle. Our interest is to study the harmonic sections of the projection π E of E into M . Our first purpose is give a characterization of harmonic sections of M into E regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of π E .

How to cite

top

Stelmastchuk, Simão. "A characterization of harmonic sections and a Liouville theorem." Archivum Mathematicum 048.2 (2012): 149-162. <http://eudml.org/doc/246902>.

@article{Stelmastchuk2012,
abstract = {Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ an associated fiber bundle. Our interest is to study the harmonic sections of the projection $\pi _\{E\}$ of $E$ into $M$. Our first purpose is give a characterization of harmonic sections of $M$ into $E$ regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of $\pi _\{E\}$.},
author = {Stelmastchuk, Simão},
journal = {Archivum Mathematicum},
keywords = {harmonic sections; Liouville theorem; stochastic analysis on manifolds; harmonic section; Liouville theorem; stochastic analysis on manifolds},
language = {eng},
number = {2},
pages = {149-162},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A characterization of harmonic sections and a Liouville theorem},
url = {http://eudml.org/doc/246902},
volume = {048},
year = {2012},
}

TY - JOUR
AU - Stelmastchuk, Simão
TI - A characterization of harmonic sections and a Liouville theorem
JO - Archivum Mathematicum
PY - 2012
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 048
IS - 2
SP - 149
EP - 162
AB - Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ an associated fiber bundle. Our interest is to study the harmonic sections of the projection $\pi _{E}$ of $E$ into $M$. Our first purpose is give a characterization of harmonic sections of $M$ into $E$ regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of $\pi _{E}$.
LA - eng
KW - harmonic sections; Liouville theorem; stochastic analysis on manifolds; harmonic section; Liouville theorem; stochastic analysis on manifolds
UR - http://eudml.org/doc/246902
ER -

References

top
  1. Arvanitoyeorgos, A., An introduction to Lie groups and the geometry of homogeneous spaces, Student Mathematical Library, vol. 22, AMS, Providence, RI, 2003. (2003) Zbl1045.53001MR2011126
  2. Benyounes, M., Loubeau, E., Wood, C. M., 10.1016/j.difgeo.2006.11.010, Differential Geom. Appl. 25 (3) (2007), 322–334. (2007) Zbl1128.53037MR2330461DOI10.1016/j.difgeo.2006.11.010
  3. Catuogno, P., A geometric Itô formula, Workshop on Differential Geometry. Mat. Contemp., vol. 33, 2007, pp. 85–99. (2007) Zbl1156.58013MR2429603
  4. Catuogno, P., Stelmastchuk, S., 10.1007/s11118-007-9068-y, Potential Anal. 28 (2008), 61–69. (2008) Zbl1131.53033MR2366399DOI10.1007/s11118-007-9068-y
  5. Elworthy, K. D., Kendall, W. S., Factorization of harmonic maps and Brownian motions, Pitman Res. Notes Math. Ser. 150 (1985), 72–83. (1985) MR0894524
  6. Emery, M., On two transfer principles in stochastic differential geometry, Séminaire de Probabilités XXIV, 407 – 441, Lectures Notes in Math., 1426, Springer, Berlin, 1989. (1989) MR1071558
  7. Emery, M., Stochastic Calculus in Manifolds, Springer, Berlin, 1989. (1989) Zbl0697.60060MR1030543
  8. Emery, M., Martingales continues dans les variétés différentiables, Lectures on probability theory and statistics (Saint-Flour, 1998), 1–84, Lecture Notes in Math., 1738, Springer, Berlin, 2000. (2000) Zbl0969.60042MR1775639
  9. Hsu, E., Stochastic analysis on manifolds, Grad. Stud. Math. 38 (2002). (2002) Zbl0994.58019MR1882015
  10. Ishihara, S., Yano, K., Tangent and cotangent bundles: Differential geometry, Pure Appl. Math. 16 (1973). (1973) Zbl0262.53024MR0350650
  11. Ishihara, T., Harmonic sections of tangent bundles, J. Math. Tokushima Univ. 13 (1979), 23–27. (1979) Zbl0427.53019MR0563393
  12. J., Vilms, Totally geodesic maps, J. Differential Geom. 4 (1970), 73–79. (1970) Zbl0194.52901MR0262984
  13. Kendall, W. S., 10.1080/17442508608833419, Stochastics 19 (1–2) (1986), 111–129. (1986) Zbl0584.58045MR0864339DOI10.1080/17442508608833419
  14. Kendall, W. S., From stochastic parallel transport to harmonic maps, New directions in Dirichlet forms, Amer. Math. Soc., Stud. Adv. Math. 8 ed., 1998, pp. 49–115. (1998) Zbl0924.58110MR1652279
  15. Kobayashi, S., Nomizu, K., Foundations of Differential Geometry, vol. I, Interscience Publishers, New York, 1963. (1963) Zbl0119.37502MR0152974
  16. Lindvall, T., Rogers, L. C. G., 10.1214/aop/1176992442, Ann. Probab. 14 (3) (1986), 860–872. (1986) Zbl0593.60076MR0841588DOI10.1214/aop/1176992442
  17. Meyer, P. A., Géométrie stochastique sans larmes, Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), Lecture Notes in Math., 850, Springer, Berlin–New York, 1981, pp. 44–102. (1981) Zbl0459.60046MR0622555
  18. Musso, E., Tricerri, F., Riemannian metrics on tangent bundle, Ann. Mat. Pura Appl. (4) 150 (1988), 1–19. (1988) MR0946027
  19. Poor, W. A., Differential geometric structures, McGraw-Hill Book Co., New York, 1981. (1981) Zbl0493.53027MR0647949
  20. Protter, P., Stochastic integration and differential equations. A new approach, Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 1990. (1990) Zbl0694.60047MR1037262
  21. Shigekawa, I., 10.1007/BF00531745, Z. Wahrsch. Verw. Gebiete 59 (2) (1982), 211–221. (1982) Zbl0487.60056MR0650613DOI10.1007/BF00531745
  22. Wood, C. M., 10.1112/jlms/s2-33.1.157, J. London Math. Soc. (2) 33 (1) (1986), 157–168. (1986) MR0829396DOI10.1112/jlms/s2-33.1.157
  23. Wood, C. M., Harmonic sections and Yang – Mills fields, Proc. London Math. Soc. (3) 54 (3) (1987), 544–558. (1987) Zbl0616.53028MR0879397
  24. Wood, C. M., 10.1007/BF02677834, Manuscripta Math. 94 (1) (1997), 1–13. (1997) Zbl0914.58011MR1468930DOI10.1007/BF02677834
  25. Wood, C. M., 10.1016/S0926-2245(03)00021-4, Differential Geom. Appl. 19 (2) (2003), 193–210. (2003) Zbl1058.53053MR2002659DOI10.1016/S0926-2245(03)00021-4

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.