Abstract
Transversal wave maps and wave maps are different. There are wave maps which are not transversal wave maps, and vice versa. However, if f is a wave map under certain circumstance, then f is a transversal wave map. We show that if f is a transversal exponential wave map, then the associated energy–momentum is transversally conserved. We finally obtain the relationship among transversal wave maps, transversal exponential wave maps and certain second order symmetric tensors.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Álvarez López J., Masa X.: Morphisms between complete Riemannian pseudogroups. Topol. Appl. 155, 544–604 (2008)
Chiang Y.-J.: Harmonic maps of V-manifolds. Ann. Glob. Anal. Geom. 8(3), 315–344 (1990)
Chiang Y.-J.: Spectral geometry of V-manifolds and its applications to harmonic maps. Proc. Symp. Pure Math. Am. Math. Soc. 54(Part 1), 93–99 (1993)
Chiang Y.-J., Ratto A.: Harmonic maps on spaces with conical singularities. Bull. Soc. Math. Fr. 120(2), 251–262 (1992)
Chiang Y.-J., Sun H.: 2-Harmonic totally real submanifolds in a complex projective space. Bull. Inst. Math. Acad. Sin. 27(2), 99–107 (1999)
Chiang Y.-J., Sun H.: Biharmonic maps on V-manifolds. Int. J. Math. Math. Sci. 27(8), 477–484 (2001)
Chiang Y.-J., Wolak R.: Transversally biharmonic maps between foliated Riemannian manifolds. Int. J. Math. 19(8), 981–996 (2008)
Eells J., Lemaire L.: A report on harmonic maps. Bull. Lond. Math. Soc. 10, 1–68 (1978)
Eells J., Lemaire L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20, 385–524 (1988)
Eells, J., Lemaire, L.: Selected topics in harmonic maps. In: CBMS Regional Conference Series in Mathematics, Vol. 150. American Mathematical Society, Providence (1983)
Eells J., Lemaire L.: Some properties of exponential harmonic maps. Banach Center Publ. 27, 129–136 (1992)
Eells J., Sampson J.H.: Harmonic mappings between Riemannian manifolds. Am. J. Math. 86, 109–164 (1964)
Eells J., Verjovsky A.: Harmonic and Riemannian foliations. Bol. Soc. Mat. Mex. 4(1), 1–12 (1998)
El Kacimi A., Gallego Gomez E.: Foliated harmonic maps. Ill. J. Math. 40, 115–122 (1996)
lainerman S., Machedon M.: Smoothing estimates for null forms and applications. Duke Math. J. 81, 99–133 (1995)
Klainerman S., Machedon M.: On the optimal local regularity for gauge fields theories. Diff. Integral Equ. 10, 1019–1030 (1997)
Konderak J.J., Wolak R.: Transversally harmonic maps between manifolds with Riemannian foliations. Q. J. Math. 54, 335–354 (2003)
Konderak J.J., Wolak R.: Some remarks on transversally harmonic maps. Glasg. J. Math. 50(1), 1–16 (2008)
Molino P.: Riemannian Foliations. Birkhäuser, Basel (1988)
Nahmod A., Stefanov A., Uhlenbeck K.: On the well-posedness of the wave map problem in high dimension. Comm. Anal. Geom. 11, 49–83 (2003)
O’Neill B.: Semi-Riemannian Manifolds with Applications to Relativity. Academic Press, New York (1983)
Shatah, J., Struwe, M.: Geometric wave equations. Courant Lecture Notes in Mathematics, Vol. 2. American Mathematical Society, Providence (1998)
Shatah J., Struwe M.: The Cauchy problem for wave maps. Int. Math. Res. Not. 1, 555–571 (2002)
Tondeur Ph.: Geometry of Foliation. Birkhäuser, Basel (1997)
Tao T.: Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Int. Math. Res. Not. 6, 299–328 (2001)
Tao T.: Global regularity of wave maps. II. Small energy in two dimension. Comm. Math. Phys. 224, 443–544 (2001)
Tataru D.: The wave maps equations. Bull. Am. Math. Soc. 41, 185–204 (2004)
Tataru D.: Rough solutions for the wave maps equation. Am. J. Math. 127(2), 293–377 (2005)
Wolak R.: Foliated and associated geometric structures on foliated manifolds. Ann. Fac. Sc. Toulouse 10(3), 337–360 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chiang, YJ., Wolak, R.A. Transversal wave maps and transversal exponential wave maps. J. Geom. 104, 443–459 (2013). https://doi.org/10.1007/s00022-013-0185-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00022-013-0185-z