Abstract
We find a simple local criterion for the existence of conjugate points on the group of volume-preserving diffeomorphisms of a 3-manifold with the Riemannian metric of ideal fluid mechanics, in terms of an ordinary differential equation along each Lagrangian path. Using this criterion, we prove that the first conjugate point along a geodesic in this group is always pathological: the differential of the exponential map always fails to be Fredholm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold V. (1966). Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l’hydrodynamique des fluids parfaits. Ann. Inst. Grenoble 16, 319–361
Arnold V., Khesin B. (1998) Topological Methods in Hydrodynamics. New York, Springer
Biliotti, L., Exel, R., Piccione, P., Tausk, D.V.: On the singularities of the exponential map in infinite dimensional Riemannian manifolds. To appear in Math. Ann.
Brenier Y. (1993) The dual least action problem for an ideal, incompressible fluid. Arch. Rat. Mech. Anal. 122, 323–351
Ebin D., Marsden J. (1970). Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92, 102–163
Ebin, D., Misiołek, G., Preston, S.C.: Singularities of the exponential map on the volume-preserving diffeomorphism group. To appear in Geom. Funct. Anal.
Friedlander S., Vishik M.M. (1992). Instability criteria for steady flows of a perfect fluid. Chaos 2(3): 455–460
Grossman N. (1965). Hilbert manifolds without epiconjugate points. Proc. Amer. Math. Soc. 16, 1365–1371
Misiołek G. (1993). Stability of ideal fluids and the geometry of the group of diffeomorphisms. Indiana Univ. Math. J. 42, 215–235
Misiołek G. (1996). Conjugate points in \(\mathcal{D}_{\mu}(T^2)\). Proc. Amer. Math. Soc. 124, 977–982
Preston S.C. (2004).For ideal fluids, Eulerian and Lagrangian instabilities are equivalent. Geom. Funct. Anal. 14, 1044–1062
Preston S.C. (2005). Nonpositive curvature on the area-preserving diffeomorphism group. J. Geom. Phys. 53(3): 259–274
Reid W.T. (1980) Sturmian Theory for Ordinary Differential Equations. New York, Springer-Verlag
Rouchon P. (1992) The Jacobi equation, Riemannian curvature and the motion of a perfect incompressible fluid. European J. Mech. B Fluids 11(3): 317–336
Shnirelman A. (1994). Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4, 586–620
Spivak M. (1999) A comprehensive introduction to differential geometry. Volume 1. Houston, TX Publish or Perish
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Kupiainen
Much of this work was completed while the author was a Lecturer at the University of Pennsylvania. The author is grateful for their hospitality.
Rights and permissions
About this article
Cite this article
Preston, S.C. On the Volumorphism Group, the First Conjugate Point is Always the Hardest. Commun. Math. Phys. 267, 493–513 (2006). https://doi.org/10.1007/s00220-006-0070-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0070-9