61 research outputs found
Singular riemannian foliations with sections, transnormal maps and basic forms
A singular riemannian foliation F on a complete riemannian manifold M is said
to admit sections if each regular point of M is contained in a complete totally
geodesic immersed submanifold (a section) that meets every leaf of F
orthogonally and whose dimension is the codimension of the regular leaves of F.
We prove that the algebra of basic forms of M relative to F is isomorphic to
the algebra of those differential forms on a section that are invariant under
the generalized Weyl pseudogroup of this section. This extends a result of
Michor for polar actions. It follows from this result that the algebra of basic
function is finitely generated if the sections are compact.
We also prove that the leaves of F coincide with the level sets of a
transnormal map (generalization of isoparametric map) if M is simply connected,
the sections are flat and the leaves of F are compact. This result extends
previous results due to Carter and West, Terng, and Heintze, Liu and Olmos.Comment: Preprint IME-USP; The final publication is available at
springerlink.com http://www.springerlink.com/content/q48682633730t831
Boundary Conditions in Stepwise Sine-Gordon Equation and Multi-Soliton Solutions
We study the stepwise sine-Gordon equation, in which the system parameter is
different for positive and negative values of the scalar field. By applying
appropriate boundary conditions, we derive relations between the soliton
velocities before and after collisions. We investigate the possibility of
formation of heavy soliton pairs from light ones and vise versa. The concept of
soliton gun is introduced for the first time; a light pair is produced moving
with high velocity, after the annihilation of a bound, heavy pair. We also
apply boundary conditions to static, periodic and quasi-periodic solutions.Comment: 14 pages, 8 figure
Reduced dynamics of Ward solitons
The moduli space of static finite energy solutions to Ward's integrable
chiral model is the space of based rational maps from \CP^1 to itself
with degree . The Lagrangian of Ward's model gives rise to a K\"ahler metric
and a magnetic vector potential on this space. However, the magnetic field
strength vanishes, and the approximate non--relativistic solutions to Ward's
model correspond to a geodesic motion on . These solutions can be compared
with exact solutions which describe non--scattering or scattering solitons.Comment: Final version, to appear in Nonlinearit
The geometric sense of R. Sasaki connection
For the Riemannian manifold two special connections on the sum of the
tangent bundle and the trivial one-dimensional bundle are constructed.
These connections are flat if and only if the space has a constant
sectional curvature . The geometric explanation of this property is
given. This construction gives a coordinate free many-dimensional
generalization of the connection from the paper: R. Sasaki 1979 Soliton
equations and pseudospherical surfaces, Nuclear Phys., {\bf 154 B}, pp.
343-357. It is shown that these connections are in close relation with the
imbedding of into Euclidean or pseudoeuclidean -dimension
spaces.Comment: 7 pages, the key reference to the paper of Min-Oo is included in the
second versio
Darboux transformation for the modified Veselov-Novikov equation
A Darboux transformation is constructed for the modified Veselov-Novikov
equation.Comment: Latex file,8 pages, 0 figure
Symplectically-invariant soliton equations from non-stretching geometric curve flows
A moving frame formulation of geometric non-stretching flows of curves in the
Riemannian symmetric spaces and is
used to derive two bi-Hamiltonian hierarchies of symplectically-invariant
soliton equations. As main results, multi-component versions of the sine-Gordon
(SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting
invariance are obtained along with their bi-Hamiltonian
integrability structure consisting of a shared hierarchy of symmetries and
conservation laws generated by a hereditary recursion operator. The
corresponding geometric curve flows in and
are shown to be described by a non-stretching wave map and a
mKdV analog of a non-stretching Schr\"odinger map.Comment: 39 pages; remarks added on algebraic aspects of the moving frame used
in the constructio
A geometric interpretation of the spectral parameter for surfaces of constant mean curvature
Considering the kinematics of the moving frame associated with a constant
mean curvature surface immersed in S^3 we derive a linear problem with the
spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral
parameter is related to the radius R of the sphere S^3. The application of the
Sym formula to this linear problem yields constant mean curvature surfaces in
E^3. Independently, we show that the Sym formula itself can be derived by an
appropriate limiting process R -> infinity.Comment: 12 page
Multidimensional Toda type systems
On the base of Lie algebraic and differential geometry methods, a wide class
of multidimensional nonlinear systems is obtained, and the integration scheme
for such equations is proposed.Comment: 29 pages, LaTeX fil
Polar foliations and isoparametric maps
A singular Riemannian foliation on a complete Riemannian manifold is
called a polar foliation if, for each regular point , there is an immersed
submanifold , called section, that passes through and that meets
all the leaves and always perpendicularly. A typical example of a polar
foliation is the partition of into the orbits of a polar action, i.e., an
isometric action with sections. In this work we prove that the leaves of
coincide with the level sets of a smooth map if is simply
connected. In particular, we have that the orbits of a polar action on a simply
connected space are level sets of an isoparametric map. This result extends
previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter
and West, and Terng.Comment: 9 pages; The final publication is available at springerlink.com
http://www.springerlink.com/content/c72g4q5350g513n1
Hamiltonian evolutions of twisted gons in \RP^n
In this paper we describe a well-chosen discrete moving frame and their
associated invariants along projective polygons in \RP^n, and we use them to
write explicit general expressions for invariant evolutions of projective
-gons. We then use a reduction process inspired by a discrete
Drinfeld-Sokolov reduction to obtain a natural Hamiltonian structure on the
space of projective invariants, and we establish a close relationship between
the projective -gon evolutions and the Hamiltonian evolutions on the
invariants of the flow. We prove that {any} Hamiltonian evolution is induced on
invariants by an evolution of -gons - what we call a projective realization
- and we give the direct connection. Finally, in the planar case we provide
completely integrable evolutions (the Boussinesq lattice related to the lattice
-algebra), their projective realizations and their Hamiltonian pencil. We
generalize both structures to -dimensions and we prove that they are
Poisson. We define explicitly the -dimensional generalization of the planar
evolution (the discretization of the -algebra) and prove that it is
completely integrable, providing also its projective realization
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