54 research outputs found
Symplectic Applicability of Lagrangian Surfaces
We develop an approach to affine symplectic invariant geometry of Lagrangian
surfaces by the method of moving frames. The fundamental invariants of elliptic
Lagrangian immersions in affine symplectic four-space are derived together with
their integrability equations. The invariant setup is applied to discuss the
question of symplectic applicability for elliptic Lagrangian immersions.
Explicit examples are considered
Quantization of the conformal arclength functional on space curves
By a conformal string in Euclidean space is meant a closed critical curve
with non-constant conformal curvatures of the conformal arclength functional.
We prove that (1) the set of conformal classes of conformal strings is in 1-1
correspondence with the rational points of the complex domain and (2) any conformal class has a model conformal
string, called symmetrical configuration, which is determined by three
phenomenological invariants: the order of its symmetry group and its linking
numbers with the two conformal circles representing the rotational axes of the
symmetry group. This amounts to the quantization of closed trajectories of the
contact dynamical system associated to the conformal arclength functional via
Griffiths' formalism of the calculus of variations.Comment: 24 pages, 6 figures. v2: final version; minor changes in the
exposition; references update
Tableaux over Lie algebras, integrable systems, and classical surface theory
Starting from suitable tableaux over finite dimensional Lie algebras, we
provide a scheme for producing involutive linear Pfaffian systems related to
various classes of submanifolds in homogeneous spaces which constitute
integrable systems. These include isothermic surfaces, Willmore surfaces, and
other classical soliton surfaces. Completely integrable equations such as the
G/G_0-system of Terng and the curved flat system of Ferus-Pedit may be obtained
as special cases of this construction. Some classes of surfaces in projective
differential geometry whose Gauss-Codazzi equations are associated with
tableaux over sl(4,R) are discussed.Comment: 16 pages, v3: final version; changes in the expositio
Critical Robertson-Walker universes
The integral of the energy density function of a closed
Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological
constant gives rise to an action functional on the space of scale
functions of RW spacetime metrics. This paper studies closed RW spacetimes
which are critical for this functional, subject to volume-preserving variations
(critical RW spacetimes). A complete classification of critical RW spacetimes
is given and explicit solutions in terms of Weierstrass elliptic functions and
their degenerate forms are computed. The standard energy conditions (weak,
dominant, and strong) as well as the cyclic property of critical RW spacetimes
are discussed.Comment: 18 pages, LaTe
Compact surfaces with no Bonnet mate
This note gives sufficient conditions (isothermic or totally nonisothermic)
for an immersion of a compact surface to have no Bonnet mate.Comment: 7 pages, LaTeX2
Topologically embedded helicoidal pseudospherical cylinders
The class of traveling wave solutions of the sine-Gordon equation is known to be in 1–1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic functions. This solves a problem posed by Popov (2014 Lobachevsky Geometry and Modern Nonlinear Problems (Berlin: Springer)). As an application, countably many continuous families of topologically embedded pseudospherical helicoids are constructed. A (singular) pseudospherical helicoid is proved to be either a dense subset of a region bounded by two coaxial cylinders, a topologically immersed cylinder with helical self-intersections, or a topologically embedded cylinder with helical singularities, called for short a pseudospherical twisted column. Pseudospherical twisted columns are characterized by four phenomenological invariants: the helicity η ∈ Z2, the parity ε ∈ Z2, the wave number n ∈ N, and the aspect ratio d > 0, up to translations along the screw axis. A systematic procedure for explicitly determining all pseudospherical twisted columns from the invariants is provided
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