2,003 research outputs found
An Inverse Problem from Sub-Riemannian Geometry
The geodesics for a sub-Riemannian metric on a three-dimensional contact
manifold form a 1-parameter family of curves along each contact direction.
However, a collection of such contact curves on , locally equivalent to the
solutions of a fourth-order ODE, are the geodesics of a sub-Riemannian metric
only if a sequence of invariants vanish. The first of these, which was earlier
identified by Fels, determines if the differential equation is variational. The
next two determine if there is a well-defined metric on and if the given
paths are its geodesics.Comment: 13 page
Stark hypersurfaces in complex projective space
Stark hypersurfaces are a special class of austere hypersurface in where the shape operator is compatible with the -structure. In this
paper, the possible shape operators for stark hypersurfaces are completely
determined, and stark hypersurfaces in are constructed as
integrals of a Frobenius exterior differential system.Comment: 12 pages; submitted to Contemporary Mathematic
Knot types, homotopies and stability of closed elastic rods
The energy minimization problem associated to uniform, isotropic, linearly
elastic rods leads to a geometric variational problem for the rod centerline,
whose solutions include closed, knotted curves. We give a complete description
of the space of closed and quasiperiodic solutions. The quasiperiodic curves
are parametrized by a two-dimensional disc. The closed curves arise as a
countable collection of one-parameter families, connecting the m-fold covered
circle to the n-fold covered circle for any m,n relatively prime. Each family
contains exactly one self-intersecting curve, one elastic curve, and one closed
curve of constant torsion. Two torus knot types are represented in each family,
and all torus knots are represented by elastic rod centerlines.Comment: 17 pages, LaTeX, epsfig; to appear in Proc. London Math. So
Austere Submanifolds in Complex Projective Space
For an arbitrary submanifold we determine
conditions under which it is austere, i.e., the normal bundle of is special
Lagrangian with respect to Stenzel's Ricci-flat K\"ahler metric on
. We also classify austere surfaces in .Comment: 13 pages; final version, to appear in Communications in Analysis and
Geometr
Parametric Backlund transformations I: Phenomenology
We begin an exploration of parametric Backlund transformations for hyperbolic
Monge-Ampere systems. We compute invariants for such transformations and
explore the behavior of four examples regarding their invariants, symmetries,
and conservation laws. We prove some preliminary results and indicate
directions for further research.Comment: 32 page
The structure Jacobi operator for hypersurfaces in CP^2 and CH^2
Using the methods of moving frames, we study real hypersurfaces in complex
projective space CP^2 and complex hyperbolic space CH^2 whose structure Jacobi
operator has various special properties. Our results complement work of several
other authors who worked on such hypersurfaces in CP^n and CH^n for n>2.Comment: 13 page
The *-Ricci tensor for hypersurfaces in CP^n and CH^n
We update and refine the work of T. Hamada concerning *-Einstein
hypersurfaces in complex space forms CP^n and CH^n. We also address existence
questions using the methods of moving frames and exterior differential systems.Comment: 26 page
Hopf Hypersurfaces of Small Hopf Principal Curvature in CH^2
Using the methods of moving frames and exterior differential systems, we show
that there exist Hopf hypersurfaces in complex hyperbolic space CH^2 with any
specified value of the Hopf principal curvature less than or equal to the
corresponding value for the horosphere. We give a construction for all such
hypersurfaces in terms of Weierstrass-type data, and also obtain a
classification of pseudo-Einstein hypersurfaces in CH^2.Comment: 14 pages; to appear in Geometriae Dedicat
Hypersurfaces in and with two distinct principal curvatures
It is known that hypersurfaces in or for which the number
of distinct principal curvatures satisfied must belong to a standard
list of Hopf hypersurfaces with constant principal curvatures, provided that . In this paper, we construct a 2-parameter family of non-Hopf
hypersurfaces in and with and show that every non-Hopf
hypersurface with is locally of this form.Comment: 15 pages; revised version to appear in Glasgow Mathematical Journa
Backlund Transformations and Darboux Integrability for Nonlinear Wave Equations
We prove that second-order hyperbolic Monge-Ampere equations for one function
of two variables are connected to the wave equation by a Backlund
transformation if and only if they are integrable by the method of Darboux at
second order. One direction of proof, proving Darboux integrability, follows
the implications of the wave equation for the invariants of the G-structure
associated to the Backlund transformation. The other direction constructs
Backlund transformations for Darboux integrable equations as solutions of an
involutive exterior differential system. Explicit transformations are given for
several equations on the Goursat-Vessiot list of Darboux-integrable equations.Comment: 48 pages; submitted, in revised form, to Asian J. Mat
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