A split special Lagrangian calibration associated with frame vorticity

Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO(M) --> M of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso_o (M) \cong SO(M): A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.Comment: We changed the title (the nomenclature "vorticity" is more appropriate for the concept we define in the article than "helicity"). We corrected some typos and an assertion in the paragraph after display number 20. To be published in Advances in Calculus of Variation

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let M and N be two connected smooth manifolds, where M is compact and oriented and N is Riemannian. Let E be the Fréchet manifold of all embeddings of M in N, endowed with the canonical weak Riemannian metric. Let ∼ be the equivalence relation on E defined by f ∼ g if and only if f = g ◦ φ for some orientation preserving diffeomorphism φ of M. The Fréchet manifold S = E/∼ of equivalence classes, which may be thought of as the set of submanifolds of N diffeomorphic to M and is called the nonlinear Grassmannian (or Chow manifold) of N of type M, inherits from E a weak Riemannian structure. Its geodesics, although they are not good from the metric point of view, are distinguished curves and have proved to be useful in various situations. We consider the following particular case: N is a compact irreducible symmetric space and M is a reflective submanifold of N (that is, a connected component of the set of fixed points of an involutive isometry of N). Let C be the set of submanifolds of N which are congruent to M. We prove that the natural inclusion of C in S is totally geodesic.submittedVersionFil: Salvai, Marcos Luis. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Matemática Pur

THE MAGNETIC FLOW ON THE MANIFOLD OF ORIENTED GEODESICS OF A THREE DIMENSIONAL SPACE FORM

Let M be the three dimensional complete simply connected manifold of constant sectional curvature 0,10,1 or −1−1. Let L be the manifold of all (unparametrized) complete oriented geodesics of M, endowed with its canonical pseudo-Riemannian metric of signature (2,2)(2,2) and Kähler structure J. A smooth curve in L determines a ruled surface in M. We characterize the ruled surfaces of MM associated with the magnetic geodesics of LL, that is, those curves σσ in LL satisfying ∇σ˙σ˙=Jσ˙∇σ˙σ˙=Jσ˙. More precisely: a time-like (space-like) magnetic geodesic determines the ruled surface in M given by the binormal vector field along a helix with positive (negative) torsion. Null magnetic geodesics describe cones, cylinders or, in the hyperbolic case, also cones with vertices at infinity. This provides a relationship between the geometries of L and M.Fil: Godoy, Yamile Alejandra. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Córdoba. Centro de Investigación y Estudios de Matemática de Córdoba(p); ArgentinaFil: Salvai, Marcos Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Córdoba. Centro de Investigación y Estudios de Matemática de Córdoba(p); Argentin

Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form

Let G be the manifold of all (unparametrized) oriented lines of R3. We study the controllability of the control system in G given by the condition that a curve in G describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed α. Actually, we pose the analogous more general problem by means of a control system on the manifold Gκ of all the oriented complete geodesics of the three dimensional space form of curvature κ: R3 for κ= 0 , S3 for κ= 1 and hyperbolic 3-space for κ= − 1. We obtain that the system is controllable if and only if α2≠ κ. In the spherical case with α= ± 1 , an admissible curve remains in the set of fibers of a fixed Hopf fibration of S3. We also address and solve a sort of Kendall’s (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joining two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.Fil: Anarella, Mateo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Katholikie Universiteit Leuven; BélgicaFil: Salvai, Marcos Luis. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin