The submanifold geometries associated to Grassmannian systems (original) (raw)

Grassmannians via projection operators and some of their special submanifolds

We study submanifolds of a general Grassmann manifold which are of 1-type in a suitably defined Euclidean space of F−Hermitian matrices (such a submanifold is minimal in some hypersphere of that space and, apart from a translation, the immersion is built using eigenfunctions from a single eigenspace of the Laplacian). We show that a minimal 1-type hypersurface of a Grassmannian is mass-symmetric and has the type-eigenvalue which is a specific number in each dimension. We derive some conditions on the mean curvature of a 1-type hypersurface of a Grassmannian and show that such a hypersurface with a nonzero constant mean curvature has at most three constant principal curvatures. At the end, we give some estimates of the first nonzero eigenvalue of the Laplacian on a compact submanifold of a Grassmannian.

Grassmannian structures on manifolds

Bulletin of the Belgian Mathematical Society - Simon Stevin, 1994

Grassmannian structures on manifolds are introduced as subbundles of the second order framebundle. The structure group is the isotropy group of a Grassmannian. It is shown that such a structure is the prolongation of a subbundle of the first order framebundle. A canonical normal connection is constructed from a Cartan connection on the bundle and a Grassmannian curvature tensor for the structure is derived.

Geometric transformations and soliton equations

We give a survey of the following six closely related topics: (i) a general method for constructing a soliton hierarchy from a splitting of a loop algebra into positive and negative subalgebras, together with a sequence of commuting positive elements, (ii) a method---based on (i)---for constructing soliton hierarchies from a symmetric space, (iii) the dressing action of the negative loop subgroup on the space of solutions of the related soliton equation, (iv) classical B\"acklund, Christoffel, Lie, and Ribaucour transformations for surfaces in three-space and their relation to dressing actions, (v) methods for constructing a Lax pair for the Gauss-Codazzi Equation of certain submanifolds that admit Lie transforms, (vi) how soliton theory can be used to generalize classical soliton surfaces to submanifolds of higher dimension and co-dimension.

Submanifolds with . . . CURVATURE VECTOR IN RIEMANNIAN AND INDEFINITE SPACE FORMS

2013

A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important since they are critical points of some natural functionals. In this paper, we survey some classical and recent results on submanifolds with parallel mean curvature vector. Special attention is paid to the classification of space-like and Lorentz surfaces with parallel mean curvature vector in Riemannian and indefinite space forms.

Schrodinger flows on Grassmannians

1999

The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schrodinger equation (MNLS). We construct via gauge transformations

Differential Geometry Parametrized curves in Lagrange Grassmannians

2007

Curves in Lagrange Grassmannians naturally appear when one studies Jacobi equations for extremals, associated with geometric structures on manifolds. We fix integers di and consider curves Λ(t) for which at each t the derivatives of order ≤ i of all curves of vectors `(t) ∈ Λ(t) span a subspace of dimension di. We will describe the construction of a complete system of symplectic invariants for such parametrized curves, satisfying a certain genericity assumption, and give applications to geometric structures, including sub-Riemannian and sub-Finslerian

Parametrized curves in Lagrange Grassmannians

Comptes Rendus Mathematique, 2007

Curves in Lagrange Grassmannians naturally appear when one studies Jacobi equations for extremals, associated with geometric structures on manifolds. We fix integers di and consider curves Λ(t) for which at each t the derivatives of order ≤ i of all curves of vectors (t) ∈ Λ(t) span a subspace of dimension di. We will describe the construction of a complete system of symplectic invariants for such parametrized curves, satisfying a certain genericity assumption, and give applications to geometric structures, including sub-Riemannian and sub-Finslerian structures.

Geometry of Submanifolds Derived from Spin-Valued Spectral Problems

2003

We present recent results motivated by Sym's theory of soliton surfaces. Quite general assumptions about the structure of the spectral problem can lead to some specific classes of surfaces. In some cases (including pseudospherical surfaces), this approach is coordinate-independent, which seems a surprising novelty. The Darboux-Bäcklund transformation is formulated in terms of Clifford numbers, which greatly simplifies constructing explicit solutions. Cumbersome computations in matrix representations are replaced with rotations represented by elements of an appropriate Spin group. Finally, the spectral problem and the spectral parameter are derived purely geometrically in the case of isometric immersions of constant-curvature spaces in spheres and Euclidean spaces.

Jacobi equations and Comparison Theorems for corank 1 sub-Riemannian structures with symmetries

Journal of Geometry and Physics, 2011

The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical bundle of moving frames and the complete system of symplectic invariants, called curvature maps, for parametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural equation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the normal form for the Jacobi equation along this extremal and the curvature maps can be seen as the "coefficients"of this normal form. In the case of a Riemannian metric there is only one curvature map and it is naturally related to the Riemannian sectional curvature. In the present paper we study the curvature maps for a sub-Riemannian structure on a corank 1 distribution having an additional transversal infinitesimal symmetry. After the factorization by the integral foliation of this symmetry, such sub-Riemannian structure can be reduced to a Riemannian manifold equipped with a closed 2-form (a magnetic field). We obtain explicit expressions for the curvature maps of the original sub-Riemannian structure in terms of the curvature tensor of this Riemannian manifold and the magnetic field. We also estimate the number of conjugate points along the sub-Riemannian extremals in terms of the bounds for the curvature tensor of this Riemannian manifold and the magnetic field in the case of an uniform magnetic field. The language developed for the calculation of the curvature maps can be applied to more general sub-Riemannian structures with symmetries, including sub-Riemmannian structures appearing naturally in Yang-Mills fields.

Symplectic space forms and submanifolds

Banach Center Publications, 2016

This is a report on some ongoing work with Michel Cahen and Thibaut Grouy: the aim of our project is to define Radon-type transforms in symplectic geometry. The chosen framework is that of symplectic symmetric spaces whose canonical connection is of Ricci-type. These can be considered as symplectic analogues of the space forms, i.e. the spaces of constant sectional curvature, in Riemannian geometry. I shall focus here on their submanifold theory and I shall recall constructions of models of such spaces. Introduction. Radon-type transforms were constructed from the beginning of the 20th century. Funk first observed in 1913 that a symmetric function on the sphere S 2 can be described from its great circle integrals. Radon showed in 1917 that a smooth function f on the Euclidean space R 3 can be determined by its integrals over the planes in R 3 ; if J(ω, p) is the integral of f over the plane defined by x • ω = p for ω a fixed unit vector and p a fixed constant in R, then f (x) = − 1 8π 2 L x S 2 J(ω, ω • x) dω where L is the Laplacian.