The natural operators lifting vector fields to the bundle of affinors
All natural operators T ↝ T(T ⊗ T*) lifting vector fields X from n-dimensional manifolds M to vector fields B(X) on the bundle of affinors ™ ⊗ T*M are described.
The natural operators
We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor on the vector r-tangent bundle over M. This problem is reflected in the concept of natural operators . For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over and we construct explicitly a basis of this module.
The natural operators transforming affinors to tensor fields of type
The natural transformations
For natural numbers and a complete classification of natural transformations over -manifolds is given, where is the linear -tangent bundle functor.
The nonexistence of a weak solution of Dirichlet's problem for the functional of minimal surface on nonconvex domains
The Non-Existence of Branch Points in Solutions to Certain Classes of Plateau Type Variational Problems.
The Novikov-Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in .
The “octoid” with triple planar symmetry and its point orbits. (Das dreifach plansymmetrische "Oktoid" und seine Punktbahnen.)
The osculating hyperquadrics of the three-component distribution in affine space
The PDE describing constant mean curvature surfaces
We give an expository account of a Weierstrass type representation of the non-zero constant mean curvature surfaces in space and discuss the meaning of the representation from the point of view of partial differential equations.
The pedal point generation of circular curves of order three in the isotropic plane. (Die Fußpunkterzeugung der zirkulären Kurven 3. Ordnung in der isotropen Ebene.)
The pedal surfaces of -congruences with a one-parameter set of ellipses.
The Plateau problem for the prescribed mean curvature equation
The polar curve of a foliation on
We study some properties of the polar curve associated to a singular holomorphic foliation on the complex projective plane . We prove that, for a generic center , the curve is irreducible and its singular points are exactly the singular points of with vanishing linear part. We also obtain upper bounds for the algebraic multiplicities of the singularities of and for its number of radial singularities.
The polar moment of inertia of the projection curve.
The problem of regular isometric embeddings and the Monge-Ampère equation of hyperbolic type
The Projectivization of Conformal Models of Fibrations Determined by the Algebra of Quaternions
Our aim is to study the principal bundles determined by the algebra of quaternions in the projective model. The projectivization of the conformal model of the Hopf fibration is considered as example.
The reconstruction theorem
The rectifying developable and the spherical Darboux image of a space curve
In this paper we study singularities of certain surfaces and curves associated with the family of rectifying planes along space curves. We establish the relationships between singularities of these subjects and geometric invariants of curves which are deeply related to the order of contact with helices.
The regularity of the trace for minimal surfaces