Transformations and non-degenerate maps

We shall prove the equivalences of a non-degenerate circle-preserving map and a Möbius transformation in\(\hat {\mathbb{R}}^n \), of a non-degenerate geodesic-preserving map and an isometry in ?ⁿ, of a non-degenerate line-preserving map and an affine transformation in ?ⁿ. That a map is non-degenerate means that the image of the whole space under the map is not a circle, or geodesic or line respectively. These results hold without either injective or surjective, or even continuous assumptions, which are new and of a fundamental nature in geometry.     

Connectedness properties of polynomial maps between affine spaces

Let f:A^mAⁿ be a polynomial map between affine spaces. We give some sufficient conditions for the connectedness of the difference kernel of f and relate this to the Jacobian Conjecture.     

Affine minimal hypersurfaces of rotation

We give a complete list of affine minimal surfaces inA ³ with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA ⁿ ,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.     

Uniqueness of Subelliptic Harmonic Maps

Let R^m be an open set, Nⁿ a Riemannian manifold, X a collection of vector fields on , and f a smooth map from into Nⁿ. We call f a subelliptic harmonic map if it is a critical point of the energy functional with respect to X. In this paper, we calculate the first and the second variations of the energy functional, and use them to prove the partial uniqueness of a subelliptic harmonic map under the condition that Nⁿ has the non-positive curvature. Then, we utilize the maximum principle for subelliptic PDEs to verify the global uniqueness of a subelliptic harmonic map under some other conditions.     

Locally symmetric complex affine hypersurfaces

In this paper we study non-degenerate locally symmetric complex affine hypersurfaces Mⁿ of the complex affine space, i.e. hypersurfaces satisfying R=0, where is the affine connection induced on Mⁿ by the complex affine structure on the complex affine space, and R is the curvature tensor of . We classify the non-degenerate locally symmetric hypersurfaces Mⁿ, n > 2, and the minimal non-degenerate locally symmetric hypersurfaces Mⁿ, n > 1.Aspirant N.F.W.O. (Belgium)     

Four-Dimensional Simply Connected Symplectic Symmetric Spaces

A symplectic is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic which are simply connected. This classification reveals a remarkable class of affine symmetric spaces with a non-Abelian solvable transvection group. The underlying manifold M of each element (M, ) belonging to this class is diffeomorphic to Rⁿwith the property that every tensor field on M invariant by the transvection group is constant; in particular, is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on Rⁿwhich are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.     

Topologische Affine Ebenen Mit Nichtstetigem Parallelismus

In this paper, we present a general method of constructing topological affine planes having non-continuous parallelism. We prove that a topological affine plane E with point set L ^k ×L ^k , and with a special K-algebraic slope has a topological affine subplane with non-continuous parallelism (Satz 4.6). Here, K is a real-closed subfield of a real-closed field L. The crucial tools needed to make our method work are the notion of a slope and the notion of K-algebraicity, a concept which is introduced and intensively studied here. As an application of our general method, we obtain in Section 5 affine Salzmann planes with lines being bent countably infinitely often admitting a subplane with non-continuous parallelism. This provides a negative answer to a question posed by H. Salzmann [13, p. 52].     

Affine normal curvature of hypersurfaces from the point of view of singularity theory

We consider smoothly embedded hypersurfaces under the action of the special affine group . We construct a differential invariant, called affine normal curvature, which assigns to a point and a tangent direction a number. We prove some of its nice properties which connect it with affine principal directions, affine umbilics, and affine mean curvature.      

Konstruktion topologischer affiner Ebenen mit nichtstetigem Parallelismus durch Knicken von Geraden

We prove the existence of a class of topological affine planes having non-continuous parallelism by using [2, Satz 5.2]. For this, we introduce a new method of constructing affine Salzmann-planes with a monotonically increasing slope (see 2.1) by bending lines on two special curves, which are not necessary lines. Furthermore, the limit inferior of a sequence of topological planes with fixed point space is defined. As application of our new method, we construct a sequence of affine Salzmann-planes such that the limit inferior of this sequence is again an affine Salzmann-plane and fulfils the assumptions of [2, Satz 5.2]. Applying this theorem repeatedly, we get a sequence of non-isomorphic topological affine subplanes with non-continuous parallelism.     

A finite algorithm for finding the projection of a point onto the canonical simplex of ∝ n

An algorithm of successive location of the solution is developed for the problem of finding the projection of a point onto the canonical simplex in the Euclidean space ⁿ . This algorithm converges in a finite number of steps. Each iteration consists in finding the projection of a point onto an affine subspace and requires only explicit and very simple computations.     

A characterization of special laguerre planes and extended dual affine planes

A special Laguerre plane is a nondegenerate transversal 3-design such that the residue of each point is a dual affine plane. A special Laguerre plane is equivalent to an optimal code with three information digits and maximal length. An extended dual affine plane is an incidence structure (whose objects will be called points and blocks) such that the residue of each point is a dual affine plane, and each pair of points is in at least one block. Finite extended dual affine planes exist only of order 2, 4, and (dubiously) 10. We show that any finite incidence structure having the residue of each point a dual affine plane either is a transversal 3-design or has a block through each pair of points. Hence theorem: If a finite nondegenerate connected incidence structure has the residue of each point a dual affine plane, then is either an extended dual affine plane or a special Laguerre plane. This research was partially supported by NSF Grant MCS-8102361.     

A theorem about affine maps in Banach spaces

It is shown that a weakly compact convex set in a strictly convex space cannot be decomposed into two non-empty, disjoint sets which are similar to each other in the sense that one is the image of the other under a non-expansive affine map.     

On Generalized Affine Rotation Surfaces

From group-theoretical point of view, we discuss affine rotation surfaces in R³ and projective rotation surfaces in RP³. These pave a way toward generalized affine rotation surfaces in R³. We will follow closely the modern approach introduced by Nomizu in the study affine differential geometry [N]. In this paper, we have the following results on generalized affine rotation surfaces:

1. Many nice properties showing duality between x² + y² = g²(z) and x² ? y² = g²(z).
2. In this set, any surface with zero Pick invariant is a quadratic surface.
3. Excluding the Caley surface z = xy ? x³/3 and the surface z = xy + log x, any affine unimodular homogeneous surface belongs to this set.
4. In this set, the following surfaces are characterized by some affine invariants: $${\matrix{x^{2}+\epsilon\ y^{2}=z \cr \qquad \qquad \qquad \qquad \qquad \qquad \qquad y^{2}=z(x+\epsilon\ (z+a)^{2/3}(z+6a)),\cr \qquad \qquad \qquad \ \ \ y^{2}=z(x+\epsilon\ z^{3}),\cr \qquad \qquad \ \ \ \ \ \ y^{2}=x+\epsilon\ z^{2/3},\cr \qquad \qquad \ \ \ \ \ \ \ y^{2}=x+\epsilon\ z^{-2/3},\cr \qquad \qquad \ \ \ y^{2}=x+\epsilon\ {\rm log}z,\cr}} $$ where ? represents 1 or ?1 in this paper.

     

On the compactifications of contractible affine threefolds and the Zariski Cancellation Problem

In this article, we consider the compactifications of some kinds of contractible smooth affine threefolds and the characterization of the affine 3-space GIF1³ keeping the 3-dimensional Zariski Cancellation Problem in mind. We classify the compactifications of contractible smooth affine threefolds with a certain condition concerning the numerical property of boundary divisors with respect to compactifications and, then, we show that if an affine threefold X satisfies X×GIF1¹GIF1⁴ and this numerical condition, then X is isomorphic to the affine 3-space GIF1³.Mathematics Subject Classification (2000):14R10, 14E30     

Affine surfaces with constant affine curvatures

In this paper, we study affine non-degenerate Blaschke immersions from a surface M in ³. We will assume that M has constant affine curvature and constant affine mean curvature, i.e. both the determinant and the trace of the shape operator are constant. Clearly, affine spheres satisfy both these conditions. In this paper, we completely classify the affine surfaces with constant affine curvature and constant affine mean curvature, which are not affine spheres.Research Assistant of the National Fund for Scientific Research (Belgium).     

Real and Rational Forms of Certain O<Subscript>2</Subscript>(ℂ)-actions,and a Solution to the Weak Complexification Problem

The main result of this paper is that there is a non-linearizable real algebraic action of the circle S¹ on ⁴, an action which becomes linearizable over . This solves the Weak Complexification Problem. We also show that for any field k of characteristic zero, there are non-linearizable algebraic actions of the group O₂(k) on four-dimensional affine k-space, and if k contains a square root of 3, then this action restricts to a non-linearizable action of the symmetric group S₃ on four-dimensional affine k-space.     

Vector Hulls of Affine Spaces and Affine Bundles

Every affine space A can be canonically immersed as a hyperplane into a vector space  , which is called the vector hull of A. This immersion satisfies a universal property for affine functions defined on A. In the same way, every affine map between affine spaces has a linear prolongation to their vector hulls. Though not much known, this construction is greatly clarifying, both for affine geometry and for its applications. The goal of this paper is to perform a thorough study of the vector hull functor and to describe its counterpart in the framework of affine bundles. With this respect, it is shown that the vector hull of some interesting affine bundles, and more specifically some jet bundles, can be identified with certain vector bundles.      

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.     

Morse theory for some lower-C 2 functions in finite dimension

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C ² functions. For this purpose, we first recall some properties of the class of lower-C ² functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a Morse index for a lower-C ² functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C ² functions.