Affine Kähler hypersurfaces satisfying certain conditions on the curvature tensor

The notion of affine Kähler immersions has been recently introduced by Nomizu-Pinkall-Podestà ([N-Pi-Po]). This work is aimed at giving some results towards the classification of non degenerate affine Kähler hypersurfaces with symmetric and parallel Ricci tensor; this problem generalizes the classical results due to Nomizu-Smyth ([N-S]) in the theory of Kählerian hypersurfaces. In a second section we deal with the case of “semisymmetric” affine Kähler immersions, when the curvature tensor R satisfies R · R = 0 and the Ricci tensor is symmetric, providing a complete classification; for affine Kähler curves we prove that the conditions above are actually equivalent to saying that the immersion is isometric for a suitable Kähler metric in C².     

The Fundamental Theorems for Affine Immersionsinto Hyperquadrics and its Applications

 We prove the fundamental theorems for affine immersions into hyperquadrics (including affine spaces) with arbitrary codimension, which are generalizations of those for isometric immersions into space forms. As applications, the fundamental theorems for equiaffine immersions into hyperquadrics with arbitrary codimension are obtained. (Received 10 February 2000)     

Erratum to “Affine surfaces in R4 with planar geodesics with respect to the affine metric”

In this work a mistake in the paper is corrected. There is also a new proof of the main theorem which classifies the non-degenerate affine surfaces in R ⁴ having planar geodesics with respect to the affine metric.     

Parallel surfaces in affine 4- space

We study affine immersions as introduced by Nomizu and Pinkall. We classify those affine immersions of a surface in R⁴ which are degenerate and have vanishing cubic form (i.e. parallel second fundamental form). This completes the classification of parallel surfaces of which the first results were obtained in the beginning of this century by Blaschke and his collaborators.     

Analysis of constrained Willmore surfaces

This paper investigates the regularity of constrained Willmore immersions into ?^m≥3 locally around both “regular” points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion and its first and second derivatives, given in terms of residues computed as circulation integrals. We deduce explicit “point removability” conditions ensuring that the immersion is smooth. Our results apply in particular to Willmore immersions and to parallel mean curvature immersions in any codimension.     

Codimension of immersions with parallel pluri-mean curvature

Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant mean curvature immersions of surfaces to Kähler manifolds of complex dimension m. Examples are the standard embeddings of Kähler symmetric spaces into the Lie algebra of its transvection group. We give a lower bound for the codimension of arbitrary ppmc immersions. In particular we show that M is locally symmetric if the codimension is minimal.     

Codimension two Kähler submanifolds of space forms

In this article we study isometric immersions from Kähler manifolds whose (1, 1) part of the second fundamental form is parallel, theppmc isometric immersions. When the domain is a Riemann surface these immersions are precisely those with parallel mean curvature. P. J. Ryan has classified the Kähler manifolds that admit isometric immersions, as real hypersurfaces, in space forms. We classify the codimension twoppmc isometric immersions into space forms.     

Codimension one or two immersions and submersions of low dimensional manifolds

LetM be a manifold satisfying certain conditions which are weaker than those of E. Thomas^[12], andf:M→N be a map with codimension one or two. We give necessary and sufficient conditions forf to be homotopic to a map with maximal rank. As an application, we completely determine the codimension one or two immersions of Dold manifolds in real projective spaces.     

Semi-parallel surfaces in Euclidean space

Semi-parallel immersions are defined as extrinsic analogue for semi-symmetric spaces and as a direct generalization of parallel immersions. Using results of Backes on Euclidean Jordan triple systems, the totally geodesic immersions are shown to be the only minimal semi-parallel immersions into a Euclidean space. Semi-parallel immersions of surfaces into E^m are studied and a classification of semi-parallel immersions with pointwise planar normal sections of surfaces in E^m is given.Research Assistant of the National Fund of Scientific Research     

A unimodularly invariant theory for immersions into the affine space

We develop a unimodularly invariant theory for immersions with higher codimension into the affine space. Received: 6 September 2001; in final form: 22 November 2001 / Published online: 29 April 2002 RID="*" ID="*" Supported by the Deutsche Forschungsgemeinschaft     

Minimal Lagrangian submanifolds in ℂP 3 and the sinh-Gordon equation

We study 3-dimensional minimal Lagrangian submanifolds of the 3-dimensional complex projective space ?P³ (4) which admit a unit length Killing vector field whose integral curves are geodesics. We show that such Lagrangian submanifolds can be obtained from either horizontal holomorphic curves in ?P³ (4) (or equivalently superminimal immersions of surfaces in S⁴ (1)) or from solutions of the two dimensional sinh-Gordon equation. In the latter case, we explicitly obtain the immersions in terms of elliptic functions in the case that the solutions of the sinh-Gordon equation depend only on one variable.     

Graph immersions with parallel cubic form

We consider non-degenerate graph immersions into affine space ${\mathbb{A}}^{n+1}$ whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs $(J,\gamma )$, where J is an n-dimensional real Jordan algebra and γ is a non-degenerate trace form on J. Every graph immersion with parallel cubic form can be extended to an affine complete symmetric space covering the maximal connected component of zero in the set of quasi-regular elements in the algebra J. It is an improper affine hypersphere if and only if the corresponding Jordan algebra is nilpotent. In this case it is an affine complete, Euclidean complete graph immersion, with a polynomial as globally defining function. We classify all such hyperspheres up to dimension 5. As a special case we describe a connection between Cayley hypersurfaces and polynomial quotient algebras. Our algebraic approach can be used to study also other classes of hypersurfaces with parallel cubic form.     

Existence and nonexistence of skew branes

A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any two distinct points are not parallel. We show that if an oriented closed manifold has a nonzero Euler characteristic c{\chi}, then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than c²/4{\chi^2{/4}}. We give a version of this result for immersed surfaces in dimension 4. We construct examples of skew spheres of arbitrary odd dimensions, generalizing the construction of skew loops in 3-dimensional space due to Ghomi and Solomon (2002). We conclude with two conjectures that are theorems in 1-dimensional case.     

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

Continuous planar functions

This paper deals with continuous planar functions and their associated topological affine and projective planes. These associated (affine and projective) planes are the so-called shift planes and in addition to these, in the case of planar partition functions, the underlying (affine and projective) translation planes. We introduce a method that allows us to combine two continuous planar functions ? → ? into a continuous planar function ?² → ?². We prove various extension and embedding results for the associated affine and projective planes and their collineation groups. Furthermore, we construct topological ovals and various kinds of polarities in the associated topological projective planes.     

Geodesic Webs on a Two-Dimensional Manifold and Euler Equations

We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the web foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar results are obtained for planar d-webs, d>4, provided that additional d−4 second-order invariants vanish.     

Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature

In this paper we study compact submanifolds of Euclidean space with nonnegative isotropic curvature and low codimension. We determine their homology completely in the case of hypersurfaces and for some low codimensional conformally flat immersions.

     

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).     

Immersions with circular normal sections and normal sections of product immersions

We study immersions with normal sections that are circles in the ambient Euclidean space and formulate lemmas concerning normal sections of product immersions. As applications we determine all parallel immersions with planar normal sections and all immersions with planar normal sections and trivial normal connection.Aspirant N.F.W.O.     

Hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics

We prove that the stable holonomies of a proper codimension 1 attractor Λ, for a C^r diffeomorphism f of a surface, are not C^1+θ for θ greater than the Hausdorff dimension of the stable leaves of f intersected with Λ. To prove this result we show that there are no diffeomorphisms of surfaces, with a proper codimension 1 attractor, that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor.