The Lipschitz-Killing curvature for an equiaffine immersion and theorems of Gauss-Bonnet type and Chern-Lashof type

In this paper, we define an equiaffine immersion of general codimension and the Lipschitz-Killing curvature for the immersion. Furthermore, we prove theorems of Gauss-Bonnet type and Chern-Lashof type for the immersion.     

On the total absolute curvature of an immersed sphere

A recent result of Tobias Ekholm [T. Ekholm, Regular homotopy and total curvature II: Sphere immersions into 3-space, Alg. Geom. Topol. 6 (2006) 493-513] shows that for every ?>0 it is possible to construct a sphere eversion such that the total absolute curvature of the immersed spheres are always less than 8π+?. It is an open question whether this is the best possible. The paper contains results relating to this conjecture. As an interesting consequence of these methods it is shown that if during an eversion the total absolute curvature does not exceed 12π then a certain topological event must take place, namely the immersion must become non-simple at some point. An immersion f in general position is simple if for any irreducible self-intersection curve of f in 3-space, its two pre-image curves in the sphere are disjoint.     

Minimal total absolute curvature for immersions

Immersions or maps of closed manifolds in Euclidean space, of minimal absolute total curvature are called tight in this paper. (They were called convex in [25].) After the definition in Chapter 1, many examples in Chapter 2, and some special topics in Chapter 3, we prove in Chapter 4 that topological tight immersions ofn-spheres are only of the expected type, namely embeddings onto the boundary of a convexn+1-dimensional body. This generalises a theorem of Chern and Lashof in the smooth case. In Chapter 5 we show that many manifolds exist that have no tight smooth immersion in any Euclidean space.This research was partially supported by National Science Foundation grant GP-7952X1.     

On the total absolute curvature of closed curves in spheres

The total absolute curvature of a closed curve in a Euclidean space is always greater or equal to 2 (Fenchel's inequality,1929, [3], [1], [13]); especially for a knotted curve it is always greater than 4 (Fary-Milnor's inequality, 1949, [2], [7], [5], [4]).For the total absolute curvature of closed curves in spheres no such lower bounds exist because there are closed geodesies. Here we derive similar bounds which depend on the length of the curve resp.the area of surfaces of disk-type bounded by the curve.In order to prove these inequalities we start from the computation of the total absolute curvature as mean value of the number of critical points of certain level functions ([11],[12]); we use some topological considerations and Poincaré's integralgeometric formula for the computation of length resp. area.     

On bounds for total absolute curvature of surfaces in hyperbolic 3-space

We construct examples of surfaces in hyperbolic space which do not satisfy the Chern–Lashof inequality (which holds for immersed surfaces in Euclidean space). To cite this article: R. Langevin, G. Solanes, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Rigidity in the class of orientable compact surfaces of minimal total absolute curvature

Consider an orientable compact surface in three-dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic curves, we show that any other isometric surface differs by at most a Euclidean motion.     

Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero

Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of 2n‐dimensional nondegenerate hypersurfaces ruled by n‐planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with respect to which the ruling is Lagrangian.     

The quadratic slice theorem and the equiaffine tube theorem for equiaffine Dupin hypersurfaces

The main purpose of this paper is to investigate the quadraticity of slices (i.e., leaves of curvature foliations) of a non-degenerate equiaffine Dupin hypersurface, where an equiaffine Dupin hypersurface is the notion defined as the equiaffine geometrical version of a (not necessarily complete) Dupin hypersurface.     

Total absolute curvature and tightness of noncompact manifolds

In the first part we prove an extension of the Chern-Lashof inequality for noncompact immersed manifolds with finitely many ends. For this we give a lower bound of the total absolute curvature in terms of topological invariants of the manifold. In the second part we discuss tightness properties for such immersions. Finally, we give an upper bound for the substantial codimension.

     

Complex polynomial vector fields having an orbit with finite total curvature

In this paper we address the following questions: (i) Let \({C \subset \mathbb{C}^2}\) be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is C contained in an algebraic curve? (ii) What can be said of a polynomial vector field which has a finitely curved transcendent orbit? We give a positive answer to (i) under some non-degeneracy conditions on the singularities of the projective foliation induced by the vector field. For vector fields with a slightly more general class of singularities we prove a classification result that captures rational pull-backs of Poincaré-Dulac normal forms.     

The automorphism group of the augmented equiaffine group

The automorphism group of the group generated by all the affine reflections in a Desarguesian plane is isomorphic to the full collineation group of the plane.The author wishes to express his thanks to J. Wilker of the University of Toronto for suggesting a simplification and to him as well as C. Fisher of the University of Regina for their encouragement.