Some Geometry of Affine Immersion of General Co-dimension

After a careful study of some works of servaral authors on affine immersion of co-dimension one [1], co-dimension two [2], co-dimension three [3] and co-dimension four [4], we extend some of thier fundamental equations to affine immersion of genaral co-dimension p. Furthermore, we extend some theorem of Frank Dillen at el in [5] to affine immersion of general co-dimension and obtain the divisibility of the cubic forms by the second fundamental forms.

Centroaffine immersions of codimension two and projective hypersurface theory

Affine differential geometry developed by Blaschke and his school [B] has been reorganized in the last several years as geometry of affine immersions. An immersion f of an n-dimensional manifold M with an affine connection ∇ into an (n + 1)-dimensional manifold Ḿwith an affine connection ∇ is called an affine immersion if there is a transversal vector field ξ such that ∇xf*(Y) = f*(∇xY) + h(X,Y)ξ holds for any vector fields X, Y on Mn. When f: Mn→ Rn+1 is a nondegenerate hypersurface, there is a uniquely determined transversal vector field ξ, called the affine normal field, an essential starting point in classical affine differential geometry. The new point of view allows us to relax the non-degeneracy condition and gives us more freedom in choosing ξ; what this new viewpoint can accomplish in relating affine differential geometry to Riemannian geometry and projective differential geometry can be seen from [NP1], [NP2], [NS] and others. For the definitions and basic formulas on affine immersions, centroaffine immersions, conormal (or dual) maps, projective flatness, etc., the reader is referred to [NP1]. These notions will be generalized to codimension 2 in this paper.

On the geometry of affine Kähler immersions

In this paper we extend the work on affine immersions [N-Pi]-1 to the case of affine immersions between complex manifolds and lay the foundation for the geometry of affine Kähler immersions. The notion of affine Kähler immersion extends that of a holomorphic and isometric immersion between Kähler manifolds and can be contrasted to the notion of holomorphic affine immersion which has been established in the work of Dillen, Vrancken and Verstraelen [D-V-V] and that of Abe [A].