ALMOST CONTACT 3-MANIFOLDS AND RICCI SOLITONS

A Kenmotsu 3-manifold M admitting a Ricci soliton (g, w) with a transversal potential vector field w (orthogonal to the Reeb vector field) is of constant sectional curvature -1. A cosymplectic 3-manifold admitting a Ricci soliton with the Reeb potential vector field or a transversal vector field is of constant sectional curvature 0.

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 Cartan-Norden theorem for affine Kähler immersions

In [N-Pi-Po] the notion of affine Kähler immersion for complex manifolds has been introduced: if Mn is an n-dimensional complex manifold and f: Mn -→ Cn+1 is a holomorphic immersion together with an anti-holomorphic transversal vector field ζ, we can induce a connection ▽ on Mn which is Kähler-like, that is, its curvature tensor R satisfies R(Z, W) = 0 as long as Z, W are (1, 0) complex vector fields on M.