Minimal Centroaffine Immersions of Codimension Two

Abstract

In this paper we consider centroaffine immersions of codimension two. We will calculate the first and also second variational formulas of centroaffine volume integral with moving frames and then give some examples of minimal centroaffine immersions of codimension two. Mathematics Subject Classification (2000). Primary 53A15; Secondary 53C42, 58E30.

Flat centroaffine surfaces with the degenerate second fundamental form and vanishing Pick invariant inR4

“…In [15] we obtained that C k ij is symmetric for i and j and C ijk is symmetric for i, j and k. Furthermore…”

“…x is called a nondegenerate centroaffine submanifold if g is nondegenerate and x is definite or indefinite if g is definite or indefinite, respectively. The structure equations can be given by [15] …”

“…On the other hand, the third author of this paper gave another structure to study the centroaffine immersions of codimension two [13,14]. He defined the centroaffine metric g of centroaffine immersions with codimension two in a new way and then choose ∆ g x as the second transversal vector field, where ∆ g denotes the Laplacian of g. According to this structure and using the efficacious moving frame method, the authors of this paper compared these different normalizations and defined the minimal centroaffine immersions of codimension two [15]. In this paper, we further consider the centroaffine invariants of centroaffine immersion x : M = M n → R n+2 (n ≥ 2) and get the centroaffine theorema egregium.…”

Centroaffine ruled surfaces in R3

“…(cf. [4][5][6][7][8][9]11]). But there are few classification results concerning with the Weingarten centroaffine submanifolds.…”

Projective Surfaces and Pre-Normalized Blaschke Immersions of Codimension Two

“…One transversal vector field is the radial vector field, that is, the position vector field of a surface, and the other is chosen to be a pre-normalized Blaschke normal vector field, which was defined in [6]. See [4,5,11,12] for other choices of transversal vector fields. Following [6], Furuhata [3] studied surfaces in R 4 with vanishing shape operator, which can be considered from a viewpoint of a certain variation problem.…”