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Centroaffine Differential Geometry of (Positive) Definite Oriented Surfaces in ℝ4

In: New Developments in Differential Geometry, Budapest 1996

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  • Christine Scharlach

    (Technische Universität Berlin , Fachbereich Mathematik)

Abstract

We develop a centroaffine theory of (positive) definite oriented surfaces in IR 4\{0}, using E. Cartan’s method of moving frames (Cartan, 1951). An oriented surface x(M 2) in IR 4\0 is called a (positive) definite oriented surface if its affine semiconformal structure is definite. The centroaffine frame (in particular the centroaffine normal plane) and the centroaffine invariants of x will be defined in a canonical way. We characterize a surface up to centroaffine transformations by its centroaffine connection ▽ (induced by the centroaffine normal plane), its centroaffine metric g and its second fundamental forms h 3 and h 4. Then we classify all (positive) definite oriented surfaces with vanishing cubic forms (resp. with ▽h 4 = 0 or ▽g = 0). Using a characterization of complex curves by centroaffine invariants (Scharlach, 1997), they turn out to be exactly the complex curves with zero centroaffine curvature.

Suggested Citation

  • Christine Scharlach, 1999. "Centroaffine Differential Geometry of (Positive) Definite Oriented Surfaces in ℝ4," Springer Books, in: J. Szenthe (ed.), New Developments in Differential Geometry, Budapest 1996, pages 411-428, Springer.
    • Handle: RePEc:spr:sprchp:978-94-011-5276-1_30
    DOI: 10.1007/978-94-011-5276-1_30
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