SIGMA 11 (2015), 001, 24 pages      arXiv:1408.4088      https://doi.org/10.3842/SIGMA.2015.001

Geometry of Centroaffine Surfaces in $\mathbb{R}^5$

Nathaniel Bushek ^a and Jeanne N. Clelland ^b
a) Department of Mathematics, UNC - Chapel Hill, CB #3250, Phillips Hall, Chapel Hill, NC 27599, USA
^b) Department of Mathematics, 395 UCB, University of Colorado, Boulder, CO 80309-0395, USA

Received August 23, 2014, in final form December 26, 2014; Published online January 06, 2015

Abstract
We use Cartan's method of moving frames to compute a complete set of local invariants for nondegenerate, 2-dimensional centroaffine surfaces in $\mathbb{R}^5 \setminus \{0\}$ with nondegenerate centroaffine metric. We then give a complete classification of all homogeneous centroaffine surfaces in this class.

Key words: centroaffine geometry; Cartan's method of moving frames.

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References

1. Clelland J.N., From Frenet to Cartan: the method of moving frames, in preparation.
2. Fulton W., Harris J., Representation theory, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991.
3. Furuhata H., Minimal centroaffine immersions of codimension two, Bull. Belg. Math. Soc. Simon Stevin 7 (2000), 125-134.
4. Furuhata H., Kurose T., Self-dual centroaffine surfaces of codimension two with constant affine mean curvature, Bull. Belg. Math. Soc. Simon Stevin 9 (2002), 573-587.
5. Gardner R.B., Wilkens G.R., The fundamental theorems of curves and hypersurfaces in centro-affine geometry, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), 379-401.
6. Griffiths P., On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775-814.
7. Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, Vol. 61, Amer. Math. Soc., Providence, RI, 2003.
8. Laugwitz D., Differentialgeometrie in Vektorräumen, unter besonderer Berücksichtigung der unendlichdimensionalen Räume, Friedr. Vieweg & Sohn, Braunschweig, 1965.
9. Li A.M., Wang C.P., Canonical centroaffine hypersurfaces in ${\mathbb R}^{n+1}$, Results Math. 20 (1991), 660-681.
10. Liu H.L., Wang C.P., The centroaffine Tchebychev operator, Results Math. 27 (1995), 77-92.
11. Mayer O., Myller A., La géométrie centroaffine des courbes planes, Ann. Scí. de l'Universit'e de Jassy 18 (1933), 234-280.
12. Milnor J., Husemoller D., Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 73, Springer-Verlag, New York - Heidelberg, 1973.
13. Nomizu K., Sasaki T., Centroaffine immersions of codimension two and projective hypersurface theory, Nagoya Math. J. 132 (1993), 63-90.
14. Nomizu K., Vrancken L., A new equiaffine theory for surfaces in ${\mathbb R}^4$, Internat. J. Math. 4 (1993), 127-165.
15. Scharlach C., Centroaffine first order invariants of surfaces in ${\mathbb R}^4$, Results Math. 27 (1995), 141-159.
16. Scharlach C., Centroaffine differential geometry of (positive) definite oriented surfaces in ${\mathbb R}^4$, in New Developments in Differential Geometry (Budapest, 1996), Kluwer Acad. Publ., Dordrecht, 1999, 411-428.
17. Scharlach C., Simon U., Verstraelen L., Vrancken L., A new intrinsic curvature invariant for centroaffine hypersurfaces, Beiträge Algebra Geom. 38 (1997), 437-458.
18. Scharlach C., Vrancken L., A curvature invariant for centroaffine hypersurfaces. II, in Geometry and Topology of Submanifolds, VIII (Brussels, 1995/Nordfjordeid, 1995), World Sci. Publ., River Edge, NJ, 1996, 341-350.
19. Scharlach C., Vrancken L., Centroaffine surfaces in ${\mathbb R}^4$ with planar $\nabla$-geodesics, Proc. Amer. Math. Soc. 126 (1998), 213-219.
20. Wang C.P., Centroaffine minimal hypersurfaces in ${\mathbb R}^{n+1}$, Geom. Dedicata 51 (1994), 63-74.
21. Wilkens G.R., Centro-affine geometry in the plane and feedback invariants of two-state scalar control systems, in Differential Geometry and Control (Boulder, CO, 1997), Proc. Sympos. Pure Math., Vol. 64, Amer. Math. Soc., Providence, RI, 1999, 319-333.
22. Yang Y., Liu H., Minimal centroaffine immersions of codimension two, Results Math. 52 (2008), 423-437.
23. Yang Y., Yu Y., Liu H., Flat centroaffine surfaces with the degenerate second fundamental form and vanishing Pick invariant in $\mathbb{R}^4$, J. Math. Anal. Appl. 397 (2013), 161-171.