Projective Surfaces and Pre-Normalized Blaschke Immersions of Codimension Two

Year 2016, , 100 - 110, 30.04.2016

Atsushi Fujioka Hitoshi Furuhata Takeshi Sasaki

https://doi.org/10.36890/iejg.591899

Abstract

   We prove that any non-degenerate surface in the projective 3-space has a local lift as a minimal

pre-normalized Blaschke immersion into the equicentroaffine 4-space. Furthermore, an indefinite

surface in the projective 3-space has a local lift as a pre-normalized Blaschke immersion into the

equicentroaffine 4-space satisfying the Einstein condition if and only if the surface is projectively

applicable to an affine sphere.

Keywords

projective surface, affine sphere, pre-normalized Blaschke immersion, centroaffine minimality, Einstein condition

References

• [1] Ferapontov, E. V., Integrable systems in projective differential geometry, Kyushu J. Math. 54(2000), no. 1, 183–215.
• [2] Fujioka A., Furuhata H. and Sasaki T., Projective minimality for centroaffine minimal surfaces, J. Geom. 105(2014), no. 1, 87–102.
• [3] Furuhata H., Minimal centroaffine immersions of codimension two, Bull. Belg. Math. Soc. 7(2000), no. 1, 125–134.
• [4] Liu H.-L., Indefinite equi-centroaffinely homogeneous surfaces with vanishing Pick-invariant in R4, Math. J. 26(1997), no. 1, 225–251.
• [5] Lopšic, A. M., On the theory of a surface of n dimensions in an equicentroaffine space of n + 2 dimensions, (Russian) Sem. Vektor. Tenzor. Analizu. 8(1950), 286–295.
• [6] Nomizu K. and Sasaki T., Centroaffine immersions of codimension two and projective hypersurface theory, Nagoya Math. J. 132(1993), 63–90.
• [7] Nomizu K. and Sasaki T., Affine differential geometry. Geometry of affine immersions, Cambridge Tracts in Mathematics, 111, Cambridge University Press, Cambridge, 1994.
• [8] Sasaki T., Projective differential geometry and linear homogeneous differential equations, Rokko Lectures in Math. 5. Kobe University, 1999.
• [9] Sasaki T., Line congruence and transformation of projective surfaces, Kyushu J. Math. 60(2006), no. 1, 101–243.
• [10] Simon, U., Schwenk-Schellschmidt, A. and Viesel, H., Introduction to the affine differential geometry of hypersurfaces, Lecture Notes of the Science University of Tokyo, Science University of Tokyo, Tokyo, 1991.
• [11] Walter, R., Centroaffine differential geometry: submanifolds of codimension 2, Results Math. 13(1988), no. 3-4, 386-402.
• [12] Yang Y. and Liu H.-L., Minimal centroaffine immersions of codimension two, Results Math. 52(2008), no. 3-4, 423–437.