Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 097, 22 pages      arXiv:0910.3609      https://doi.org/10.3842/SIGMA.2009.097
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2

Christine Scharlach
Technische Universität Berlin, Fak. II, Inst. f. Mathematik, MA 8-3, 10623 Berlin, Germany

Received May 08, 2009, in final form October 06, 2009; Published online October 19, 2009

Abstract
An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(TpM) for all pM, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. S = HId (and thus S is trivially preserved). In Part 1 we found the possible symmetry groups G and gave for each G a canonical form of K. We started a classification by showing that hyperspheres admitting a pointwise Z2 × Z2 resp. R-symmetry are well-known, they have constant sectional curvature and Pick invariant J < 0 resp. J = 0. Here, we continue with affine hyperspheres admitting a pointwise Z3- or SO(2)-symmetry. They turn out to be warped products of affine spheres (Z3) or quadrics (SO(2)) with a curve.

Key words: affine hyperspheres; indefinite affine metric; pointwise symmetry; affine differential geometry; affine spheres; warped products.

pdf (344 kb)   ps (227 kb)   tex (24 kb)

References

  1. Bryant R.L., Second order families of special Lagrangian 3-folds, in Perspectives in Riemannian Geometry, CRM Proc. Lecture Notes, Vol. 40, Amer. Math. Soc., Providence, RI, 2006, 63-98, math.DG/0007128.
  2. Calabi E., Complete affine hyperspheres. I, in Convegno di Geometria Differenziale (INDAM, Rome, 1971), Symposia Mathematica, Vol. X, Academic Press, London, 1972, 19-38.
  3. Dillen F., Equivalence theorems in affine differential geometry, Geom. Dedicata 32 (1989), 81-92.
  4. Dillen F., Katsumi N., Vrancken L., Conjugate connections and Radon's theorem in affine differential geometry, Monatsh. Math. 109 (1990), 221-235.
  5. Dillen F., Vrancken L., Calabi-type composition of affine spheres, Differential Geom. Appl. 4 (1994), 303-328.
  6. Hu Z., Li H.Z., Vrancken L, Characterizations of the Calabi product of hyperbolic affine hyperspheres, Results Math. 52 (2008), 299-314.
  7. Li A.M., Simon U., Zhao G.S., Global affine differential geometry of hypersurfaces, de Gruyter Expositions in Mathematics, Vol. 11, Walter de Gruyter & Co., Berlin, 1993.
  8. Lu Y., Scharlach C., Affine hypersurfaces admitting a pointwise symmetry, Results Math. 48 (2005), 275-300, math.DG/0510150.
  9. Nölker S., Isometric immersions of warped products, Differential Geom. Appl. 6 (1996), 1-30.
  10. Nomizu K., Sasaki T., Affine differential geometry. Geometry of affine immersions, Cambridge Tracts in Mathematics, Vol. 111, Cambridge University Press, Cambridge, 1994.
  11. O'Neill B., Semi-Riemannian geometry. With applications to relativity, Pure and Applied Mathematics, Vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983.
  12. Ponge R., Reckziegel H., Twisted products in pseudo-Riemannian geometry, Geom. Dedicata 48 (1993), 15-25.
  13. Scharlach C., Indefinite affine hyperspheres admitting a pointwise symmetry. Part 1, Beiträge Algebra Geom. 48 (2007), 469-491, math.DG/0510531.
  14. Vrancken L., Special classes of three dimensional affine hyperspheres characterized by properties of their cubic form, in Contemporary Geometry and Related Topics, World Sci. Publ., River Edge, NJ, 2004, 431-459.

Previous article   Next article   Contents of Volume 5 (2009)