Affine surfaces with parallel shape operators

Włodzimierz Jelonek

Annales Polonici Mathematici (1992)

  • Volume: 56, Issue: 2, page 179-186
  • ISSN: 0066-2216

Abstract

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We study affine nondegenerate Blaschke hypersurfaces whose shape operators are parallel with respect to the induced Blaschke connections. We classify such surfaces and thus give an exact classification of extremal locally symmetric surfaces, first described by F. Dillen.

How to cite

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Włodzimierz Jelonek. "Affine surfaces with parallel shape operators." Annales Polonici Mathematici 56.2 (1992): 179-186. <http://eudml.org/doc/262305>.

@article{WłodzimierzJelonek1992,
abstract = {We study affine nondegenerate Blaschke hypersurfaces whose shape operators are parallel with respect to the induced Blaschke connections. We classify such surfaces and thus give an exact classification of extremal locally symmetric surfaces, first described by F. Dillen.},
author = {Włodzimierz Jelonek},
journal = {Annales Polonici Mathematici},
keywords = {Blaschke structure; affine locally symmetric surface; affine shape operator; equiaffine structure; affine normal field; extremal locally symmetric surfaces; affine Blaschke surfaces; shape operator},
language = {eng},
number = {2},
pages = {179-186},
title = {Affine surfaces with parallel shape operators},
url = {http://eudml.org/doc/262305},
volume = {56},
year = {1992},
}

TY - JOUR
AU - Włodzimierz Jelonek
TI - Affine surfaces with parallel shape operators
JO - Annales Polonici Mathematici
PY - 1992
VL - 56
IS - 2
SP - 179
EP - 186
AB - We study affine nondegenerate Blaschke hypersurfaces whose shape operators are parallel with respect to the induced Blaschke connections. We classify such surfaces and thus give an exact classification of extremal locally symmetric surfaces, first described by F. Dillen.
LA - eng
KW - Blaschke structure; affine locally symmetric surface; affine shape operator; equiaffine structure; affine normal field; extremal locally symmetric surfaces; affine Blaschke surfaces; shape operator
UR - http://eudml.org/doc/262305
ER -

References

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  1. [1] F. Dillen, Locally symmetric complex affine hypersurfaces, J. Geom. 33 (1988), 27-38. Zbl0656.53012
  2. [2] F. Dillen, Equivalence theorems in affine differential geometry, Geom. Dedicata 32 (1989), 81-92. Zbl0684.53012
  3. [3] M. Magid and K. Nomizu, On affine surfaces whose cubic forms are parallel relative to the affine metric, Proc. Nat. Acad. Sci. U.S.A. 65 (1989), 215-218. Zbl0705.53010
  4. [4] K. Nomizu, On completeness in affine differential geometry, Geom. Dedicata 20 (1986), 43-49. Zbl0587.53010
  5. [5] K. Nomizu and U. Pinkall, On the geometry of affine immersions, Math. Z. 195 (1987), 165-178. Zbl0629.53012
  6. [6] K. Nomizu and U. Pinkall, Cayley surfaces in affine differential geometry, Tôhoku Math. J. 41 (1989), 589-596. 

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