Curvature tensors in dimension four which do not belong to any curvature homogeneous space

Oldřich Kowalski; Friedbert Prüfer

Archivum Mathematicum (1994)

  • Volume: 030, Issue: 1, page 45-57
  • ISSN: 0044-8753

Abstract

top
A six-parameter family is constructed of (algebraic) Riemannian curvature tensors in dimension four which do not belong to any curvature homogeneous space. Also a general method is given for a possible extension of this result.

How to cite

top

Kowalski, Oldřich, and Prüfer, Friedbert. "Curvature tensors in dimension four which do not belong to any curvature homogeneous space." Archivum Mathematicum 030.1 (1994): 45-57. <http://eudml.org/doc/247559>.

@article{Kowalski1994,
abstract = {A six-parameter family is constructed of (algebraic) Riemannian curvature tensors in dimension four which do not belong to any curvature homogeneous space. Also a general method is given for a possible extension of this result.},
author = {Kowalski, Oldřich, Prüfer, Friedbert},
journal = {Archivum Mathematicum},
keywords = {Riemannian manifolds; curvature tensor; curvature homogeneous spaces; locally homogeneous spaces; curvature homogeneous; algebraic curvature tensors},
language = {eng},
number = {1},
pages = {45-57},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Curvature tensors in dimension four which do not belong to any curvature homogeneous space},
url = {http://eudml.org/doc/247559},
volume = {030},
year = {1994},
}

TY - JOUR
AU - Kowalski, Oldřich
AU - Prüfer, Friedbert
TI - Curvature tensors in dimension four which do not belong to any curvature homogeneous space
JO - Archivum Mathematicum
PY - 1994
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 030
IS - 1
SP - 45
EP - 57
AB - A six-parameter family is constructed of (algebraic) Riemannian curvature tensors in dimension four which do not belong to any curvature homogeneous space. Also a general method is given for a possible extension of this result.
LA - eng
KW - Riemannian manifolds; curvature tensor; curvature homogeneous spaces; locally homogeneous spaces; curvature homogeneous; algebraic curvature tensors
UR - http://eudml.org/doc/247559
ER -

References

top
  1. Chern S.S., On the Curvature and Characteristic Classes of a Riemannian manifold, Abh. Math. Sem. Univ. Hamburg 20 (1955), 117-126. (1955) MR0075647
  2. Kowalski O., An explicit classification of 3-dimensional Riemannian spaces satisfying R ( X , Y ) · R = 0 , Preprint, 1991. (1991) MR1408298
  3. Kowalski O., A classification of Riemannian 3-manifolds with constant principal Ricci curvatures ρ 1 = ρ 2 ρ 3 , To appear in Nagoya Math. J. 132 (1993). (1993) MR1253692
  4. Kowalski O., Nonhomogeneous Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Comment. Math. Univ. Carolinae, 34, 3 (1993), 451-457. (1993) Zbl0789.53024MR1243077
  5. Klinger R., A Basis that Reduces to Zero as many Curvature Components as Possible, Abh. Math. Sem. Univ. Hamburg 61 (1991), 243-248. (1991) Zbl0753.53012MR1138290
  6. Kobayashi S., Nomizu K., Foundations of Differential geometry I, Interscience Publishers, New York 1963. (1963) Zbl0119.37502MR0152974
  7. Kowalski O., Prüfer F., On Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Preprint, 1993. (1993) MR1289828
  8. Kowalski O., Tricerri F., Vanhecke L., New examples of non-homogeneous Riemannian manifolds whose curvature tensor is that of a Riemannian symmetric space, C. R. Acad. Sci. Paris, Sér. I, 311 (1990), 355-360 (1990) Zbl0713.53028MR1071643
  9. Kowalski O., Tricerri F., Vanhecke L., Curvature homogeneous Riemannian manifolds, J. Math. Pures Appl. 71 (1992), 471 - 501. (1992) Zbl0836.53029MR1193605
  10. Kowalski O., Tricerri F., Vanhecke L., Curvature homogeneous spaces with a solvable Lie group as a homogeneous model, J. Math. Soc. Japan, 44 (1992), 461-484. (1992) MR1167378
  11. Kowalski O., Vanhecke L., Ball-Homogeneous and Disk-Homogeneous Riemannian manifolds, Math. Z. 180 (1982), 429-444. (1982) Zbl0476.53023MR0666999
  12. Nicolodi L., Tricerri F., On two Theorems of I.M. Singer about Homogeneous Spaces, Ann. Global Anal. Geom. 8 (1990), 193-209. (1990) Zbl0676.53058MR1088511
  13. Milnor J., Curvatures of left invariant metrics on Lie groups, Adv. in Math. 21 (1976), 293-329. (1976) Zbl0341.53030MR0425012
  14. Sekigawa K., On some 3-dimensional Riemannian manifolds, Hokkaido Math. J. 2 (1973), 259-270. (1973) Zbl0266.53034MR0353204
  15. Singer I.M., Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13 (1960), 685-697. (1960) Zbl0171.42503MR0131248
  16. Singer I.M., Thorpe J.A., The curvature of 4-dimensional Einstein spaces, In: Global Analysis (Papers in honor of K. Kodaira, pp. 355-366) Princeton, New Jersey, Princeton University Press 1969. (1969) Zbl0199.25401MR0256303
  17. Spiro A., Tricerri F., 3-dimensional Riemannian metrics with prescribed Ricci principal curvatures, Preprint, 1993. (1993) MR1327884
  18. Tsukada T., Curvature homogeneous hypersurfaces immersed in a real space form, Tôhoku Math. J. 40 (1988), 221-244. (1988) Zbl0651.53037MR0943821
  19. Tricerri F., Vanhecke L., Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365-398. (1981) Zbl0484.53014MR0626479
  20. Yamato K., A characterization of locally homogeneous Riemannian manifolds of dimension three, Nagoya Math. J. 123 (1991), 77-90. (1991) MR1126183

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.