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

SIGMA 3 (2007), 095, 13 pages      arXiv:0708.0957
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Stanilov-Tsankov-Videv Theory

Miguel Brozos-Vázquez a, Bernd Fiedler b, Eduardo García-Río a, Peter Gilkey c, Stana Nikcevic d, Grozio Stanilov e, Yulian Tsankov e, Ramón Vázquez-Lorenzo a and Veselin Videv f
a) Department of Geometry and Topology, Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela 15782, Spain
b) Eichelbaumstr. 13, D-04249 Leipzig, Germany
c) Mathematics Department, University of Oregon, Eugene Oregon 97403-1222, USA
d) Mathematical Institute, SANU, Knez Mihailova 35, p.p. 367, 11001 Belgrade, Serbia
e) Sofia University ''St. Kl. Ohridski'', Sofia, Bulgaria
f) Mathematics Department, Thracian University, University Campus, 6000 Stara Zagora, Bulgaria

Received August 07, 2007, in final form September 22, 2007; Published online September 28, 2007

We survey some recent results concerning Stanilov-Tsankov-Videv theory, conformal Osserman geometry, and Walker geometry which relate algebraic properties of the curvature operator to the underlying geometry of the manifold.

Key words: algebraic curvature tensor; anti-self-dual; conformal Jacobi operator; conformal Osserman manifold; Jacobi operator; Jacobi-Tsankov; Jacobi-Videv; mixed-Tsankov; Osserman manifold; Ricci operator; self-dual; skew-symmetric curvature operator; skew-Tsankov; skew-Videv; Walker manifold; Weyl conformal curvature operator.

pdf (267 kb)   ps (193 kb)   tex (16 kb)


  1. Blazic N., Gilkey P., Conformally Osserman manifolds and conformally complex space forms, Int. J. Geom. Methods Mod. Phys. 1 (2004), 97-106, math.DG/0311263.
  2. Blazic N., Gilkey P., Conformally Osserman manifolds and self-duality in Riemannian geometry, in Proceedings of the Conference "Differential Geometry and Its Applications" (August 30 - September 3, 2004, Charles University, Prague, Czech Republic), Editors J. Bures, O. Kowalski, D. Krupka and J. Slovak, MATFYZPRESS, 2005, 15-18, math.DG/0504498.
  3. Blazic N., Gilkey P., Nikcevic S., Simon U., The spectral geometry of the Weyl conformal tensor, Banach Center Publ. 69 (2005), 195-203, math.DG/0310226.
  4. Brozos-Vázquez M., García-Río E., Gilkey P., Vázquez-Lorenzo R., Examples of signature (2,2) manifolds with commuting curvature operators, J. Phys. A: Math. Theor., to appear, arXiv:0708.2770.
  5. Brozos-Vázquez M., García-Río E., Vázquez-Lorenzo R., Conformally Osserman four-dimensional manifolds whose conformal Jacobi operators have complex eigenvalues, Proc. Royal Soc. A 462 (2006), 1425-1441.
  6. Brozos-Vázquez M., García-Río E., Gilkey P., Vázquez-Lorenzo R., Completeness, Ricci blowup, the Osserman and the conformal Osserman condition for Walker signature (2,2) manifolds, in Proceedings of XV International Workshop on Geometry and Physics, to appear, math.DG/0611279.
  7. Brozos-Vázquez M., Gilkey P., Pseudo-Riemannian manifolds with commuting Jacobi operators, Rend. Circ. Mat. Palermo 55 (2006), 163-174, math.DG/0608707.
  8. Brozos-Vázquez M., Gilkey P., The global geometry of Riemannian manifolds with commuting curvature operators, J. Fixed Point Theory Appl. 1 (2007), 87-96, math.DG/0609500.
  9. Brozos-Vázquez M., Gilkey P., Manifolds with commuting Jacobi operators, J. Geom. 86 (2007), 21-30, math.DG/0507554.
  10. Brozos-Vázquez M., Gilkey P., Nikcevic S., Jacobi-Tsankov manifolds which are not 2-step nilpotent, in Proceedings of the Conference "Contemporary Geometry and Related Topics" (June 26 - July 2, 2005, Belgrade, Serbia and Montenegro), Editors N. Bokan, M. Djoric, A.T. Fomenko, Z. Rakic, B. Wegner and J. Wess, University of Belgrade, Serbia, 2006, 63-79, math.DG/0609565.
  11. Chaichi M., García-Río E., Matsushita Y., Curvature properties of four-dimensional Walker metrics, Classical Quantum Gravity 22 (2005), 559-577.
  12. Díaz-Ramos J.C., García-Río E., A note on the structure of algebraic curvature tensors, Linear Algebra Appl. 382 (2004), 271-277.
  13. Fiedler B., Determination of the structure of algebraic curvature tensors by means of Young symmetrizers, Seminaire Lotharingien de Combinatoire B48d (2003), 20 pages, math.CO/0212278.
  14. Fiedler B., Gilkey P., Nilpotent Szabó, Osserman and Ivanov-Petrova pseudo Riemannian manifolds, Contemp. Math. 337 (2003), 53-64, math.DG/0211080.
  15. García-Río E., Kupeli D.N., Vázquez-Abal M.E., Vázquez-Lorenzo R., Affine Osserman connections and their Riemann extensions, Differential Geom. Appl. 11 (1999), 145-153.
  16. García-Río E., Kupeli D., Vázquez-Lorenzo R., Osserman manifolds in semi-Riemannian geometry, Lecture Notes in Mathematics, Vol. 1777, Springer-Verlag, Berlin, 2002.
  17. Gilkey P., Geometric properties of natural operators defined by the Riemann curvature tensor, World Scientific, 2001.
  18. Gilkey P., The geometry of curvature homogeneous pseudo Riemannian manifolds, Imperial College Press, 2007.
  19. Gilkey P., Puffini E., Videv V., Puffini-Videv models and manifolds, J. Geom., to appear, math.DG/0605464.
  20. Gilkey P., Nikcevic S., Pseudo-Riemannian Jacobi-Videv manifolds, Int. J. Geom. Methods Mod. Phys. 4 (2007), 727-738, arXiv:0708.1096.
  21. Gilkey P., Stanilov G., Videv V., Pseudo Riemannian manifolds whose generalized Jacobi operator has constant characteristic polynomial, J. Geom. 62 (1998) 144-153.
  22. Ivanova M., Videv V., Zhelev Z., Four-dimensional Riemannian manifolds with commuting higher order Jacobi operators, math.DG/0701090.
  23. Osserman R., Curvature in the eighties, Amer. Math. Monthly 97 (1990), 731-756.
  24. Stanilov G., Videv V., On a generalization of the Jacobi operator in the Riemannian geometry, God. Sofij. Univ., Fak. Mat. Inform. 86 (1994) 27-34.
  25. Stanilov G., Videv V., Four dimensional pointwise Osserman manifolds, Abh. Math. Sem. Univ. Hamburg 68 (1998), 1-6.
  26. Stanilov G., Videv V., On the commuting of curvature operators, in Proceedings of the 33rd Spring Conference of the Union of Bulgarian Mathematicians Borovtes "Mathematics and Education in Mathematics" (April 1-4, 2004, Sofia), Sofia, 2004, 176-179.
  27. Tsankov Y., A characterization of n-dimensional hypersurface in Euclidean space with commuting curvature operators, Banach Center Publ. 69 (2005), 205-209.
  28. Videv V., A characterization of the 4-dimensional Einstein Riemannian manifolds using curvature operators, Preprint.
  29. Walker A.G., Canonical form for a Riemannian space with a parallel field of null planes, Quart. J. Math., Oxford Ser. (2) 1 (1950), 69-79.
  30. Walker A.G., Canonical forms. II. Parallel partially null planes, Quart. J. Math., Oxford Ser. (2) 1 (1950), 147-152.

Previous article   Next article   Contents of Volume 3 (2007)