The Jordan normal form of higher order Osserman algebraic curvature tensors

Peter Gilkey; Raina Ivanova

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 2, page 231-242
  • ISSN: 0010-2628

Abstract

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We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type ( r , s ) in a vector space of signature ( p , q ) . We then use these examples to establish some results concerning higher order Osserman and higher order Jordan Osserman algebraic curvature tensors.

How to cite

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Gilkey, Peter, and Ivanova, Raina. "The Jordan normal form of higher order Osserman algebraic curvature tensors." Commentationes Mathematicae Universitatis Carolinae 43.2 (2002): 231-242. <http://eudml.org/doc/249007>.

@article{Gilkey2002,
abstract = {We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type $(r,s)$ in a vector space of signature $(p,q)$. We then use these examples to establish some results concerning higher order Osserman and higher order Jordan Osserman algebraic curvature tensors.},
author = {Gilkey, Peter, Ivanova, Raina},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {higher order Jacobi operator; Osserman algebraic curvature tensors; Jordan Osserman algebraic curvature tensors; higher order Jacobi operator; Osserman algebraic curvature tensors; Jordan Osserman algebraic curvature tensors},
language = {eng},
number = {2},
pages = {231-242},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Jordan normal form of higher order Osserman algebraic curvature tensors},
url = {http://eudml.org/doc/249007},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Gilkey, Peter
AU - Ivanova, Raina
TI - The Jordan normal form of higher order Osserman algebraic curvature tensors
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 2
SP - 231
EP - 242
AB - We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type $(r,s)$ in a vector space of signature $(p,q)$. We then use these examples to establish some results concerning higher order Osserman and higher order Jordan Osserman algebraic curvature tensors.
LA - eng
KW - higher order Jacobi operator; Osserman algebraic curvature tensors; Jordan Osserman algebraic curvature tensors; higher order Jacobi operator; Osserman algebraic curvature tensors; Jordan Osserman algebraic curvature tensors
UR - http://eudml.org/doc/249007
ER -

References

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  10. Gilkey P., Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor, World Scientific, 2002. Zbl1007.53001MR1877530
  11. Gilkey P., Ivanova R., The Jordan normal form of Osserman algebraic curvature tensors, Results Math. 40 (2001), 192-204. (2001) Zbl0999.53014MR1860368
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  13. Gilkey P., Stanilov G., Videv V., Pseudo-Riemannian manifolds whose generalized Jacobi operator has constant characteristic polynomial, J. Geom. 62 (1998), 144-153. (1998) Zbl0906.53046MR1631494
  14. Gilkey P., Swann A., Vanhecke L., Isoparametric geodesic spheres and a conjecture of Osserman regarding the Jacobi operator, Quart. J. Math. Oxford Ser. 46 (1995), 299-320. (1995) MR1348819
  15. Osserman R., Curvature in the eighties, Amer. Math. Monthly 97 (1990), 731-756. (1990) Zbl0722.53001MR1072814
  16. Stanilov G., Curvature operators based on the skew-symmetric curvature operator and their place in Differential Geometry, preprint, 2000. 
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  18. Stanilov G., Videv V., Four-dimensional pointwise Osserman manifolds, Abh. Math. Sem. Univ. Hamburg 68 (1998), 1-6. (1998) Zbl0980.53058MR1658408

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