Riemannian manifolds in which certain curvature operator has constant eigenvalues along each helix
Archivum Mathematicum (1999)
- Volume: 035, Issue: 2, page 129-140
- ISSN: 0044-8753
Access Full Article
topAbstract
topHow to cite
topAlexieva, Yana, and Ivanov, Stefan. "Riemannian manifolds in which certain curvature operator has constant eigenvalues along each helix." Archivum Mathematicum 035.2 (1999): 129-140. <http://eudml.org/doc/248353>.
@article{Alexieva1999,
abstract = {Riemannian manifolds for which a natural skew-symmetric curvature operator has constant eigenvalues on helices are studied. A local classification in dimension three is given. In the three dimensional case one gets all locally symmetric spaces and all Riemannian manifolds with the constant principal Ricci curvatures $r_1 = r_2 = 0, r_3 \ne 0$, which are not locally homogeneous, in general.},
author = {Alexieva, Yana, Ivanov, Stefan},
journal = {Archivum Mathematicum},
keywords = {helix; constant eigenvalues of the curvature operator; locally symmetric spaces; curvature homogeneous spaces; helix; constant eigenvalues of the curvature},
language = {eng},
number = {2},
pages = {129-140},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Riemannian manifolds in which certain curvature operator has constant eigenvalues along each helix},
url = {http://eudml.org/doc/248353},
volume = {035},
year = {1999},
}
TY - JOUR
AU - Alexieva, Yana
AU - Ivanov, Stefan
TI - Riemannian manifolds in which certain curvature operator has constant eigenvalues along each helix
JO - Archivum Mathematicum
PY - 1999
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 035
IS - 2
SP - 129
EP - 140
AB - Riemannian manifolds for which a natural skew-symmetric curvature operator has constant eigenvalues on helices are studied. A local classification in dimension three is given. In the three dimensional case one gets all locally symmetric spaces and all Riemannian manifolds with the constant principal Ricci curvatures $r_1 = r_2 = 0, r_3 \ne 0$, which are not locally homogeneous, in general.
LA - eng
KW - helix; constant eigenvalues of the curvature operator; locally symmetric spaces; curvature homogeneous spaces; helix; constant eigenvalues of the curvature
UR - http://eudml.org/doc/248353
ER -
References
top- Berndt J., Vanhecke L., Two natural generalizations of locally symmetric spaces, Diff.Geom. and Appl. 2 (1992), 57–80. (1992) Zbl0747.53013MR1244456
- Berndt J., Prüfer F., Vanhecke L., Symmetric-like Riemannian manifolds and geodesic symmetries, Proc. Royal Soc. Edinburg A 125 (1995), 265–282. (1995) Zbl0830.53036MR1331561
- Berndt J., Vanhecke L., Geodesic sprays and -and -spaces, Rend. Sem. Politec. Torino 50(1992), no.4, 343–358. (1992) MR1261447
- Berndt J., Vanhecke L., Geodesic spheres and generalizations of symmetric spaces, Boll. Un. Mat. Ital. A(7), 7 (1993), no. 1, 125–134. (1993) Zbl0778.53043MR1215106
- Chi Q. S., A curvature characterization of certain locally rank-one symmetric spaces, J. Diff. Geom. 28 (1988), 187–202. (1988) Zbl0654.53053MR0961513
- Gilkey P., Manifolds whose curvature operator has constant eigenvalues at the basepoint, J. Geom. Anal. 4 2 (1992), 157–160. (1992) MR1277503
- Gilkey P., Manifolds whose higher odd order curvature operators have constant eigenvalues at the basepoint, J. Geom. Anal. 2, 2 (1992), 151–156. (1992) Zbl0739.53011MR1151757
- Gilkey P., Swann A., Vanhecke L., Isoparametric geodesic spheres and a Conjecture of Osserman concerning the Jacobi operator, Quart. J. Math. Oxford (2), 46 (1995), 299–320. (1995) Zbl0848.53023MR1348819
- Gilkey P., Leahy J., Sadofsky H., Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues, preprint. Zbl0990.53011MR1722810
- Ivanov S., Petrova I., Riemannian manifold in which certain curvature operator has constant eigenvalues along each circle, Ann. Glob. Anal. Geom. 15 (1997), 157–171. (1997) MR1448723
- Ivanov S., Petrova I., Curvature operator with parallel Jordanian basis on circles, Riv. Mat. Univ. Parma, 5 (1996), 23–31. (1996) Zbl0877.53031MR1456394
- Ivanov S., Petrova I., Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues, Geometrie Dedicata, 70 (1998), 269–282. (1998) Zbl0903.53016MR1624814
- Ivanov S., Petrova I., Locally conformal flat Riemannian manifolds with constant principal Ricci curvatures and locally conformal flat -spaces, E-print dg-ga/9702009.
- Ivanov S., Petrova I., Conformally flat Einstein-like manifolds and conformally flat Riemannian 4-manifolds all of whose Jacobi operators have parallel eigenspaces along every geodesic, E-print dg-ga/9702019.
- Kowalski O., A classification of Riemannian manifolds with constant principal Ricci curvatures , Nagoya Math. J. 132 (1993), 1–36. (1993) MR1253692
- Kowalski O., [unknown], private communication Zbl1235.35187
- Kowalski O., Prüfer F., On Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Math. Ann. 300 (1994), 17–28. (1994) MR1289828
- Milnor J., Curvature of left-invariant metrics on Lie groups, Adv. in Math. 21 (1976), 163–170. (1976) MR0425012
- Osserman R., Curvature in the 80’s, Amer. Math. Monthly, (1990), 731–756. (1990) MR1072814
- Shabó Z., Structure theorems on Riemannian spaces satisfying , I. The local version, J. Diff. Geom. 17 (1982), 531–582. (1982) MR0683165
- Spiro A., Tricerri F., 3-dimensional Riemannian metrics with prescribed Ricci principal curvatures, J. Math. Pures Appl. 74 (1995), 253–271. (1995) Zbl0851.53022MR1327884
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.