# Article

Full entry | PDF   (0.3 MB)
Keywords:
helix; constant eigenvalues of the curvature operator; locally symmetric spaces; curvature homogeneous spaces
Summary:
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.
References:
[1] Berndt J., Vanhecke L.: Two natural generalizations of locally symmetric spaces. Diff.Geom. and Appl. 2 (1992), 57–80. MR 1244456 | Zbl 0747.53013
[2] Berndt J., Prüfer F., Vanhecke L.: Symmetric-like Riemannian manifolds and geodesic symmetries. Proc. Royal Soc. Edinburg A 125 (1995), 265–282. MR 1331561 | Zbl 0830.53036
[3] Berndt J., Vanhecke L.: Geodesic sprays and $\mathcal C$-and $\mathcal B$-spaces. Rend. Sem. Politec. Torino 50(1992), no.4, 343–358. MR 1261447
[4] Berndt J., Vanhecke L.: Geodesic spheres and generalizations of symmetric spaces. Boll. Un. Mat. Ital. A(7), 7 (1993), no. 1, 125–134. MR 1215106 | Zbl 0778.53043
[5] Chi Q. S.: A curvature characterization of certain locally rank-one symmetric spaces. J. Diff. Geom. 28 (1988), 187–202. MR 0961513 | Zbl 0654.53053
[6] Gilkey P.: Manifolds whose curvature operator has constant eigenvalues at the basepoint. J. Geom. Anal. 4 2 (1992), 157–160. MR 1277503
[7] Gilkey P.: Manifolds whose higher odd order curvature operators have constant eigenvalues at the basepoint. J. Geom. Anal. 2, 2 (1992), 151–156. MR 1151757 | Zbl 0739.53011
[8] 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. MR 1348819 | Zbl 0848.53023
[9] Gilkey P., Leahy J., Sadofsky H.: Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues. preprint. MR 1722810 | Zbl 0990.53011
[10] 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. MR 1448723
[11] Ivanov S., Petrova I.: Curvature operator with parallel Jordanian basis on circles. Riv. Mat. Univ. Parma, 5 (1996), 23–31. MR 1456394 | Zbl 0877.53031
[12] Ivanov S., Petrova I.: Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues. Geometrie Dedicata, 70 (1998), 269–282. MR 1624814 | Zbl 0903.53016
[13] Ivanov S., Petrova I.: Locally conformal flat Riemannian manifolds with constant principal Ricci curvatures and locally conformal flat ${\mathcal C}$-spaces. E-print dg-ga/9702009.
[14] 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.
[15] Kowalski O.: A classification of Riemannian manifolds with constant principal Ricci curvatures $r_1 = r_2 \ne r_3$. Nagoya Math. J. 132 (1993), 1–36. MR 1253692
[16] Kowalski O.: private communication Zbl 1235.35187
[17] Kowalski O., Prüfer F.: On Riemannian 3-manifolds with distinct constant Ricci eigenvalues. Math. Ann. 300 (1994), 17–28. MR 1289828
[18] Milnor J.: Curvature of left-invariant metrics on Lie groups. Adv. in Math. 21 (1976), 163–170. MR 0425012
[19] Osserman R.: Curvature in the 80’s. Amer. Math. Monthly, (1990), 731–756. MR 1072814
[20] Shabó Z.: Structure theorems on Riemannian spaces satisfying $R(X,Y)\circ R = 0$, I. The local version. J. Diff. Geom. 17 (1982), 531–582. MR 0683165
[21] Spiro A., Tricerri F.: 3-dimensional Riemannian metrics with prescribed Ricci principal curvatures. J. Math. Pures Appl. 74 (1995), 253–271. MR 1327884 | Zbl 0851.53022

Partner of