Previous |  Up |  Next


Riemannian manifold; linear frame bundle; orthonormal frame bundle; $g$-natural metrics; homogeneity
In this paper we prove that each $g$-natural metric on a linear frame bundle $LM$ over a Riemannian manifold $(M, g)$ is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define $g$-natural metrics on the orthonormal frame bundle $OM$ and we prove the same invariance result as above for $OM$. Hence we see that, over a space $(M, g)$ of constant sectional curvature, the bundle $OM$ with an arbitrary $g$-natural metric $\tilde{G}$ is locally homogeneous.
[1] Abbassi, M. T. K.: Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifolds $(M,g)$. Comment. Math. Univ. Carolin. 45 (2004), 591–596. MR 2103077
[2] Cordero, L. A., de León, M.: Lifts of tensor fields to the frame bundle. Rend. Circ. Mat. Palermo 32 (1983), 236–271. DOI 10.1007/BF02844834 | MR 0729099
[3] Cordero, L. A., de León, M.: On the curvature of the induced Riemannian metric on the frame bundle of a Riemannian manifold. J. Math. Pures Appl. 65 (1986), 81–91. MR 0844241
[4] Cordero, L. A., Dodson, C. T. J., de León, M.: Differential Geometry of Frame Bundles. Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1989. MR 0980716
[5] Jensen, G.: The scalar curvature of left invariant Riemannian metrics. Indiana Univ. Math. J. 20 (1971), 1125–1143. DOI 10.1512/iumj.1971.20.20104 | MR 0289726 | Zbl 0219.53044
[6] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer-Verlag, Berlin-Heidelberg-New York, 1993. MR 1202431
[7] Kowalski, O., Sekizawa, M.: Invariance of $g$-natural metrics on tangent bundles. to appear in Proceedings of 10th International Conference on Differential Geometry and Its Applications, World Scientific. MR 2462791
[8] Kowalski, O., Sekizawa, M.: On the geometry of orthonormal frame bundles. to appear in Math. Nachr. MR 2473330 | Zbl 1158.53015
[9] Kowalski, O., Sekizawa, M.: On the geometry of orthonormal frame bundles II. to appear in Ann. Global Anal. Geom. MR 2395192 | Zbl 1141.53023
[10] Kowalski, O., Sekizawa, M.: Natural transformations of Riemannian metrics on manifolds to metrics on linear frame bundles—a classification. Differential Geometry and its Applications, Proceeding of the Conference, August 24–30, 1986, Brno, Czechoslovakia, D. Reidel Publ. Comp., pp. 149-178, 1987. MR 0923348 | Zbl 0632.53040
[11] Kowalski, O., Sekizawa, M.: On curvatures of linear frame bundles with naturally lifted metrics. Rend. Sem. Mat. Univ. Politec. Torino 63 (2005), 283–295. MR 2202049 | Zbl 1141.53020
[12] Krupka, D.: Elementary theory of differential invariants. Arch. Math. (Brno) 4 (1978), 207–214. MR 0512763 | Zbl 0428.58002
[13] Krupka, D.: Differential invariants. Lecture Notes, Faculty of Science, Purkyně University, Brno (1979).
[14] Krupka, D., Janyška, J.: Lectures on Differential Invariants. University J. E. Purkyně in Brno, 1990. MR 1108622
[15] Krupka, D., V. Mikolášová, : On the uniqueness of some differential invariants: $d$, [ , ], $\nabla $. Czechoslovak Math. J. 34 (1984), 588–597. MR 0764440
[16] Mok, K. P.: On the differential geometry of frame bundles of Riemannian manifolds. J. Reine Angew. Math. 302 (1978), 16–31. MR 0511689 | Zbl 0378.53016
[17] O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469. DOI 10.1307/mmj/1028999604 | MR 0200865
[18] Zou, X.: A new type of homogeneous spaces and the Einstein metrics on $O(n+1)$. Nanjing Univ. J. Math. Biquarterly 23 (2006), 70–78. MR 2245416 | Zbl 1192.53055
Partner of
EuDML logo