zbMATH — the first resource for mathematics

On some hereditary properties of Riemannian $$g$$-natural metrics on tangent bundles of Riemannian manifolds. (English) Zbl 1068.53016
Summary: It is well known that if the tangent bundle $$TM$$ of a Riemannian manifold $$(M,g)$$ is endowed with the Sasaki metric $$g^s$$, then the flatness property on $$TM$$ is inherited from the base manifold [O. Kowalski, J. Reine Angew. Math. 250, 124–129 (1971; Zbl 0222.53044)]. This motivates us to the general question if the flatness and also other simple geometrical properties remain “hereditary” if we replace $$g^s$$ by the most general Riemannian “$$g$$-natural metric” on $$TM$$ [see O. Kowalski and M. Sekizawa, Bull. Tokyo Gakugei Univ., Sect. IV, Ser. Math. Nat. Sci. 40, 1–29 (1988; Zbl 0656.53021), M. T. K. Abbassi and M. Sarih, Arch. Math. (Brno), submitted for publication]. In this direction, we prove that if $$(TM,G)$$ is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then $$(M,g)$$ possesses the same property, respectively. We also give explicit examples of $$g$$-natural metrics of arbitrary constant scalar curvature on $$TM$$.

MSC:
 53B20 Local Riemannian geometry 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 53A55 Differential invariants (local theory), geometric objects 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
Full Text:
References:
 [1] K.M.T. Abbassi, Note on the classification theorems of g-natural metrics on the tangent bundle of a Riemannian manifold (M,g), Comment. Math. Univ. Carolinae, submitted for publication · Zbl 1097.53013 [2] Abbassi, K.M.T.; Sarih, M., Killing vector fields on tangent bundles with cheeger – gromoll metric, Tsukuba J. math., 27, 2, 295-306, (2003) · Zbl 1060.53019 [3] K.M.T. Abbassi, M. Sarih, On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. (Brno), submitted for publication · Zbl 1114.53015 [4] K.M.T. Abbassi, M. Sarih, The Levi-Civita connection of Riemannian natural metrics on the tangent bundle of an oriented Riemannian manifold, preprint · Zbl 1167.53304 [5] Fuglede, B., A criterion of non-vanishing differential of a smooth map, Bull. London math. soc., 14, 98-102, (1982) · Zbl 0491.53010 [6] S. Gudmundsson, The geometry of harmonic morphisms, Ph.D. Thesis in Pure Mathematics, University of Leeds, England, April 1992 · Zbl 0715.53029 [7] Gudmundsson, S.; Kappos, E., On the geometry of the tangent bundle with the cheeger – gromoll metric, Tokyo J. math., 14, 2, 407-717, (2002) [8] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Intersci. Pub. New York (I, 1963 and II, 1967) [9] Kolář, I.; Michor, P.W.; Slovák, J., Natural operations in differential geometry, (1993), Springer-Verlag Berlin · Zbl 0782.53013 [10] Kowalski, O., Curvature of the induced riemannian metric of the tangent bundle of Riemannian manifold, J. reine angew. math., 250, 124-129, (1971) · Zbl 0222.53044 [11] Kowalski, O.; Sekizawa, M., Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles—a classification, Bull. Tokyo gakugei univ. (4), 40, 1-29, (1988) · Zbl 0656.53021 [12] Krupka, D.; Janyška, J., Lectures on differential invariants, (1990), University J.E. Purkyně Brno · Zbl 0752.53004 [13] Mok, K.P.; Patterson, E.M.; Wong, Y.C., Structure of symmetric tensors of type $$(0, 2)$$ and tensors of type $$(1, 1)$$ on the tangent bundle, Trans. amer. math. soc., 234, 253-278, (1977) · Zbl 0363.53016 [14] Musso, E.; Tricerri, F., Riemannian metrics on tangent bundles, Ann. math. pura appl. (4), 150, 1-20, (1988) · Zbl 0658.53045 [15] Oproiu, V., Some new geometric structures on the tangent bundle, Publ. math. debrecen, 55, 261-281, (1999) · Zbl 0992.53053 [16] Oproiu, V., A locally symmetric Kähler Einstein structure on the tangent bundle of a space form, Beiträge zur algebra und geometrie/contributions to algebra and geometry, 40, 363-372, (1999) · Zbl 0951.53032 [17] Oproiu, V., A Kähler Einstein structure on the tangent bundle of a space form, Int. J. math. math. sci., 25, 183-195, (2001) · Zbl 0981.53063 [18] V. Oproiu, Some Kähler structures on the tangent bundle of a space form, preprint · Zbl 0934.53044 [19] Oproiu, V.; Papaghiuc, N., Locally symmetric space structures on the tangent bundle, (), 99-109 · Zbl 0960.53027 [20] Oproiu, V.; Papaghiuc, N., A locally symmetric Kähler Einstein structure on a tube in the tangent bundle of a space form, Rev. roum. math. pures appl., 45, 863-871, (2000) · Zbl 1004.53024 [21] Oproiu, V.; Papaghiuc, N., A Kähler structure on the nonzero tangent bundle of a space form, Differential geom. appl., 11, 1-12, (1999) · Zbl 0926.53014 [22] V. Oproiu, D.D. Poroşniuc, A Kähler Einstein structure on the cotangent bundle of a Riemannian manifold, An. Ştiinţ. Univ. Al. I. Cuza, Iaşi, submitted for publication · Zbl 1066.53064 [23] Papaghiuc, N., Another Kähler structure on the tangent bundle of a space form, Demonstratio math., 31, 855-866, (1998) · Zbl 0966.53016 [24] Sekizawa, M., Curvatures of tangent bundles with cheeger – gromoll metric, Tokyo J. math., 14, 2, 407-717, (1991) · Zbl 0768.53020 [25] Tahara, M.; Marchiafava, S.; Watanabe, Y., Quaterrnion Kähler structures on the tangent bundle of a complex space form, Rend. istit. mat. univ. trieste (suppl.), 30, 163-175, (1999) · Zbl 0969.53021 [26] Tahara, M.; Vanhecke, L.; Watanabe, Y., New structures on tangent bundles, Note di matematica (lecce), 18, 131-141, (1998) · Zbl 0964.53021 [27] Tahara, M.; Watanabe, Y., Natural almost Hermitian, Hermitian and Kähler metrics on the tangent bundles, Math. J. toyama univ., 20, 149-160, (1997) · Zbl 1076.53516 [28] Wong, Y.C.; Mok, K.P., Connections and M-tensors on the tangent bundle TM, (), 157-172
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.