Abstract
We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle \(E\longrightarrow M\), over a Riemannian manifold M, when E is endowed with a metric connection. The tangent bundle of E admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of E; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant–Salamon type \({\mathrm {G}_{2}}\) manifolds.
Similar content being viewed by others
Notes
These interactions are important, specially for the above problem when we think of the pseudo-Riemannian case. However, even for this situation, for the weighted Sasaki pseudo-Riemannian structures, defined by obvious sign change in (6), we believe the metric completeness of the base manifold is still the sufficient condition, with the same arguments as above.
Indeed, after reading both articles, the present author does not find the significant reason for this attribution and he seems not to be the only; in [20] the choice is referred as a matter of inspiration.
We remark that the structures of generalized Sasaki type with weight functions dependent of the base point \(x\in M\), rather than the squared-radius r, have been studied by the author in [5].
One may say the close relations between metric and complex structures start with the twist \(\varphi _1,\varphi _2\mapsto \psi ,\overline{\psi }\).
Everywhere possible, we omit the ± which is attached to each 2-form and vector bundle.
The \(e^i,\ i=1,2,3\), have norm 2 for the usual metric on 2-forms, but indeed it is \(\frac{1}{2}\) of this that is used in [7]. In particular, the notation here for r refers to the half squared-radius r mentioned there.
References
Abbassi, M.T.K.: Note on the classification theorems of \(g\)-natural metrics on the tangent bundle of a Riemannian manifold \((M, g)\). Comment. Math. Univ. Carolinae 45(4), 591–596 (2004)
Abbassi, M.T.K., Calvaruso, G.: \(g\)-natural contact metrics on unit tangent sphere bundles. Monatshefte für Math. 151, 89–109 (2006)
Abbassi, M.T.K., Sarih, M.: On natural metrics on tangent bundles of Riemannian manifolds, Archivum Mathematicum (Brno). Tomus 41, 71–92 (2005)
Albuquerque, R.: Hermitian and quaternionic Hermitian structures on tangent bundles. Geometriae Dedicata 133(1), 95–110 (2008)
Albuquerque, R.: Weighted metrics on tangent sphere bundles. Q. J. Math. 63(2), 259–273 (2012)
Albuquerque, R.: Homotheties and topology of tangent sphere bundles. J. Geom. 105(2), 327–342 (2014)
Albuquerque, R.: Self-duality and associated parallel or cocalibrated \({\text{G}}_{2}\) structures. http://arxiv.org/abs/1401.7314
Albuquerque, R.: An invariant Kähler metric on the tangent disk bundle of a space-form. http://arxiv.org/abs/1609.03125
Belegradek, I., Wei, G.: Metrics of positive Ricci curvature on vector bundles over nimanifolds. Geom. Funct. Anal. 12(7), 56–72 (2002)
Benyounes, M., Loubeau, E., Wood, C.M.: Harmonic sections of Riemannian bundles and metrics of Cheeger–Gromoll type. Differ. Geom. Appl. 25, 322–334 (2007)
Benyounes, M., Loubeau, E., Wood, C.M.: The geometry of generalized Cheeger–Gromoll metrics. Tokyo J. Math. 32, 287–312 (2009)
Bérard Bergery, L.: Quelques exemples de variétés riemanniennes complètes non compactes à courbure de Ricci positive. C. R. Acad. Sci. Paris Sér. I Math. 302(4), 159–161 (1986). http://gallica.bnf.fr/ark:/12148/bpt6k5744719j/f26.image
Berndt, J., Boeckx, E., Nagy, P., Vanhecke, L.: Geodesics on the unit tangent bundle. Proc. R. Soc. Edinburgh Sect. A 133, 1209–1229 (2003)
Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)
Bryant, R.L.: Metrics with exceptional holonomy. Ann. Math. (2) 126(3), 525–576 (1987)
Bryant, R.L., Salamon, S.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58(3), 829–850 (1989)
Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 96, 413–443 (1972)
Gibbons, G.W., Page, D.N., Pope, C.N.: Einstein metrics on \(S^3,\;{{{\mathbb{R}}}}^3\) and \({{{\mathbb{R}}}}^4\) bundles. Commun. Math. Phys. 127(3), 529–553 (1990)
Joyce, D.: Riemannian Holonomy Groups and Calibrated Geometry. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (2009)
Kazimova, S., Salimov, A.A.: Geodesics of the Cheeger–Gromoll metric. Turk. J. Math. 33, 99–105 (2009)
Kobayashi, S.: Differential Geometry of Complex Vector Bundles, Kanô Memorial Lectures 5. Publishers and Princeton University Press, Iwanami Shoten (1987)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1, 2. Wiley Classics Library (1996)
Kowalski, O.: Curvature of the induced Riemannian metric of the tangent bundle of a Riemannian manifold. J. Reine Angew. Math. 250, 124–129 (1971)
Kozlowski, W., Niedzialomski, K.: Differential as a harmonic morphism with respect to Cheeger–Gromoll-type metrics. Ann. Global Anal. Geom. 37, 327–337 (2010)
O’Brian, N., Rawnsley, J.: Twistor spaces. Ann. Global Anal. Geom. 3(1), 29–58 (1985)
Munteanu, M.I.: Some aspects on the geometry of the tangent bundles and tangent sphere bundles of a riemannian manifold. Mediter. J. Math. 5, 43–59 (2008)
Murad, Ö., Walschap, G.: Vector bundles with no soul. Proc. Am. Math. Soc. 120(2), 565–567 (1994)
Musso, E., Tricerri, F.: Riemannian metrics on tangent bundles. Annali Mate. Pura ed Appl. 150(4), 1–20 (1988)
Sekizawa, M.: Curvatures of tangent bundles with Cheeger–Gromoll metric. Tokyo J. Math. 14, 407–417 (1991)
Tahara, M., Vanhecke, L., Watanabe, Y.: New structures on tangent bundles. Note Mat. 18(1), 131–141 (1998)
Tapp, K.: Conditions for nonnegative curvature on vector bundles and sphere bundles. Duke Math. J. 116(1), 77–101 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA Grant Agreement No. PIEF-GA-2012-332209.
Rights and permissions
About this article
Cite this article
Albuquerque, R. On vector bundle manifolds with spherically symmetric metrics. Ann Glob Anal Geom 51, 129–154 (2017). https://doi.org/10.1007/s10455-016-9528-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-016-9528-y