No CrossRef data available.
Article contents
On pull-backs of the universal connection
Published online by Cambridge University Press: 10 December 2021
Abstract
Narasihman and Ramanan proved in [Amer. J. Math. 83(1961), 563–572] that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to relate geometric properties of the classifying map to geometric properties of the pulled-back connection. More specifically, we describe conditions on the classifying map under which the pulled-back connection: (1) is fat (in the sphere bundle), (2) has a parallel curvature tensor, and (3) induces a connection metric with nonnegative sectional curvature on the vector bundle (or positive sectional curvature on the sphere bundle).
MSC classification
Secondary:
53B15: Other connections
- Type
- Article
- Information
- Copyright
- © Canadian Mathematical Society, 2021
References
Calabi, E.,
Isometric imbeddings of complex manifolds. Ann. of Math. (2) 58(1953), 1–23.CrossRefGoogle Scholar
Chen, B., Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Leuvain, 1990.Google Scholar
Derdzinski, A. and Rigas, A.,
Unflat connections in the
$3$
-sphere bundles over
${S}^4$
. Trans. Amer. Math. Soc. 265(1981), 485–495.Google Scholar
$3$
-sphere bundles over
${S}^4$
. Trans. Amer. Math. Soc. 265(1981), 485–495.Google ScholarDi Scala, A. J., Ishi, H., and Loi, A., Kähler immersions of Homogeneous Kähler manifolds into complex space forms. Preprint, 2010. Asian J. Math. 16(2012), no. 3, 479–487 arXiv:1009.4045
Google Scholar
Guijarro, L., Sadun, L., and Walschap, G.,
Parallel connections over symmetric spaces. J. Geom. Anal. 11(2001), no. 2, 265–281.CrossRefGoogle Scholar
Maeda, S., Ohnita, Y., and Udagawa, S.,
On slant immersions into Kähler manifolds. Kodai Math J. 16(1993), 205–219.CrossRefGoogle Scholar
Nakagawa, H. and Takagi, R.,
On locally symmetric Kähler submanifolds in a complex projective space. J. Math. Soc. Japan 28(1976), no. 4, 638–667.CrossRefGoogle Scholar
Narasihman, M. S. and Ramanan, S.,
Existence of universal connections. Amer. J. Math. 83(1961), 563–572.CrossRefGoogle Scholar
Rigas, A., Geodesic spheres as generators of the homotopy groups of
$O$
,
$\mathrm{BO}$
. J. Differential Geom. 13(1978), 527–545.CrossRefGoogle Scholar
$O$
,
$\mathrm{BO}$
. J. Differential Geom. 13(1978), 527–545.CrossRefGoogle ScholarRos, A.,
A characterization of seven compact Kähler submanifolds by holomorphic pinching, Ann. of Math. (2) 121(1985), 377–382.CrossRefGoogle Scholar
Ros, A. and Verstraelen, L.,
On a conjecture of Ogiue. J. Differential Geom. 19(1984) 561–566.CrossRefGoogle Scholar
Shankar, K., Tapp, K., and Tuschmann, W.,
Nonnegatively and positively curved invariant metrics on circle bundles. Proc. Amer. Math. Soc. 133(2005), no. 8, 2449–2459.CrossRefGoogle Scholar
Shen, Y.,
On compact submanifolds in
${\mathbb{CP}}^{n+p}$
with nonnegative sectional curvature. Proc. Amer. Math Soc. 123(1995), no. 11, 3507–3512.Google Scholar
${\mathbb{CP}}^{n+p}$
with nonnegative sectional curvature. Proc. Amer. Math Soc. 123(1995), no. 11, 3507–3512.Google ScholarStrake, M. and Walschap, G.,
Connection metrics of nonnegative curvature on vector bundles. Manuscripta Math. 66(1990), 309–318.CrossRefGoogle Scholar
Tapp, K.,
Conditions of nonnegative curvature on vector bundles and sphere bundles. Duke Math. J. 116(2003), no. 1, 77–101.CrossRefGoogle Scholar
Weinstein, A.,
Fat bundles and symplectic manifolds. Adv. Math. 37(1980), 239–250.CrossRefGoogle Scholar
Ziller, W., Fatness Revisited, Lecture Notes, 1999.Google Scholar