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On pull-backs of the universal connection

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  10 December 2021

Kristopher Tapp
Kristopher Tapp*
Affiliation:
Department of Mathematics, Saint Joseph’s University, 5600 City Avenue Philadelphia, PA 19131, USA
*
e-mail: ktapp@sju.edu
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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).

Keywords

Universal connectionfat connectionpositive curvature

MSC classification

Primary: 53C20: Global Riemannian geometry, including pinching
Secondary: 53B15: Other connections
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 4 , December 2022 , pp. 845 - 859
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Copyright
© Canadian Mathematical Society, 2021

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