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On the Pontrjagin classes of spray manifolds

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  11 December 2023

Zhongmin Shen  and
Runzhong Zhao Open the ORCID record for Runzhong Zhao [Opens in a new window]
Zhongmin Shen
Affiliation:
Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, Indianapolis, IN 46202-3216, United States e-mail: zshen@math.iupui.edu
Runzhong Zhao*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
*
e-mail: runzzhao@xmu.edu.cn
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Abstract

The characterization of projectively flat Finsler metrics on an open subset in $R^n$ is the Hilbert’s fourth problem in the regular case. Locally projectively flat Finsler manifolds form an important class of Finsler manifolds. Every Finsler metric induces a spray on the manifold via geodesics. Therefore, it is a natural problem to investigate the geometric and topological properties of manifolds equipped with a spray. In this paper, we study the Pontrjagin classes of a manifold equipped with a locally projectively flat spray and show that such manifold must have zero Pontrjagin classes.

Keywords

spray geometryPontrjagin classprojectively flat spraysDouglas sprays

MSC classification

Primary: 53C60: Finsler spaces and generalizations (areal metrics) 53B40: Finsler spaces and generalizations (areal metrics)

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 3 , September 2024 , pp. 544 - 553
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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