Metric structures on fibered manifolds through partitions of unity

Abstract

The notion of partitions of unity is extremely useful as it allows one to extend local constructions on Euclidean patches to global ones. It is widely used in many fields in mathematics. Therefore, prolongation of this useful tool to another manifold may help constructing many geometric structures. In this paper, we construct a partition of unity on a fiber bundle by using a given partition of unity on the base manifold. On the other hand we show that the converse is also possible if it is a vector bundle. As an application, we define a Riemannian metric on the fiber bundle by using induced partition of unity on the fiber bundle.

Keywords

• Partitions of Unity,fiber bundles

References

1. Dold, A., Partitions of Unity in the Theory of Fibrations, Annals of Math, 78 (2), pp: 223-255, 1963.
2. Lovett, S., Differential Geometry of Manifolds, AK Peters, Ltd., Natick, Massachusetts, 2010.
3. Lee, J. M., Introduction to Smooth Manifolds, Springer Science+Business, Newyork, 2003.
4. Melenk, J. M. and Babuska, I, The Partition of Unity Finite Element Method: Basic Theory and Applications, Computer Methods in Applied Mechanics and Engineering, 139 (No:1-4) 289-314, 1996.
5. Morimoto, A., Prolongations of G-Structures To Tangent Bundles, Nagoya Math. J., 12, pp: 67-108, 1968.
6. Pemantle, R. and Wilson, M.C, Asymptotic expansions of oscillatory integrals with complex phase, Contemporary Mathematics: 520, pp: 220- 240, 2010.
7. Richter, Christian, A chain of controllable partitions of unity on the cube and the approximation of Holder continuous functions, llinois J. Math. : 43 (1), pp: 159-191, 1999.
8. Saunders D.J., The Geometry of Jet Bundles, Cambridge University Press, Cambridge-New York, 1989.

9. Walschap, Gerard, Metric Structures in Differential Geometry, Springer-Verlag, New York, 2004.