Pullback of measures on Riemannian manifolds. (English) Zbl 1219.58003
The aim of this paper consists in connecting the different approaches for the pullback of distributions. Furthermore, one relates these approaches to the more general concept of single-layer distributions, which allows a generalization to pullbacks on submanifolds of Riemannian manifolds. In this paper, only the pullback of measures are studied. A general formula for the pullback of measures on submanifolds of Riemannian manifolds is derived. One proves the equivalence of three different approaches to distributions supported on submanifolds, originating from Gelfand and Shilov, and Friedlander, and a characterization using single-layer distributions, respectively. As a consequence, one presents an alternative formula for the pullback of an arbitrary distribution. Finally, one applies the obtained formulas to an illustrative example in order to give an intuitive appeal to these abstract concepts.
Reviewer: Kun Soo Chang (Seoul)
MSC:
| 58C35 | Integration on manifolds; measures on manifolds |
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
| 46F25 | Distributions on infinite-dimensional spaces |
Keywords:
distribution theory; differential geometry; pullback; Riemannian manifold; single-layer distributionsReferences:
| [1] | Abraham, R.; Marsden, J. E., Foundation of Mechanics (1978), Benjamin/Cummings: Benjamin/Cummings Reading · Zbl 0393.70001 |
| [2] | Dieudonné, J., Éléments dʼAnalyse, vol. III (1970), Gauthiers-Villars: Gauthiers-Villars Paris · Zbl 0208.31802 |
| [3] | Friedlander, F. G., Introduction to the Theory of Distributions (1982), Cambridge University Press · Zbl 0499.46020 |
| [4] | Gelfand, I. M.; Shilov, G. E., Generalized Functions, vol. I (1964), Academic Press · Zbl 0115.33101 |
| [5] | Hörmander, L., The Analysis of Linear Partial Differential Operators, vol. I: Distribution Theory and Fourier Analysis (1983), Springer: Springer Berlin · Zbl 0521.35001 |
| [6] | Li, C. K., Several products of distributions on manifolds, Novi Sad J. Math., 39, 1, 31-46 (2009) · Zbl 1284.46033 |
| [7] | de Rham, G., Variétés Différentiables (1973), Hermann: Hermann Paris · Zbl 0284.58001 |
| [8] | Schwartz, L., Théorie des Distributions (1966), Hermann: Hermann Paris · Zbl 0149.09501 |
| [9] | R.T. Seeley, Distributions on surfaces, Report T.W. 78, Math. Centre Amsterdam, 1962.; R.T. Seeley, Distributions on surfaces, Report T.W. 78, Math. Centre Amsterdam, 1962. |
| [10] | Shilov, G. E., Linear Algebra (1977), Dover Publications · Zbl 0218.15003 |
| [11] | Wagner, P., Distributions supported by hypersurfaces, Appl. Anal., 89, 8, 1183-1199 (2010) · Zbl 1201.58001 |
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