Divergence of multivector fields on infinite-dimensional manifolds

Yu.  Bogdanskii; V.  Shram

doi:10.37863/umzh.v74i12.6522

Yu. Bogdanskii Nat. Techn. Univ. Ukraine ``Igor Sikorsky Kyiv Polytechnic Institute’’
V. Shram Univ. Bonn, Germany

DOI: https://doi.org/10.37863/umzh.v74i12.6522

Keywords: Banach manifold, Radon measure, multivector field, divergence, surface measure

Abstract

UDC 514.763.2+515.164.17

We study the divergence of multivector fields on Banach manifolds with a Radon measure. We propose an infinite-dimensional version of divergence consistent with the classical divergence from finite-dimensional differential geometry. We then transfer certain natural properties of the divergence operator to the infinite-dimensional setting. Finally, we study the relation between the divergence operator ${\rm div}_M$ on a manifold $M$ and the divergence operator ${\rm div}_S$ on a submanifold $S \subset M.$

References

Yu. Bogdanskii, Banach manifolds with bounded structure and Gauss–Ostrogradskii formula, Ukr. Math. J., 64, № 10, 1299–1313 (2012). DOI: https://doi.org/10.1007/s11253-013-0730-9

Yu. Bogdanskii, Stokes' formula on Banach manifolds, Ukr. Math. J., 72, № 11, 1455–1468 (2020). DOI: https://doi.org/10.37863/umzh.v72i11.2295

Yu. Bogdanskii, E. Moravetskaya, Surface measures on Banach manifolds with uniform structure, Ukr. Math. J., 69, № 8, 1030–1048 (2017). DOI: https://doi.org/10.1007/s11253-017-1425-4

Yu. Bogdanskii, A. Potapenko, Laplacian with respect to a measure on a Riemannian manifold and Dirichlet problem.~I, Ukr. Math. J., 68, № 7, 897–907 (2016). DOI: https://doi.org/10.1007/s11253-016-1274-6

N. Bourbaki, Éléments de mathématique. Algèbre, chap.~I-III, Hermann (1970); chap.~IV-VII, Masson (1981); chap.~VIII-X, Springer (2007, 2012).

N. Broojerdian, E. Peyghan, A. Heydari, Differentiation along multivector fields, Iran. J. Math. Sci. and Inform., 6, № 1, 79–96 (2011); DOI 10.7508/ijmsi.2011.01.007.

Yu. Daletskii, Ya. Belopolskaya, Stochastic equations and differential geometry, Vyshcha Shkola, Kyiv (1989).

Yu. Daletskii, B. Maryanin, Smooth measures on infinite-dimensional manifolds, Dokl. Akad. Nauk SSSR, 285, № 6, 1297–1300 (1985).

D. H. Fremlin, Measurable functions and almost continuous functions, Manuscripta Math., 33, № 3-4, 387–405 (1981); DOI 10.1007/BF01798235. DOI: https://doi.org/10.1007/BF01798235

R. Fry, S. McManus, Smooth bump functions and the geometry of Banach spaces: a brief survey, Expo. Math., 20, № 2, 143–183 (2002); DOI 10.1016/S0723-0869(02)80017-2. DOI: https://doi.org/10.1016/S0723-0869(02)80017-2

S. Lang, Fundamentals of differential geometry, Springer-Verlag, New York etc. (1999). DOI: https://doi.org/10.1007/978-1-4612-0541-8

C.-M. Marle, Schouten–Nijenhuis bracket and interior products, J. Geom. and Phys., 23, № 3-4, 350–359 (1997); DOI 10.1016/S0393-0440(97)80009-5. DOI: https://doi.org/10.1016/S0393-0440(97)80009-5