Abstract: Helmholtz decomposition of a vector field on potential and solenoidal parts is well known [1]. The decomposition is much more natural from physical and geometric point of view then representations through the components over coordinate systems in Euclidean space. A profound generalization of Helmholtz decomposition to a case of the symmetric tensor fields, given in a compact Riemannian manifold, was suggested in [2]. The structure, representation through potentials and detailed decomposition for 2D and 3D symmetric m-tensor fields in the case of the Euclidean metric was established in [3, 4]. In the case of the Riemannian metric similar results are known partially only for vector fields. We investigate here the properties of the solenoidal vector and 2-tensor fields given in the Riemannian domain with conformal metric and established the connections between the fields and the metric. The solenoidal fields of the general type and their partial case of toroidal fields is considered. This work was partially supported by RFBR according to the research project RFBR-DFG No. 19-51-12008. References 1. Weyl H. The method of orthogonal projection in potential theory // Duke Math. J. 1940. V. 7. P. 411-444. 2. Sharafutdinov V.A. Integral Geometry of Tensor Fields. Utrecht: VSP, 1994. 3. Derevtsov E.Yu., Svetov I.E. Tomography of tensor fields in the plane // Eurasian J. Math. Comp. Appl. 2015. V. 3, N. 2. P. 24-68. 4. Svetov I.E., Polyakova A.P. A detailed decomposition of 3D tensor fields. International Conference "Marchuk Scientific Readings 2020" (MSR-2020), dedicated to the 95th anniversary of the birthday of Academician Guri I. Marchuk October 19 - 23, 2020, Novosibirsk, Russia. Abstracts. P. 133.

Cite: Derevtsov E.Y.
Properties of solenoidal vector and 2-tensor fields given in domains with conformal Riemannian metric
In compilation International Conference «Marchuk Scientific Readings 2020» (MSR-2020), dedicated to the 95th anniversary of the birthday of RAS Academician Guri. I. Marchuk October 19 - 23, 2020, Akademgorodok, Novosibirsk, Russia. 2021. – Т.1715. – C.135. DOI: 10.24412/CL-35064-2021-189

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