Geodesic
A difference approximation of the covariant derivative and other operators and geometric objects given in a Riemannian domain
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 973-990.

Voir la notice de l'article provenant de la source Math-Net.Ru

Difference approximations for covariant derivatives of tensor fields of arbitrary rank given in Riemannian domain, retaining main geometrical properties, are suggested. An approach to its construction is based on certain difference analogs for Christoffel symbols. The main criterium here is exact vanishing for a difference covariant derivative of fundamental tensor and, in addition, an exact approximation of commutation relations which is possible only at a certain developed in the paper difference approximations for the curvature tensor.
Keywords: difference approximation, Riemannian domain, covariant derivative, tensor of curvature, Christoffel symbols
Mots-clés : Ricci formulae. 
@article{SEMR_2015_12_a65,
     author = {E. Yu. Derevtsov},
     title = {A difference approximation of the covariant derivative and other operators and geometric objects given in a {Riemannian} domain},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {973--990},
     publisher = {mathdoc},
     volume = {12},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2015_12_a65/}
}
TY  - JOUR
AU  - E. Yu. Derevtsov
TI  - A difference approximation of the covariant derivative and other operators and geometric objects given in a Riemannian domain
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2015
SP  - 973
EP  - 990
VL  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2015_12_a65/
LA  - ru
ID  - SEMR_2015_12_a65
ER  - 
%0 Journal Article
%A E. Yu. Derevtsov
%T A difference approximation of the covariant derivative and other operators and geometric objects given in a Riemannian domain
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2015
%P 973-990
%V 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2015_12_a65/
%G ru
%F SEMR_2015_12_a65
E. Yu. Derevtsov. A difference approximation of the covariant derivative and other operators and geometric objects given in a Riemannian domain. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 973-990. http://geodesic.mathdoc.fr/item/SEMR_2015_12_a65/

[1] A. N. Konovalov, “Numerical methods for static problems of elasticity”, Sib. Math. Journal, 36:3 (1995), 491–505 | DOI | MR | Zbl

[2] M. M. Karchevsky, S. N. Voloshanskaia, “On approximation of the strain tensor in curvilinear coordinates. The difference scheme for the problem on the equilibrum of elastic cylinder”, Izv. vuzov: Mathematics, 1977, no. 10, 70–80 (in Russian) | MR

[3] V. N. Timashev, “An approximation of differential operators in curvilinear coordinates”, Prikladnuie problemu prochnosti i plastichnosti, 6, Gorkyi, 1977, 47–54 (in Russian)

[4] N. V. Tsurikov, “On approximtion of covariant derivatives of vectors and tensors in arbitrary curvilinear coordinate system”, Variatsionnuie metodui v zadachakh chislennogo analiza, VC SO AN SSSR, Novosibirsk, 1986, 150–156 (in Russian) | Zbl

[5] V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, 1994 | MR

[6] N. E. Kochin, Vector Calculus and Fundamentals of Tensor Calculus, Gos. Tech.-Teor. Izd., L.–M., 1934 (in Russian)

[7] H. Weyl, “The method of orthogonal projection in potential theory”, Duke Math. J., 7 (1940), 411–444 | DOI | MR

[8] E. Yu. Derevtsov, V. V. Pikalov, “Reconstruction of vector fields and their singularities from ray transforms”, Numerical Analysis and Applications, 4:1 (2011), 21–35 | DOI | Zbl

[9] E. Yu. Derevtsov, “Some problems of non-scalar tomography”, Proceedings of the First International Cientific Conference and Young Scientists School “Theory and Computational Methods for Inverse and Ill-posed Problems”, Part I, Siberian Electronic Mathematical Reports, 7, 2010, 81–111 (in Russian)

[10] E. Yu. Derevtsov, Tomography of complecated media: models, methods, algorithms, v. II, Models of vector and tensor tomography, GASU, Gorno-Altaisk, 2010 (in Russian)

[11] M. A. Bezuglova, E. Yu. Derevtsov, S. B. Sorokin, “The reconstruction of a vector field by finite difference methods”, J. Inverse Ill-posed Problems, 10:2 (2002), 125–154 | DOI | MR | Zbl

[12] E. Yu. Derevtsov, “An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems”, J. Inverse Ill-Posed Problems, 13:3–6 (2005), 213–246 | DOI | MR | Zbl

[13] A. A. Samarsky, V. B. Andreev, Difference methods for elliptic equations, Nauka, M., 1976 (in Russian) | MR

[14] K. Jano, S. Bochner, Curvature and Betti numbers, Princeton University Press, Princeton, New Jersey, 1953 | MR

[15] B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry. Methods and Applications, v. 1, GTM, 93, Springer-Verlag, 1984 ; v. 2, GTM, 104, 1985 | MR | Zbl | MR | Zbl