To geometry of integrable distributions in En
DOI:
https://doi.org/10.26577/JMMCS-2019-4-m7Keywords:
Λn ij is the tensor, Rij pq is the scalar curvature of the hyperdistribution, geodesic unidistributions of the line Δ(y)Abstract
The proposed work is devoted to the identification and study of multidimensional networks
constructively connected by distribution. In the initial approach to the selection of networks,
the vector of average distribution curvature is essentially used. Therefore, such a separation is
feasible only in metric spaces (in the work this question is studied in Euclidean n-space). The
article investigates the conditions for the existence of canonical distributions of planes belonging
to the tangent plane of the surface of Euclidean space. This article introduces the concept of
parallel site transfer along integral unidistribution curves. The statement is proved that vector
fields are collinear if and only if the geometric object has a zero curvature tensor.
Differential equations of unidistribution are derived and a necessary and sufficient condition is
found for the geodesic line of unidistribution to be flat, and conditions are found under which the
integral curves of unidistribution are curvature lines with respect to unidistribution. The statement
is proved that the line will be geodesic if and only if its main normals coincide with the normals of
the surface on which this line is located. Differential equations of a geodesic flat line are obtained.
References
inform. No 6 (1974): 189-205.
[2] Bazyilev V.T. "O V -sopryazhennyih setyah v prostranstve affinnoy svyaznosti [About V -adjoint networks in a space of
affine connection]" , Izv. vyissh. uchebn. zavedeniy. Matematika. No 5 (1974): 25-30.
[3] Bazyilev V.T. "O mnogomernyih setyah i ih preobrazovaniyah [About multidimensional networks and their
transformations]" , Tr. Geometr, seminara. Vses. in-t nauch. i tehn. inform. No 3 (1965): 138-164.
[4] Bazyilev V.T. "Ob odnom zamechatelnom klasse setey [On one great class of networks]" , Problemyi geometrii (Itogi nauki
i tehniki) No 7 (1975): 105-116.
[5] Ostianu N.M. "Raspredeleniya m-mernyih lineynyih elementov v proektivnoy svyaznosti [Distributions of m-dimensional
linear elements in projective connection]" , Trudyi geometricheskogo seminara. Vsesoyuznyiy institut nauchnoy i
tehnicheskoy informatsii Akkademii Nauk SSSR No 3 (1971): 40-47.
[6] Norden A.P. Prostranstva affinnoy svyaznosti [Spaces of affine connection] Izd. 2-e, ispravl. (M.: Nauka, 1976): 432.
[7] Finikov S.P. Metod vneshnih form Kartana v differentsialnoy geometrii [The method of external Cartan forms in
differential geometry] (Gostehizdat, 1948): 482.
[8] Tihonov V.A. "Ob odnom preobrazovanii ploskih setey, prisoedinennyih k giperraspredeleniyam v affinnom prostranstve
[About a transformation of planar networks connected to hyperdistributions in an affine space]" , Tezisyi dokl. 6-y Vses.
geometr, konf. po sovr. probl. geometrii. Vilnyus (1975): 238-239.
[9] Kuzmin M.K. "O kanonicheskih setyah raspredeleniy na poverhnostyah evklidova prostranstva [About canonical
distribution networks on the surfaces of Euclidean space]" , V sb. «Probl. geometrii (Itogi nauki i tehn. VINITI AN
SSSR)» Vol. 7 (1975): 231-248.
[10] Gudz L.P. "Ob odnom klasse ortogonalnyih setey na giperpoverhnosti chetyirehmernogo evklidova prostranstva [On a class
of orthogonal networks on hypersurfaces of a four-dimensional Euclidean space]" , Sbornik "Geometriya pogruzhennyih
mnogoobraziy" , M. (1972): 39-45.
[11] Ostianu N.M. "Differentsialno-geometricheskie strukturyi na differentsiruemyih mnogoobraziyah [Differential-geometric
structures on differentiable manifolds]" , Problemyi geometrii (Itogi nauki i tehniki) Vol. 8 (1976): 89-111.
[12] Tihonov V.A. "Seti, opredelyaemyie giperraspredeleniyami v affinnom prostranstve i ih obobscheniya [Networks defined
by hyperdistributions in affine space and their generalizations]" , Problemyi geometrii (Itogi nauki i tehniki) Vol. 8 (1976):
199-225.
[13] Eli Kartan. Rimanova geometriya v ortogonalnom repere [Riemannian geometry in the orthogonal frame] (Izdatelstvo
Moskovskogo universiteta, 1970).
[14] Vasilev A.M., Solovev Yu.P. Differentsialnaya geometriya [Differential geometry] (Izdatelstvo Moskovskogo universiteta,
1981).
