2020 №51

Six-dimensional submanifolds of Cayley algebra equipped with an almost Hermitian structure of class W1 W2 W4 defined by means of three-fold vector cross products are considered. As it is known, the class W1 W2 W4 contains all Kählerian, nearly Kählerian, almost Kählerian, locally conformal Kählerian, quasi-Kählerian and Vaisman — Gray manifolds. The Cartan structural equations of the W1 W2 W4 -structure on such six-dimensional submanifolds of the octave algebra are obtained. A criterion in terms of the configuration tensor for an arbitrary almost Hermitian structure on a six-dimensional submanifold of Cayley algebra to belong to the W1 W2 W4 -class is established. It is proved that if a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the quasi-Sasakian hypersurfaces axiom (i.e. a hypersurface with a quasi-Sasakian structure passes through every point of such submanifold), then it is an almost Kählerian manifold. It is also proved that a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the eta-quasi-umbilical quasi-Sasakian hypersurfaces axiom, then it is a Kählerian manifold.

The properties of almost Hermitian manifolds belonging to the Gray — Hervella class W4 are considered. The almost Hermitian manifolds of this class were studied by such outstanding geometers like Alfred Gray, Izu Vaisman, and Vadim Feodorovich Kirichenko. Using the Cartan structural equations of an almost contact metric structure induced on an arbitrary oriented hypersurface of a W4-manifold, some results on totally umbilical and totally geodesic hypersurfaces of W4-manifolds are presented. It is proved that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a W4-manifold is either homothetic to a Sasakian structure or cosymplectic. Moreover, the quasi-Sasakian structure is cosymplectic if and only if the hypersurface is a to­tally geodesic submanifold of the considered W4-manifold. From the present result it immediately follows that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a locally confor­mal Kählerian (LCK-) manifold also is either homothetic to a Sasakian structure or cosymplectic.

We consider a surface as a variety of centered planes in a multidi­mensional projective space. A fiber bundle of the linear coframes appears over this manifold. It is important to emphasize the fiber bundle is not the principal bundle. We called it a glued bundle of the linear coframes. A connection is set by the Laptev — Lumiste method in the fiber bundle. The ifferential equations of the connection object components have been found. This leads to a space of the glued linear connection. The expres­sions for a curvature object of the given connection are found in the pa­per. The theorem is proved that the curvature object is a tensor. A condi­tion is found under which the space of the glued linear connection turns into the space of Cartan projective connection. The study uses the Cartan — Laptev method, which is based on cal­culating external differential forms. Moreover, all considerations in the article have a local manner.

The space  of centered planes is considered in the Cartan projec­ti­ve connection space . The space  is important because it has con­nec­tion with the Grassmann manifold, which plays an important role in geometry and topology, since it is the basic space of a universal vector bundle.
The space  is an n-dimensional differentiable manifold  with each point of which an n-dimensional projective space  containing this point is associated. Thus, the manifold  is the base, and the space  is the n-dimensional fiber “glued” to the points of the base.
A projective space  is a quotient space  of a linear space  with respect to the equivalence (collinearity) of non-zero vectors, that is . The projective space  is a manifold of di­men­sion n.
In this paper we use the Laptev — Lumiste invariant analytical meth­od of differential geometric studies of the space  of centered planes and introduce a fundamental-group connection in the associated bundle . The bundle  contains four quotient bundles. It is show that the connection object  is a quasi-tensor containing four subquasi-tensors that define connections in the corresponding quotient bundles.

Let M be an almost contact metric manifold of dimension n = 2m + 1. The distribution D of the manifold M admits a natural structure of a smooth manifold of dimension n = 4m + 1. On the manifold M, is defined a linear connection  that preserves the distribution D; this connection is determined by the interior connection that allows parallel transport of admissible vectors along admissible curves. The assigment of the linear connection  is equivalent to the assignment of a Riemannian metric of the Sasaki type on the distribution D. Certain tensor field of type (1,1) on D defines a so-called prolonged almost contact metric structure. Each section  of the distribution D defines a morphism  of smooth manifolds. It is proved that if a semi-invariant sub­manifold of the manifold M and  is a covariantly constant vec­tor field with respect to the N-connection , then  is a semi-invariant submanifold of the manifold D with respect to the prolonged almost contact metric structure.

The hypercentered planes family, whose dimension coincides with dimension of generating plane, is considered in the projective space. Two principal fiber bundles arise over it. Typical fiber for one of them is the stationarity subgroup for hypercentered plane, for other — the linear group operating in each tangent space to the manifold. The latter bundle is called the principal bundle of linear coframes. The structural forms of two bundles are related by equations.
It is proved that hypercentered planes family is a holonomic smooth manifold.
In the principal bundle of linear coframes the coaffine connection is given. From the differential equations it follows that the coaffine connec­tion object forms quasipseudotensor. It is proved that the curvature and torsion objects for the coaffine connection in the linear coframes bundle form pseudotensors

On a sub-Riemannian manifold M of contact type, is considered an N-connection  defined by the pair (, N), where  is an interior metric connection,  is an endomorphism of the distribution D. It is proved that there exists a unique N-connection  such that its torsion is skew-symmetric as a contravariant tensor field. A construction of the endomorphism corresponding to such connection is found. The sufficient conditions for the obtained connection to be a metric connec­tion with parallel torsion are given.

In the projective space, we continue to study a hypersurface with three strongly mutual distributions. For equipping distributions of a hypersurface, normalization in the sense of Norden — Chakmazyan is introduced internally. The distribution of the equipping planes is normal­ized in the sense of Norden — Chakmazyan if the fields of normals of the 1st kind and normals of the 2nd kind are attached to it in an invariant way. For each equipping distribution, the fields of normals of the 1st and 2nd kind are defined by the corresponding fields of quasitensors. At each point of the hypersurface, the normal of the 1st kind of the equipping dis­tribution of the hypersurface passes through the characteristic of the tan­gent hypersurface. This characteristic was obtained with displacements of the point along the integral curves of the equipping distribution.
For equipping distributions, the coverage of quasitensors is found un­der which the conditions of invariance of the normals of the 1st and 2nd kind are satisfied.
Coverage of quasitensors is found for which normalization in the sense of Norden — Chakmazyan is attached to the equipping distributions of the hypersurface in a second-order differential neighborhood.

In three-dimensional equiaffine space, we consider a differentiable map generated by complexes with three-parameter families of elliptic paraboloids according to the method proposed by the author in the mate­rials of the international scientific conference on geometry and applica­tions in Bulgaria in 1986, as well as in works published earlier in the sci­entific collection of Differ. Geom. Mnogoobr. Figur. The study is carried out in the canonical frame, the vertex of which coincides with the top of the generating element of the manifold, the first two coordinate vectors are conjugate and lie in the tangent plane of the elliptic paraboloid at its vertex, the third coordinate vector is directed along the main diameter of the generating element so that the ends are, respectively, the sums of the first and third, and also the sums of the second and third coordinate vec­tors lay on a paraboloid, while the indicatrixes of all three coordinate vec­tors describe lines with tangents, parallel to the first coordinate vector. The existence theorem of the mapping under study is proved, according to which it exists and is determined with the arbitrariness of one function of one argument. The systems of equations of the indicatrix and the main directions of the mapping under consideration are obtained. The indicatrix and the cone of the main directions of the indicated mapping are geomet­rically characterized.

It is shown how to define one or several prime numbers following af­ter given prime number without using computer only by calculating sev­eral arithmetic progressions. Five examples of finding such prime num­bers are given.

The linear frame bundle over a smooth manifold is considered. The mapping dе defined by the differentials of the first-order frame e is a lift to the normal N, i. e., a space complementing the first-order tangent space to the second-order tangent space to this bundle. In particular, the map­ping defined by the differentials of the vertical vector of this frame is a vertical lift into normal N.
The lift dе allows us to construct a prolongation both of the tangent space and its vertical subspace into the second-order tangent space, more precisely into the normal N. The normal lift dе defines the normal prolon­gation of the tangent space (i. e. the normal N) and its vertical subspace. The vertical lift defines the vertical prolongation of the tangent space and its vertical subspace. The differential of an arbitrary vector field on the linear frame bundle is a complete lift from the first-order tangent space to the second-order tangent space to this bundle.
It is known that the action of vector fields as differential operators on functions coincides with action of the differentials of these functions as 1-forms on these vector fields. Horizontal vectors played a dual role in the fibre bundle. On the one hand, the basic horizontal vectors serve as opera­tors for the covariant differentiation of geometric objects in the bundle. On the other hand, the differentials of these geometric objects can be con­sidered as forms (including tangential-valued ones) and their values on basic horizontal vectors give covariant derivatives of these geometric ob­jects.
For objects which covariant derivatives require the second-order con­nection, the covariant derivatives are equal to the values of the differen­tials of these objects on horizontal vectors in prolonged affine connectivi­ty. Prolongations of the basic horizontal vectors, i. e., the second-order horizontal vectors for prolonged connection, were constructed. The sec­ond-order tangent space is represented as a straight sum of the first-order tangent space, vertical prolongations of the vertical and horizontal sub­spaces, and horizontal prolongation of the horizontal subspace.

A compiled hyperplane distribution  is considered in an n-dimensional projective space . We will briefly call it a -distribution. Note that the plane L(A) is the distribution characteristic obtained by displacement in the center belonging to the L-subbundle. The following results were obtained:
a) The existence theorem is proved: -distribution exists with arbitrary (3n – 5) functions of n arguments.
b) A focal manifold  is constructed in the normal plane  of the 1st kind of L-subbundle. It was obtained by shifting the cen­ter A along the curves belonging to the L-distribution. A focal manifold  is also given, which is an analog of the Koenigs plane for the distribution pair (L, L).
c) It is shown that a framed -distribution in the 1st kind normal field of H-distribution induces tangent    and  normal bundles.
d) Six connection theorems induced by a framed -distri­bu­tion in these bundles are proved.
In each of the bundles ,  the framed -distribution induces an intrin­sic torsion-free affine connection in the tangent bundle and a centro-affine connection in the corresponding normal bundle.
e) In each of the bundles (d) in the differential neighborhood of the 2nd order, the covers of 2-forms of curvature and curvature tensors of the corresponding connections are constructed.

Conformal Killing form is a natural generalization of con­formal Killing vector field. These forms were exten­si­vely studied by many geometricians. These considerations we­re motivated by existence of various applications for the­se forms. The vector space of conformal Killing p-forms on an n-dimensional  closed Riemannian mani­fold M has a finite dimension  na­med the Tachibana number. These numbers are conformal scalar invariant of M and satisfy the duality theorem: .

In the present article we prove two vanishing theorems. According to the first theorem, there are no nonzero Tachi­bana numbers on an n-dimensional  closed Rie­mannian manifold with pinched negative sectional curva­ture such that  for some pinching con­stant . From the second theorem we conc­lude that there are no nonzero Tachibana numbers on a three-dimensional closed Riemannian manifold with ne­gative sectional curvature.

Quasi-hyperbolic spaces are projective spaces with decaying abso­lute. This work is a continuation of the author's work [7], in which surfac­es in one of these spaces are examined by methods of external forms and a moving frame. The semi-Chebyshev and Chebyshev net­works of lines on the surface in quasi-hyperbolic space  are considered. In this pa­per we use the definition of parallel transfer adopted in [6]. By analogy with Euclidean geometry, the semi-Chebyshev network of lines on the surface is the network in which the tangents to the lines of one family are carried parallel along the lines of another family. A Che­byshev network is a network in which tangents to the lines of each family are carried parallel along the lines of another family.
We proved three theorems. In Theorem 1, we obtain a natural equa­tion for non-geodesic lines that are part of a conjugate semi-Chebyshev network on the surface so that tangents to lines of another family are transferred in parallel along them. In Theorem 2, the natural equation of non-geodesic lines in the Chebyshev network is obtained. In Theorem 3 we prove that conjugate Chebyshev networks, one family of which is nei­ther geodesic lines, nor Euclidean sections, exist on surfaces with the lati­tude of four functions of one argument.

The work is devoted to the study of the Bianchi transform for surfac­es of revolution of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, the pseudosphere (Beltrami surface). The study of surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudospherical surfaces with the theory of networks, with the theory of solitons, with nonlinear differential equations and sin-Gordon equations is established. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transform for the Minding top is constructed. Using a mathematical package, Minding's top and its Bianchi transform are constructed.

In this paper, we study the classification of three-dimensional Lie al­gebras over a field of complex numbers up to isomorphism. The proposed classification is based on the consideration of objects invariant with re­spect to isomorphism, namely such quantities as the derivative of a subal­gebra and the center of a Lie algebra. The above classification is distin­guished from others by a more detailed and simple presentation.
Any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras. Six classes of three-dimensional Lie algebras not isomorphic to each other over a field of complex numbers are presented. In each of the classes, its properties are described, as well as structural equations defining each of the Lie alge­bras. One of the reasons for considering these low dimensional Lie alge­bras that they often occur as subalgebras of large Lie algebras

We consider a space with Laptev's fundamental group connection generalizing spaces with Cartan connections. Laptev structural equations are reduced to a simpler form. The continuation of the given structural equations made it possible to find differential comparisons for the coeffi­cients in these equations. It is proved that one part of these coefficients forms a tensor, and the other part forms is quasitensor, which justifies the name quasitensor of torsion-curvature for the entire set. From differential congruences for the components of this quasitensor, congruences are ob­tained for the components of the Laptev curvature-torsion tensor, which contains 9 subtensors included in the unreduced structural equations.
In two special cases, a space with a fundamental connection is a spa­ce with a Cartan connection, having a quasitensor of torsion-curvature, which contains a quasitensor of torsion. In the reductive case, the space of the Cartan connection is turned into such a principal bundle with connec­tion that has not only a curvature tensor, but also a torsion tensor.