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Six-dimensional planar submanifolds of Cayley algebra equipped with almost Hermitian structures induced by Brown — Gray three-fold vector cross products in  are considered.

We select the case when the almost Hermitian structures on such six-dimensional planar submanifolds of Cayley algebra are Hermitian, i. e. these structures are integrable. We study almost contact metric structures on totally umbilical hypersurfaces in such six-dimensional Hermitian pla­nar submanifolds of the octave algebra.

We prove that if these almost contact metric structures on a totally umbilical hypersurface of a six-dimensional Hermitian planar submani­fold of Cayley algebra are quasi-Sasakian, then they are Sasakian.

We consider posed in 1960s Alfred Gray problem on the existence of a six-dimensional non-Kählerian almost Kählerian manifold.

We study six-dimensional almost Hermitian locally symmetric sub­manifolds of Ricci type of Cayley algebra (the notion of such six-dimensional submanifolds of the octave algebra was introduced by Vadim Feodorovich Kirichenko).

Our main result is the following: it is proved that a six-dimensional almost Hermitian locally symmetric submanifold of Ricci type of Cayley algebra does not admit a non-Kählerian almost Kählerian structure.

We continue to study the space of centered planes in projective space . In this paper, we use E. Cartan's method of external forms and the group-theoretical method of G. F. Laptev to study the space of centered planes of the same dimension. These methods are successfully applied in physics.

In a generalized bundle, a bilinear connection associated with a space is given. The connection object contains two simplest subtensors and subquasi-tensors (four simplest and three simple subquasi-tensors). The object field of this connection defines the objects of torsion, curvature-torsion, and curvature. The curvature tensor contains six simplest and four simple subtensors, and curvature-torsion tensor contains three simplest and two simple subtensors.

The canonical case of a generalized bilinear connection is considered.

The concept of a generalized nonholonomic Kenmotsu manifold is introduced. In contrast to the previously defined nonholonomic Kenmotsu manifold, the manifold studied in the article is an almost normal almost contact metric manifold of odd rank. The manifold is equipped with a metric connection with torsion, which is called the canonical connection in this work. The main properties of the canonical connection are studied. The canonical connection is an analogue of the generalized Tanaka-Webster connection. In this paper, we prove that the canonical connection is the only metric connection with torsion of a special structure that pre­serves the structural 1-form and the Reeb vector field. We study the in­trinsic geometry of a generalized nonholonomic Kenmotsu manifold equipped with a canonical connection. It is proved that if a generalized nonholonomic Kenmotsu manifold is an Einstein manifold with respect to a canonical connection, then it is Ricci-flat with respect to this connec­tion. An example of a generalized nonholonomic Kenmotsu manifold that is not a nonholonomic Kenmotsu manifold is given.

The study of hyperbands and their generalizations in spaces with dif­ferent fundamental groups is of great interest in connection with numer­ous applications in mathematics and physics. In this paper, we study a special class of hyperbands, i. e., centrally equipped hyperbands. A hy­perband Hm (m ≥ 2) is said to be centrally rigged if the rigging lines in the normals of the 1st kind of the base surface pass through one (the center of the rigging).

The article gives a task of a centrally equipped hyperband in the 1st order frame. A sequence of fundamental geometric objects of a hyperstrip with central framing is constructed. An existence theorem for a hyperband with a central framing is proved. It is proved that a hyperstrip with central framing and framing in the sense of Cartan induces a projective connec­tion obtained by projection, where the projection center at each point is the Cartan plane. The spans of the components of the curvature-torsion tensor of the constructed connection are found.

Smooth manifold with almost contact metric structure (i. e., almost contact metric manifold) was considered in this paper. We used a modern version of Cartan’s method of external forms to conduct our study. We assume that its Reeb vector field is affine motion. We got formulas for components of second covariant differential of contact form for an arbi­trary almost contact metric manifold. Criterion for affine motion of Reeb vector field has been obtained for arbitrary almost contact metric mani­fold in this paper. It is proved that if Reeb vector field of almost contact structure is affine motion then sixth structural tensor of almost contact metric structure is vanishing. It is proved that if Reeb vector field is affine motion and torse-forming vector field then Reeb vector field is Killing vector field. It is proved that if Reeb vector field of almost contact metric structure is torse-forming vector field and it is not Killing vector field then it is not affine motion.

A projective structure on a smooth manifold is a maximal atlas such that all its transition maps are the fractional linear transformations. Our aim is to interpret this notion in terms of the higher order frame bundles and their structure forms. It is shown that the projective structure gener­ates the sequence of differential geometric structures. The construction is following:

Step 1. For a smooth manifold the so-called quotient frame bundle as­sociated to the 2nd order frame bundle on the manifold is constructed.

Step 2. Given projective structure on the manifold, the mappings from the quotient frame bundle to the higher order frame bundles are con­structed. These mappings are the differential geometric structures.

Step 3. The pullbacks of the structure forms of the frame bundles via the mappings are considered. These are called structure forms of the pro­jective structure. The expressions of their exterior differentials in terms of the forms themselves are found. These expressions coincide with the structure equations of a projective space.

A model of linear fractional transformations of the complex plane in the form of points of the complex three-dimensional projective space without a linear “forbidden” quadric is presented. A model of real linear fractional transformations of the complex plane in the form of points of the real three-di­mensional projective space without a linear “forbidden” quadric is presented. A geometric separation of points corresponding to parabolic, hyperbolic and elliptic real linear fractional transformations by a “parabolic” cone touching the forbidden quadric is found. Some pro­per­ties of model points corresponding to real linear fractional transfor­ma­tions are found. Some properties of model points corresponding to fun­da­men­tal groups transformations of biconnected domains of the complex pla­ne are found.

This paper relates to differential geometry, and the research technique is based on G. F. Laptev’s method of extensions and envelopments, which generalizes E. Cartan’s method of moving frame and exterior forms. We consider a smooth m-dimensional manifold, its tangent and cotangent spaces, as well as the second-order frames and coframes on this manifold.

Using the perturbation of the exterior derivative and ordinary diffe­ren­tial, mappings are introduced that enable us to construct non-sym­met­rical second-order frames and coframes on a smooth manifold. It is shown that the extension of the second order tangent space to a smooth m-dimen­sional manifold is carried out by adding the vertical vectors to the linear frame bundle over the manifold to the second order tangent vectors to this manifold.

A deformed external differential is widely used, which is a differen­tial, i. e., its reapplication vanishes. We introduce a deformed external dif­ferential being a differential along the curves on the manifold, i. e., its re­peated application along the curves on the manifold gives zero.

The purpose of this paper is to prove of Liouville type theorems, i. e., theorems on the non-existence of Killing — Ric­ci and Codazzi — Ricci tensors on complete non-com­pact Riemannian manifolds. Our results complement the two classical vanishing theorems from the last chapter of fa­mous Besse’s monograph on Einstein manifolds.

Synectic extensions of complete lifts of linear connections in tangent bundles were introduced by A. P. Shirokov in the seventies of the last century [1; 2]. He established that these connections are linear and are real realizations of linear connections on first-order tangent bundles en­do­wed with a smooth structure over the algebra of dual numbers. He also pro­ved the existence of a smooth structure on tangent bundles of arbitrary or­der  on a smooth manifold M over the algebra  of plu­ral numbers. Studying holomorphic linear connections on  over an algebra , A. P. Shirokov obtained real realizations of these con­nec­tions, which he called Synectic extensions of a linear connection defi­ned on M. A natural generalization of the algebra of plural numbers is the A. Weyl algebra, and a generalization of the tangent bundle is the A. Weyl bundle. It was shown in [3] that a synectic extension of linear connections defined on M a smooth manifold can also be constructed on A. Weyl bundles , where is the A. Weyl algebra. The geometry of these bundles has been studied by many authors — A. Morimoto, V. V. Shu­rygin and others. A detailed analysis of these works can be found in [3].

In this paper, we study synectic lifts of linear connections defined on A. Weyl bundles.

Finding the conditions for the invariance of geometric objects under the action of transformation groups is one of the main objects of geomet­ric research. Almost Hermitian structures and structures of the Gray — Her­vella classification on smooth manifolds are considered in this paper. All arguments are given using invariant Koszul’s calculus. Conditions for the invariance of the Kähler form in type structures are investigated and it is shown that the Kähler form is covariantly constant with respect to the Lie vector field. Conditions for the invariance of the Riemannian metric under the action of a one-parameter group of diffeomorphisms generated by a Lie vector field are studied. A criterion for the invariance of an al­most complex structure with respect to the local group of diffeomor­phisms generated by the Lie vector field in the class W4 is proved.

Conditions for the invariance of an operator of an almost complex structure, a tensor of a Riemannian metric, are proved. It is established that the invariance of the Riemannian structure g implies the invariance of the operator of an almost complex structure for some class of manifolds according to the Gray — Hervella classification, and conditions for the covariant constancy of the Lie form in certain classes of manifolds of dimensions above four were obtained. It is proved that the Lie form is covariantly constant in some classes of the type of dimensions above four.

T. Brooke Benjamin and P. J. Olver “Hamiltonian structure, symmet­ries and conservation laws for water waves” study the behavior of Hamil­to­nian systems with an infinite-dimensional phase space. The methods of va­riational problems and infinite-dimensional differential geometry are applicable to this problem. A special case of the problem is an abstract prob­lem of hydrodynamics for an ideal fluid. Its configuration space is the group of volume-preserving diffeomorphisms of some manifold in  or  filled with fluid. Even more special is the problem of waves on water. Its non-standard nature is due to the presence of boundary con­di­tions on the free surface. These boundary conditions can be interpreted in terms of the functional derivatives of the energy integral, which plays the role of the Hamiltonian. Here we consider in detail the case of this prob­lem in R2, taking into account surface tension, and find symmetries for it, which was not considered in detail in the article. Finding symmet­ries can be achieved without recourse to the Hamiltonian structure of the gi­ven problem.

It is well-known Levi-Chivita’s construction of object for affine connection (in modern terminology — linear connection) by the field of non-degenerate metric on a smooth manifold. An inverse problem (a construction of metric by given linear connection) is solved ambiguously, besides, the metric may turn out to be degenerate and indefinite. On the one hand, two metrics differing in a sign are obviously build: by curvature tensor contractionwith subsequent symmetrization. Оn the other hand, Vranceanu’s metric is a double contraction of multiplication of a torsion tensor’s components. In this paper Levi-Chivita’s inverse problem is solved in other way using the field of connection object.

It is proved that in the general case, when the linear connection is not semi-symmetric, six metrics can be constructed. In the special case, when the linear connection is semi-symmetric (in particular, torsion-free), the constructed metrics vanish.

The investigation is done on a semi-holonomic smooth manifold by means of two prolongation its structure equations.