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We study a $$D$$-dimensional Einstein–Gauss–Bonnet gravitational model including the Gauss-Bonnet term and the cosmological constant $$\Lambda$$. We find a class of cosmological type solutions with exponential dependence of two scale factors on the variable $$u$$ (either cosmological time or a spatial coordinate), governed by two Hubble-like parameters $$H\neq 0$$ and $$h$$, corresponding to factor spaces of dimensions $$m>2$$ and $$l>2$$, respectively, and depending on the sign parameter $$\varepsilon=\pm 1$$ ($$\varepsilon=1$$ corresponds to cosmological solutions and $$\varepsilon=-1$$ to static ones). These solutions are governed by a certain master equation $$\Lambda\alpha=\lambda(x)$$ and the restriction $$\alpha\varepsilon(x-x_{+})(x-x_{-})

A ( m + 3 ) -dimensional Einstein–Gauss–Bonnet gravitational model including the Gauss–Bonnet term and the cosmological term Λ is considered. Exact solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters H > 0 and h ≠ H , corresponding to factor spaces of dimensions m > 2 and l = 2 , respectively, are found. Under certain restrictions on x = h / H , the stability of the solutions in a class of cosmological solutions with diagonal metrics is proved. A subclass of solutions with small enough variation of the effective gravitational constant G is considered and the stability of all solutions from this subclass is shown. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.

We consider $10$-dimensional gravitational model with $SO(6)$ Yang-Mills field, Gauss-Bonnet term and $\Lambda$-term. We study so-called cosmological type solutions defined on product manifold $M = R \times R^3 \times K$, where $K$ is $6d$ Calabi-Yau manifold. By putting the gauge field 1-form to be coinciding with 1-form spin connection on $K$, we obtain exact cosmological solutions with exponential dependence of scale factors (upon $t$-variable), governed by two non-coinciding Hubble-like parameters: $H >0$, $h$, obeying $ H + 2 h \neq 0$. We also present static analogs of these cosmological solutions (for $H \neq 0$, $h \neq H$ and $ H + 2 h \neq 0$). The islands of stability for both classes of solutions are outlined., Comment: 17 pages, 3 figures, LaTex, Revised version: 3 paragraphs are added into Introduction, new references are included and few references (self-citations) are omitted

Dilatonic black hole dyon-like solutions are overviewed in the gravitational 4D model with a scalar field, two 2-forms, two dilatonic coupling constants λi ≠ 0, i = 1, 2, obeying λ1 ≠ − λ2, and the sign parameter e = ±1 before the scalar field kinetic term. Here e = −1 corresponds to a phantom scalar field. The solutions are defined up to solutions of two master equations for two moduli functions, when $$\lambda_i^2 \neq 1/2$$ for e = −1. Several integrable cases, corresponding to the Lie algebras A1 + A1, A2, B2 = C2 and G2 are considered. Some physical parameters of the solutions are derived: the gravitational mass, scalar charge, Hawking temperature, black hole area entropy and PPN parameters β and γ. Bounds on the gravitational mass and scalar charge, based on a certain conjecture, are presented.

We deal with Einstein-Gauss-Bonnet model in dimension $D$ with a $\Lambda$-term. We obtain three stable cosmological solutions with exponential behavior (in time) of three scale factors corresponding to subspaces of dimensions $(l_0,l_1,l_2) = (3, 4, 4), (3, 3, 2), (3, 4, 3)$ and $D = 12, 9, 11$, respectively. Any solution may describe an exponential expansion of $3$-dimensional subspace governed by Hubble parameter $H$. Two of them may also describe a small enough variation of the effective gravitational constant $G$ (in Jordan frame) for certain values of $\Lambda$., Comment: 9 pages, Latex, certain editing of the text (mainly in Abstract, Introduction and Conclusions) is done; few phrases and words are deleted; one reference is added

We deal with generalized Melvin-like solutions associated with Lie algebras of rank 4 ( $$A_4$$ , $$B_4$$ , $$C_4$$ , $$D_4$$ , $$F_4$$ ). Any solution has static cylindrically symmetric metric in D dimensions in the presence of four Abelian two-form and four scalar fields. The solution is governed by four moduli functions $$H_s(z)$$ ( $$s = 1,\ldots ,4$$ ) of squared radial coordinate $$z=\rho ^2$$ obeying four differential equations of the Toda chain type. These functions are polynomials of powers $$(n_1,n_2, n_3, n_4) = (4,6,6,4), (8,14,18,10), (7,12,15,16), (6,10,6,6), (22,42,30,16)$$ for Lie algebras $$A_4$$ , $$B_4$$ , $$C_4$$ , $$D_4$$ , $$F_4$$ , respectively. The asymptotic behaviour for the polynomials at large z is governed by an integer-valued $$4 \times 4$$ matrix $$\nu $$ connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in $$A_4$$ case) the matrix representing a generator of the $${\mathbb {Z}}_2$$ -group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are studied. We also present two-form flux integrals over a two-dimensional submanifold. Dilatonic black hole analogs of the obtained Melvin-type solutions, e.g. “phantom” ones, are also considered. The phantom black holes are described by fluxbrane polynomials under consideration.

We consider a class of solutions in multidimensional gravity which generalize Melvin’s well-known cylindrically symmetric solution, originally describing the gravitational field of a magnetic flux tube. The solutions considered contain the metric, two Abelian 2-forms and two scalar fields, and are governed by two moduli functions H1(z) and H2(z) (z = ρ2, ρ is a radial coordinate) which have a polynomial structure and obey two differential (Toda-like) master equations with certain boundary conditions. These equations are governed by a certain matrix A which is a Cartan matrix for some Lie algebra. The models for rank-2 Lie algebras A2, C2 and G2 are considered. We study a number of physical and geometric properties of these models. In particular, duality identities are proved, which reveal a certain behavior of the solutions under the transformation ρ → 1/ρ; asymptotic relations for the solutions at large distances are obtained; 2-form flux integrals over 2-dimensional regions and the corresponding Wilson loop factors are calculated, and their convergence is demonstrated. These properties make the solutions potentially applicable in the context of some dual holographic models. The duality identities can also be understood in terms of the Z2 symmetry on vertices of the Dynkin diagram for the corresponding Lie algebra.

A D-dimensional gravitational model with Gauss–Bonnet and cosmological terms is considered. When an ansatz with a diagonal cosmological metric is adopted, we find new examples of solutions for Λ Λ ≠ 0 and D = 8 with an exponential dependence of the scale factors, which describe expansion of our 3D factor-space and contraction of 4D internal space. We also study the stability of the solutions with static Hubble-like parameters hi and prove that two solutions with Λ = 0 in dimensions D = 22, 28, which were found earlier, are stable. For both solutions we find asymptotic relations for the effective gravitational constant.

The gravitational D-dimensional model is considered, with l scalar fields, a cosmological constant and several forms. When a cosmological block-diagonal metric, defined on a product of an 1-dimensional interval and n oriented Einstein spaces, is chosen, an electromagnetic composite brane ansatz is adopted, and certain restrictions on the branes are imposed, the conformally covariant Wheeler–DeWitt (WDW) equation for the model is studied. Under certain restrictions, asymptotic solutions to the WDWequation are found in the limit of the formation of billiard walls which reduce the problem to the socalled quantum billiard on (n + l - 1)-dimensional hyperbolic space. Several examples of billiards in the model with {pmn} non-intersecting electric branes, e.g., corresponding to hyperbolic Kac–Moody algebras, are considered. In the classical case, any of these billiards describe a never-ending oscillating behavior of scale factors while approaching to the singularity, which is either spacelike or timelike. For n = 2 the model is completely integrable in the asymptotic regime in the clasical and quantum cases.

We obtain new examples of partially supersymmetric M-brane solutions defined on products of Ricci-flat manifolds, which contain two-dimensional Lorentzian submanifold R^{1,1}_{*}/Z_2 with one parallel spinor. The examples belong to the following configurations: M2, M5, M2-M5 and M5-M5. Among them a M2 solution with N =1/32 fractional number of preserved supersymmetries is presented. The examples with three M-branes were considered earlier in our paper with A.A. Golubtsova., Comment: 8 pages, LaTeX, no figures, 14 refs. are added

Recent results of measurements of the Avogadro constant by the method of crystal silicon spheres in connection with the planned transition to a new definition of the basic SI unit, the mole, are discussed.

The role of the fundamental constants and of measurement data on the Planck and Avogadro constants, the kelvin, and the electrical charge in the planned transition to new definitions of the four SI base units (kilogram, mole, ampere, and kelvin) based on fixed values of these constants is discussed.

A multidimensional generalization of Melvin’s solution for an arbitrary simple Lie algebra $${\mathcal {G}}$$ is considered. The gravitational model in D dimensions, $$D \ge 4$$ , contains n 2-forms and $$l \ge n$$ scalar fields, where n is the rank of $${\mathcal {G}}$$ . The solution is governed by a set of n functions $$H_s(z)$$ obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). The polynomials $$H_s(z)$$ , $$s = 1,\ldots ,6$$ , for the Lie algebra $$E_6$$ are obtained and a corresponding solution for $$l = n = 6$$ is presented. The polynomials depend upon integration constants $$Q_s$$ , $$s = 1,\ldots ,6$$ . They obey symmetry and duality identities. The latter ones are used in deriving asymptotic relations for solutions at large distances. The power-law asymptotic relations for $$E_6$$ -polynomials at large z are governed by the integer-valued matrix $$\nu = A^{-1} (I + P)$$ , where $$A^{-1}$$ is the inverse Cartan matrix, I is the identity matrix and P is a permutation matrix, corresponding to a generator of the $$Z_2$$ -group of symmetry of the Dynkin diagram. The 2-form fluxes $$\Phi ^s$$ , $$s = 1,\ldots ,6$$ , are calculated.

A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra $\cal G$ is considered. The solution contains a metric, $n$ Abelian 2-forms and $n$ scalar fields, where $n$ is the rank of $\cal G$. It is governed by a set of $n$ moduli functions $H_s(z)$ obeying $n$ ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials - the so-called fluxbrane polynomials. These polynomials depend upon integration constants $q_s$, $s = 1,\dots,n$. In the case when the conjecture on the polynomial structure for the Lie algebra $\cal G$ is satisfied, it is proved that 2-form flux integrals $\Phi^s$ over a proper $2d$ submanifold are finite and obey the relations: $q_s \Phi^s = 4 \pi n_s h_s$, where $h_s > 0$ are certain constants (related to dilatonic coupling vectors) and $n_s$ are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, $s = 1,\dots,n$. The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra $\cal G$. Examples of polynomials and fluxes for the Lie algebras $A_1$, $A_2$, $A_3$, $C_2$, $G_2$ and $A_1 + A_1$ are presented., Comment: 10 pages, Latex, no figures, prepared for a talk at RUSGRAV-16 conference, 2nd revised version, several typos (mainly grammar ones) are eliminated. arXiv admin note: text overlap with arXiv:1706.06621

Two families of composite black brane solutions are overviewed, fluxbrane and black brane ones, in a model with scalar fields and fields of forms. The metric of any solution is defined on a manifold which contains a product of several Ricci-flat “internal” spaces. The solutions are governed by moduli functions $\mathcal{H}_s $ (for fluxbranes) and H s (for black branes), obeying nonlinear differential equations with certain boundary conditions. Themaster equations for $\mathcal{H}_s $ and H s are equivalent to Toda-like equations and depend on a nondegenerate matrix A related to brane intersection rules. The functions H s and $\mathcal{H}_s $ , as was conjectured and confirmed (at least partly) earlier, should be polynomials in proper variables if A is a Cartan matrix of some semisimple finite-dimensional Lie algebra. The fluxbrane polynomials $\mathcal{H}_s $ were shown to be used for the construction of black brane polynomials H s . This approach is illustrated by examples of nonextremal electric black p-brane solutions related to Lie algebras A 2, C 2, and G 2.

Generalized Melvin solutions for rank-$3$ Lie algebras $A_3$, $B_3$ and $C_3$ are considered. Any solution contains metric, three Abelian 2-forms and three scalar fields. It is governed by three moduli functions $H_1(z),H_2(z),H_3(z)$ ($z = \rho^2$ and $\rho$ is a radial variable), obeying three differential equations with certain boundary conditions imposed. These functions are polynomials with powers $(n_1,n_2, n_3) = (3,4,3), (6,10,6), (5,8,9)$ for Lie algebras $A_3$, $B_3$, $C_3$, respectively. The solutions depend upon integration constants $q_1, q_2, q_3 \neq 0$. The power-law asymptotic relations for polynomials at large $z$ are governed by integer-valued $3 \times 3$ matrix $\nu$, which coincides with twice the inverse Cartan matrix $2 A^{-1}$ for Lie algebras $B_3$ and $C_3$, while in the $A_3$ case $\nu = A^{-1} (I + P)$, where $I$ is the identity matrix and $P$ is a permutation matrix, corresponding to a generator of the $\mathbb{Z}_2$-group of symmetry of the Dynkin diagram. The duality identities for polynomials and asymptotic relations for solutions at large distances are obtained. 2-form flux integrals over a $2$-dimensional disc of radius $R$ and corresponding Wilson loop factors over a circle of radius $R$ are presented., Comment: 10 pages, Latex, 1 figure; 5th version: the abstract in the Latex file is corrected. arXiv admin note: text overlap with arXiv:1706.07856

A [Formula: see text]-dimensional Einstein–Gauss–Bonnet (EGB) model with the cosmological term [Formula: see text] is considered. We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters [Formula: see text] and [Formula: see text], corresponding to two factor spaces of dimension [Formula: see text] and obeying [Formula: see text]. We prove that the solutions, obeying [Formula: see text], are stable (in a class of cosmological solutions with diagonal metrics) for [Formula: see text] and they are unstable for [Formula: see text]. A subclass of solutions with small enough variation of the effective gravitational constant [Formula: see text] is considered. It is shown that all solutions from this subclass are stable.

We investigate all standard triple composite M-brane intersections defined on products of Ricci-flat manifolds for preserving supersymmetries in eleven-dimensional N =1 supergravity. The explicit formulae for computing the numbers of preserved supersymmetries are obtained, which generalize the relations for topologically trivial flat factor spaces presented in the classification by Bergshoeff et al. We obtain certain examples of configurations preserving some fractions of supersymmetries, e.g. containing such factor spaces as K3, C^2_{*}/Z_2, a four-dimensional pp-wave manifold and the two-dimensional pseudo-Euclidean manifold R^{1,1}_{*}/Z_2., 26 pages, LaTex, 7 figures (are changed), typos and few words are eliminated, certain notations are simplified, several phrases are added. To be published in Nuclear Physics B

Bianchi type I cosmological model in (n+1)-dimensional gravity with several forms is considered. When the electric non-composite brane ansatz is adopted, the Wheeler-DeWitt (WDW) equation for the model, written in a conformally covariant form, is analyzed. Under certain restrictions, asymptotic solutions to the WDW equation near the singularity are found, which reduce the problem to the so-called quantum billiard on the (n-1)-dimensional Lobachevsky space H^{n-1}. Two examples of quantum billiards are considered: a 2-dimensional quantum billiard for a 4D model with three 2-forms and a 9D quantum billiard for an 11D model with 120 4-forms which mimics SM2-brane sector of D=11 supergravity. For certain solutions, vanishing of the wave function at the singularity is proved., Comment: 14 pages, LaTex, no figures, several typos are eliminated

A D-dimensional gravitational model with Gauss-Bonnet and cosmological term is considered. When ansatz with diagonal cosmological metrics is adopted, we overview recent solutions for zero cosmological term and find new examples of solutions for non-zero cosmological term and D = 8 with exponential dependence of scale factors which describe an expansion of our 3-dimensional factor-space and contraction of 4-dimensional internal space., Comment: 7 pages, Latex, no figures. To be published in: Proceedings of the XII International Conference on Gravitation, Cosmology and Astrophysics, (ICGAC-12, June 28-July 5, 2015, Moscow, PFUR), Eds. J.P. Hsu and V.N. Melnikov, World Scientific Publ., Singapore, 2016

We consider a D-dimensional model of gravity with non-linear "scalar fields" as a matter source. The model is defined on the product manifold M, which contains n Einstein factor spaces. General cosmological type solutions to the field equations are obtained when n-1 factor spaces are Ricci-flat, e.g. when one space M_1 of dimension d_1 > 1 has nonzero scalar curvature. The solutions are defined up to solutions to geodesic equations corresponding to a sigma model target space. Several examples of sigma models are presented. A subclass of spherically-symmetric solutions is studied and a restricted version of "no-hair theorem" for black holes is proved. For the case d_1 =2 a subclass of latent soliton solutions is singled out., 22 pages, Latex, several phrases and 4 refs. are addes, few typos are eliminated

The latest high-precision experimental results and theoretical calculations to determine the value of the fine-structure constant α are analyzed. These indicate a possible increase in its value by approximately 0.5·10−8α compared with the CODATA data (2006). It is shown that this shift in the value of α may play an important role in the new system of units, including the solution of the problem of matching the results of measurements of the Avogadro and Planck constants obtained by different methods (in the Avogadro project and by means of the watt balance).

We review spherically symmetric solutions with a horizon in two models: (i) with scalar fields and fields of forms, and (ii) with a multi-component anisotropic fluid. The metrics of the solutions are defined on a manifold that contains a product of n − 1 Ricci-flat “internal” spaces. The solutions are governed by functions H s obeying nonlinear differential equations with certain boundary conditions. Simulation of black-brane solutions is considered, and the Hawking temperature is calculated. For the fluid solution, the post-Newtonian parameters β and Γ corresponding to the 4-dimensional section of the metric are found.

A (n + 1)-dimensional Einstein-Gauss-Bonnet (EGB) model is considered. For diagonal cosmological-type metrics, the equations of motion are reduced to a set of Lagrange equations. The effective Lagrangian contains two "minisuperspace" metrics on R^n. The first one is the well-known 2-metric of pseudo-Euclidean signature and the second one is the Finslerian 4-metric that is proportional to n-dimensional Berwald-Moor 4-metric. When a "synchronous-like" time gauge is considered the equations of motion are reduced to an autonomous system of first-order differential equations. For the case of the "pure" Gauss-Bonnet model, two exact solutions with power-law and exponential dependence of scale factors (with respect to "synchronous-like" variable) are obtained. (In the cosmological case the power-law solution was considered earlier in papers of N. Deruelle, A. Toporensky, P. Tretyakov and S. Pavluchenko.) A generalization of the effective Lagrangian to the Lowelock case is conjectured. This hypothesis implies existence of exact solutions with power-law and exponential dependence of scale factors for the "pure" Lowelock model of m-th order., Comment: 24 pages, Latex, typos are eliminated

A family of spherically symmetric solutions in the model with m-component anisotropic fluid is considered. The metric of the solution depends on parameters q_s, s = 1,...,m, relating radial pressures and the densities and contains (n -1)m parameters corresponding to Ricci-flat "internal space" metrics and obeying certain m(m-1)/2 ("orthogonality") relations. For q_s = 1 (for all s) and certian equations of state (p_i^s = \pm \rho^s) the metric coincides with the metric of intersecting black brane solution in the model with antisymmetric forms. A family of solutions with (regular) horizon corresponding to natural numbers q_s = 1,2,... is singled out. Certain examples of "generalized simulation" of intersecting M-branes in D=11 supergravity are considered. The post-Newtonian parameters \beta and \gamma corresponding to the 4-dimensional section of the metric are calculated., Comment: 16 pages, no figures

A family of generalized $S$-brane solutions with orthogonal intersection rules and $n$ Ricci-flat factor spaces in the theory with several scalar fields and antisymmetric forms is considered. Two subclasses of solutions with power-law and exponential behaviour of scale factors are singled out. These subclasses contain sub-families of solutions with accelerated expansion of certain factor spaces. The solutions depend on charge densities of branes, their dimensions and intersections, dilatonic couplings and the number of dilatonic fields., To appear in GRG

Black p-brane solutions for a wide class of intersection rules and Ricci-flat ``internal'' spaces are considered. They are defined up to moduli functions H_s obeying non-linear differential equations with certain boundary conditions imposed. A new solution with intersections corresponding to the Lie algebra A_3 is obtained. The functions H_1, H_2 and H_3 for this solution are polynomials of degree 3, 4 and 3, correspondingly. An example of A_3-solution with three 3-branes in 12-dimensional model (suggested by N. Khviengia et al) is presented., 11 pages, Latex, to appear in Phys. Lett. B

This topical review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the form M = M_0 x M_1 x ...x M_n, where M_i are Einstein spaces (i >0). The sigma-model approach and exact solutions in the model are reviewed and the solutions with p-branes (e.g. Majumdar-Papapetrou-type, cosmological, spherically symmetric, black-brane and Freund-Rubin-type ones) are considered., 63 pages, Latex, topical review, few typos are eliminated

Solutions in multidimensional gravity with m p-branes related to Toda-like systems (of general type) are obtained. These solutions are defined on a product of n+1 Ricci-flat spaces M_0 x M_1 x...x M_n and are governed by one harmonic function on M_0. The solutions are defined up to the solutions of Laplace and Toda-type equations and correspond to null-geodesics of the (sigma-model) target-space metric. Special solutions relating to A_m Toda chains (e.g. with m =1,2) are considered., Comment: 20 pages, Latex, to be submit. to JMP

The block-orthogonal generalization of the Majumdar-Papapetrou type solutions for the sigma-model studied earlier are obtained and corresponding solutions with p-branes are considered. The existence of solutions and the number of independent harmonic functions is defined by the matrix of scalar products of vectors $U^s$, governing the sigma-model target space metric. For orthogonal $U^s$, when target space is a symmetric homogeneous space, the solutions reduce to the previous ones. Two special classes of solutions with $U^s$ related to finite dimensional Lie algebras and hyperbolic (Kac-Moody) algebras are singled out and investigated. The affine Cartan matrices do not arise in the scheme under consideration. Some examples of obtained solutions and intersection rules for D=11 supergravity, related D=12 theory and extending them $B_D$-models are considered. For special multicenter solutions criterions for the existence of horizon and curvature singularity are found., Comment: 22 pages, Latex. Resubmit. to Class. and Quantum Grav. Revised version

A family of Majumdar-Papapetrou type solutions in sigma-model of p-brane origin is obtained for all direct sums of finite-dimensional simple Lie algebras. Several examples of p-brane dyonic configurations in D=10 (IIA) and D=11 supergravities corresponding to the Lie algebra sl(3,C) are considered., Comment: 13 pages, LaTeX, submitted to Phys. Lett. B

Multidimensional gravitational model on the manifold M = M0 × ∏i=1n Mi, where Mi are Einstein spaces (i ≥ 1), is considered. The action contains m = 2n − 1 dilatonic scalar fields ϕ1 and m (antisymmetric) forms A1. When all fields and scale factors of the metric depend (essentially) on the point of M0 and any A1 is “proportional” to the volume form of submanifold Mi1 × … × Mik, 1 ≤ i1 < … < ik ≤ n, the σ-model representation is obtained. A family of “Majumdar-Papapetrou type” solutions are obtained, when all Mν are Ricci-flat. Relation of our generalized p-branes to usual intersecting p-branes is discussed.

The multidimensional cosmological model describing the evolution of n Einstein spaces in the presence of multicomponent perfect fluid is considered. We define vectors related to the equations of state of the components. If they are orthogonal with respect to the minisuperspace metric, the Einstein equations are integrable and a Kasner-like form of the solutions is presented. For special sets of parameters the cosmological model is reduced to the Euclidean Toda-like system connected with some Lie algebra. The integrable vacuum (1+5+5)-model with two 5-dimensional Einstein spaces and non-zero Ricci tensors is obtained. Its reduction to a (1+5+3+2)-solution is given. For a special choice of the integration constant and one of the spaces (M1 = S5) a non-singular solution with the topology \(R^6 \times {\text{ }}M_2\) is obtained.

The vacuum cosmological model on the manifold $R \times M_1 \times \ldots \times M_n$ describing the evolution of $n$ Einstein spaces of non-zero curvatures is considered. For $n = 2$ the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when $(N_1 = $dim $ M_1, N_2 = $ dim$ M_2) = (6,3), (5,5), (8,2)$. The Kasner-like behaviour of the solutions near the singularity $t_s \to +0$ is considered ($t_s$ is synchronous time). The exceptional ("Milne-type") solutions are obtained for arbitrary $n$. For $n=2$ these solutions are attractors for other ones, when $t_s \to + \infty$. For dim $ M = 10, 11$ and $3 \leq n \leq 5$ certain two-parametric families of solutions are obtained from $n=2$ ones using "curvature-splitting" trick. In the case $n=2$, $(N_1, N_2)= (6,3)$ a family of non-singular solutions with the topology $R^7 \times M_2$ is found., Comment: 21 pages, LaTex. 5 figures are available upon request (hard copy). Submitted to Classical and Quantum Gravity

A multidimensional cosmological model describing the evolution of n Einstein spaces in the presence of a multicomponent perfect fluid is considered. When vectors corresponding to the equations of state of the components are orthogonal with respect to the minisuperspace metric, the Einstein equations are integrated and a Kasner-like form of the solutions is presented. For special sets of parameters the cosmological model is reduced to the Euclidean Toda-like system connected with some Lie algebra. PACS numbers: 04.20.J, 04.60.+n, 03.65.Ge

The multidimensional cosmological model describing the evolution of n Einstein spaces in the presence of multicomponent perfect fluid is considered. When the vectors corresponding to the equations of state of the components are orthogonal with respect to the minisuperspace metric, the Einstein equations are integrated and a Kasner‐like form of the solutions is presented. For special sets of parameters the cosmological model is reduced to the Euclidean Toda‐like system connected with some Lie algebra G. For G=A2 exact solutions are explicitly written. A certain family of wormhole solutions is also obtained.

The $n$-time generalization of Schwarzschild solution is considered. The equations of geodesics for the metric are integrated. The multitemporal analogues of Newton laws for the extended objects described by the solution are suggested. The scalar-vacuum generalization of the solution is also presented., 7 pages

Two families of exact solutions in multidimensional gravity with scalar fields and fields of forms are considered: fluxbrane and black brane ones. A brief overview of main results on billiard approach for cosmological-type models with branes is also presented.

The multidimensional charged dilatonic black hole solution with $n$ internal Ricci-flat spaces is considered. The bound on the mass of the black hole is obtained. In the strong dilatonic coupling limit the critical mass becomes zero. The case $n = \infty$ is also considered., 11 pages, 1 figure available on request from ubleyer@aip.de, LaTeX, Potsdam University, AIP 94-05

A cosmological model describing the evolution of n Einstein spaces is considered. For n 1 ⩾2, n − n 1 ⩾1, where n 1 is the number of spaces with non-zero curvature, the model is reduced to a Toda-like system that does not satisfy the Adler-van Moerbeke criterion of integrability. In the case n 1 =0, 1 ( n ⩾2) the Eistein equations are integrated and tree-generalizations of the solutions are considered.

It is shown that the tensor Banach functor of projective type $$\hat{T}_{K} $$ [1] corresponding to the complete normed fieldK is quasiidempotent on infinite-dimensionall 1 spaces, i.e., $$\hat{T}_{K} (\theta _{K} (\hat{T}_{K} (l_1 (M.K)))) \cong \hat{T}_K (l_1 (M.K)).$$ whereM is an infinite set and θ K is the forgetful functor. Anl 1 realization of the Banach algebra $$\hat{T}_{K} (l_1 (M.K))$$ is constructed.

We construct a tensor Banach functorTκ that establishes a correspondence between every projectively admissible Banach spaceE over a complete normed fieldK with a tensor Banach algebra of projective typeTκ.